researchseminars.org

Lexing Ying (Stanford University)
2020-04-08 23:10 - 2020-04-09 00:00 UTC | https://researchseminars.org/talk/BerekelyApplied/1/
Title: Solving inverse problems with deep learning (https://researchseminars.org/talk/BerekelyApplied/1/)
by Lexing Ying (Stanford University) as part of Berkeley applied mathematics seminar

Abstract
This talk is about some recent progress on solving inverse problems using deep learning. Compared to traditional machine learning problems, inverse problems are often limited by the size of the training data set. We show how to overcome this issue by incorporating mathematical analysis and physics into the design of neural network architectures. We first describe neural network representations of pseudodifferential operators and Fourier integral operators. We then continue to discuss applications including electric impedance tomography, optical tomography, inverse acoustic/EM scattering, and travel-time tomography.

Michael Mahoney (UC Berkeley)
2020-04-22 23:10 - 2020-04-23 00:00 | https://researchseminars.org/talk/BerekelyApplied/2/
Title: Determinantal point processes and randomized numerical linear algebra (https://researchseminars.org/talk/BerekelyApplied/2/)
by Michael Mahoney (UC Berkeley) as part of Berkeley applied mathematics seminar

Abstract
Randomized Numerical Linear Algebra (RandNLA) is an area which uses randomness, most notably random sampling and random projection methods, to develop improved algorithms for ubiquitous matrix problems, such as those that arise in scientific computing, data science, machine learning, etc. A seemingly different topic, but one which has a long history in pure and applied mathematics, is that of Determinantal Point Processes (DPPs), which are stochastic point processes, the probability distribution of which is characterized by sub-determinants of some matrix. Recent work has uncovered deep and fruitful connections between DPPs and RandNLA. For example, random sampling with a DPP leads to new kinds of unbiased estimators for classical RandNLA tasks, enabling more refined statistical and inferential understanding of RandNLA algorithms; a DPP is, in some sense, an optimal randomized method for many RandNLA problems; and a standard RandNLA technique, called leverage score sampling, can be derived as the marginal distribution of a DPP. This work will be reviewed, as will recent algorithmic developments, illustrating that, while not quite as efficient as simply applying a random projection, these DPP-based algorithms are only moderately more expensive. Joint work with Michal Derezinski.

Ken Kamrin (MIT)
2020-04-29 23:10 - 2020-04-30 00:00 | https://researchseminars.org/talk/BerekelyApplied/3/
Title: Toward reduced-order models for flowing grains: Surprising complexity meets surprising simplicity (https://researchseminars.org/talk/BerekelyApplied/3/)
by Ken Kamrin (MIT) as part of Berkeley applied mathematics seminar

Abstract
Despite the commonality of granular materials in day-to-day life, modeling systems of millions or more flowing particles has proven historically difficult. This has direct real-world ramifications owing to the prominent role granular media play in multiple industries and in terrain dynamics. One can attempt to track every grain with discrete particle methods, but realistic systems are often too large for this approach and a continuum model is desired. However, granular media display unusual behaviors that complicate the continuum treatment: they can behave like solid, flow like liquid, or separate into a “gas”, and the rheology of the flowing state displays remarkable subtleties.

To address these challenges, in this talk we develop a family of continuum models and solvers, permitting quantitative modeling capabilities. We discuss a variety of applications, ranging from general problems to specific techniques for problems of intrusion, impact, driving, and locomotion in granular media. To calculate flows in general cases, a rather significant nonlocal effect is evident, which is well-described with our recent nonlocal model accounting for grain cooperativity within the flow rule. On the other hand, to model only intrusion forces on submerged objects, we will show, and explain why, many of the experimentally observed results can be captured from a much simpler tension-free frictional plasticity model. This approach gives way to some surprisingly simple general tools, including the granular Resistive Force Theory, and a broad set of scaling laws inherent to the problem of granular locomotion. These scalings are validated directly and suggest a new down-scaled paradigm for granular locomotive design, on earth and beyond, to be used much like scaling laws in fluid mechanics.

We close with a brief discussion of ongoing modeling efforts for wet granular systems, including those with non-trivial grain-grain interactions and those with highly deformable particles.

Tamara Kolda (Sandia National Laboratory)
2020-05-06 23:10 - 2020-05-07 00:00 | https://researchseminars.org/talk/BerekelyApplied/4/
Title: Practical leverage-based sampling for low-rank tensor decomposition (https://researchseminars.org/talk/BerekelyApplied/4/)
by Tamara Kolda (Sandia National Laboratory) as part of Berkeley applied mathematics seminar

Abstract
Conventional algorithms for finding low-rank canonical polyadic (CP) tensor decompositions are unwieldy for large sparse tensors. The CP decomposition can be computed by solving a sequence of overdetermined least problems with special Khatri-Rao structure. In this work, we present an application of randomized algorithms to fitting the CP decomposition of sparse tensors, solving a significantly smaller sampled least squares problem at each iteration with probabilistic guarantees on the approximation errors. Prior work has shown that sketching is effective in the dense case, but the prior approach cannot be applied to the sparse case because a fast Johnson-Lindenstrauss transform (e.g., using a fast Fourier transform) must be applied in each mode, causing the sparse tensor to become dense. Instead, we perform sketching through leverage score sampling, crucially relying on the fact that the structure of the Khatri-Rao product allows sampling from overestimates of the leverage scores without forming the full product or the corresponding probabilities. Naïve application of leverage score sampling is infective because we often have cases where a few scores are quite large, so we propose a novel hybrid of deterministic and random leverage-score sampling which is more efficient and effective. Numerical results on real-world large-scale tensors show the method is faster than competing methods without sacrificing accuracy. This is joint work with Brett Larsen at Stanford University.

Linfeng Zhang (Princeton University)
2020-05-13 23:10 - 2020-05-14 00:00 | https://researchseminars.org/talk/BerekelyApplied/5/
Title: Symmetry preserving neural network models for molecular modelling (https://researchseminars.org/talk/BerekelyApplied/5/)
by Linfeng Zhang (Princeton University) as part of Berkeley applied mathematics seminar

Abstract
We discuss how to leverage the fitting ability of neural networks to accurately and efficiently represent two types of maps in molecular modelling problems. The first type takes as input the coordinates of atoms and their associated chemical species, and outputs physical observables such as the interatomic potential energy (a scalar), the electric polarization (a vector) and polarizability (a tensor), and the charge density (a field). The second type, like post–Hartree–Fock methods, uses the ground-state electronic orbitals as the input, and predicts the energy difference between results of highly accurate models such as the coupled-cluster method and low accuracy models such as the Hartree-Fock (HF) method. Special attentions are paid to how the neural network models take care of physical properties like symmetry and locality, so that models trained with small-size systems can be transferred to different and large-size ones; and how they are made end-to-end, so that little human intervention is required for various complex tasks. This is joint work with Yixiao Chen, Han Wang, and Weinan E.

Yanhe Huang (UC Berkeley)
2020-08-18 23:10 - 2020-08-19 00:00 | https://researchseminars.org/talk/BerekelyApplied/6/
Title: Axisymmetric bubbles rising in 3D and a new accurate algorithm for evaluating orthogonal polynomials (https://researchseminars.org/talk/BerekelyApplied/6/)
by Yanhe Huang (UC Berkeley) as part of Berkeley applied mathematics seminar

Abstract
n the high Reynolds number regime, under what conditions do there exist steadily rising bubbles? This question has been studied extensively both experimentally and numerically, but current mathematical models and numerical discretizations suffer from large numerical errors that make the results less convincing. In the first part of this talk, we build an inviscid model for the steady rising problem and find different solution branches of bubble shapes characterized by the number of humps. These only exist when there is no gravity. When there is gravity, viscous potential flow is used to find different steady shapes. The corresponding dynamic problem is also studied. Techniques such as axisymmetric potential theory, Hou-Lowengrub-Shelley framework, and weak/hyper-singularity removal are applied to guarantee spectral accuracy.

Due to the importance of accurate evaluation of orthogonal polynomials in the boundary integral method used in the first part, in the second part of the talk I will introduce a new way to evaluate orthogonal polynomials more accurately near the endpoints of the integration interval. An associated family of orthogonal polynomials is evaluated at interior points to determine the values of the original polynomials near endpoints. The new method can achieve round-off error accuracy even for end-point evaluation of generic high-degree Jacobi polynomials and generalized Laguerre polynomials. More accurate quadrature abscissas and weights can be achieved accordingly.

Stefan Steinerberger (University of Washington)
2020-08-26 23:10 - 2020-08-27 00:00 | https://researchseminars.org/talk/BerekelyApplied/7/
Title: Solving Linear Systems of Equations via Randomized Kaczmarz/Stochastic Gradient Descent (https://researchseminars.org/talk/BerekelyApplied/7/)
by Stefan Steinerberger (University of Washington) as part of Berkeley applied mathematics seminar

Abstract
The Randomized Kaczmarz method is a classical iterative method to solve linear systems: the solution of a system Ax = b is simply the point of intersection of several hyperplanes. The Kaczmarz method (also known as the Projection Onto Convex Sets Method) proceeds by simply starting with a point and then iteratively projecting it on these hyperplanes. If the hyperplanes (=rows of the matrix) are picked in random order, the algorithm was analyzed by Strohmer & Vershynin and has linear convergence. We show that the method, as a byproduct, also computes small singular vectors and, in fact, the iterates tend to approach the true solution from the direction of the smallest singular vector in a meta-stable way. This also explains why the algorithm has such wonderful regularization properties. The arguments are all fairly self-contained, elementary and nicely geometric. This gives a pretty clear picture – the question is: can this picture be used to improve the method?

Chao Ma (Stanford University)
2020-09-02 23:10 - 2020-09-03 00:00 | https://researchseminars.org/talk/BerekelyApplied/8/
Title: The Slow Deterioration of the Generalization Error of the Random Feature Model (https://researchseminars.org/talk/BerekelyApplied/8/)
by Chao Ma (Stanford University) as part of Berkeley applied mathematics seminar

Abstract: TBA

Weile Jia (UC Berkeley)
2020-09-09 23:10 - 2020-09-10 00:00 | https://researchseminars.org/talk/BerekelyApplied/9/
Title: HPC+AI: pushing ab initio MD to 100 million atoms on the Summit supercomputer (https://researchseminars.org/talk/BerekelyApplied/9/)
by Weile Jia (UC Berkeley) as part of Berkeley applied mathematics seminar

Abstract: TBA

Alex Townsend (Cornell University)
2020-09-16 23:10 - 2020-09-17 00:00 | https://researchseminars.org/talk/BerekelyApplied/10/
Title: The ultraspherical spectral method (https://researchseminars.org/talk/BerekelyApplied/10/)
by Alex Townsend (Cornell University) as part of Berkeley applied mathematics seminar

Abstract
Pseudospectral methods, based on high degree polynomials, have spectral accuracy when solving differential equations but typically lead to dense and ill-conditioned matrices. The ultraspherical spectral method is a numerical technique to solve ordinary and partial differential equations, leading to almost banded well-conditioned linear systems while maintaining spectral accuracy. In this talk, we introduce the ultraspherical spectral method and develop it into a spectral element method using a modification to a hierarchical Poincare-Steklov domain decomposition method.

Martina Bukac (University of Notre Dame)
2020-09-23 23:10 - 2020-09-24 00:00 | https://researchseminars.org/talk/BerekelyApplied/11/
Title: A computational framework for fluid-structure interaction problems (https://researchseminars.org/talk/BerekelyApplied/11/)
by Martina Bukac (University of Notre Dame) as part of Berkeley applied mathematics seminar

Abstract
Fluid-structure interaction (FSI) problems arise in many applications, such as aerodynamics, geomechanics and hemodynamics. They are moving domain, multiphysics problems characterized by nonlinear coupling between a fluid and structure. As a result, FSI problems are challenging to numerically solve and analyze. A popular approach is to solve the fluid and structure sub-problems in a partitioned manner, allowing the use of solvers specifically designed for the physics of each subproblem. However, stability issues often arise as a result of FSI coupling unless the design and implementation of a partitioned scheme is carefully developed. We will present a family of partitioned numerical schemes for the interaction between an incompressible, viscous fluid and an elastic structure. We will consider cases where the structure is thick, i.e., described using the same number of spatial dimensions as the fluid, and when the structure is thin, i.e., described using a lower-dimensional model. We will present stability and convergence results, as well as numerical examples where the presented methods are compared to other methods in the literature.

Andrew Childs (University of Maryland)
2020-09-30 23:10 - 2020-10-01 00:00 | https://researchseminars.org/talk/BerekelyApplied/12/
Title: Symmetries, graph properties, and quantum speedups (https://researchseminars.org/talk/BerekelyApplied/12/)
by Andrew Childs (University of Maryland) as part of Berkeley applied mathematics seminar

Abstract
Aaronson and Ambainis (2009) and Chailloux (2018) showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a large quantum speedup?

In this work, we prove that hypergraph symmetries in the adjacency matrix model allow at most a polynomial separation between randomized and quantum query complexities. We also show that, remarkably, permutation groups constructed out of these symmetries are essentially the only permutation groups that prevent super-polynomial quantum speedups. We prove this by fully characterizing the primitive permutation groups that allow super-polynomial quantum speedups.

In contrast, in the adjacency list model for bounded-degree graphs (where graph symmetry is manifested differently), we exhibit a property testing problem that shows an exponential quantum speedup. These results resolve open questions posed by Ambainis, Childs, and Liu (2010) and Montanaro and de Wolf (2013).

This is joint work with Shalev Ben-David, András Gilyén, William Kretschmer, Supartha Podder, and Daochen Wang. https://arxiv.org/abs/2006.12760

Franziska Weber (Carnegie Mellon University)
2020-10-14 23:10 - 2020-10-15 00:00 | https://researchseminars.org/talk/BerekelyApplied/13/
Title: Numerical approximation of statistical solutions of hyperbolic systems of conservation laws (https://researchseminars.org/talk/BerekelyApplied/13/)
by Franziska Weber (Carnegie Mellon University) as part of Berkeley applied mathematics seminar

Abstract
Statistical solutions are time-parameterized probability measures on spaces of integrable functions, which have been proposed recently as a framework for global solutions for multi-dimensional hyperbolic systems of conservation laws. We develop a numerical algorithm to approximate statistical solutions of conservation laws and show that under the assumption of ‘weak statistical scaling’, which is inspired by Kolmogorov’s 1941 turbulence theory, the approximations converge in an appropriate topology to statistical solutions. Numerical experiments confirm that the assumption might hold true.

Rolando Somma (Los Alamos National Laboratory)
2020-10-21 23:10 - 2020-10-22 00:00 | https://researchseminars.org/talk/BerekelyApplied/14/
Title: Quantum linear systems problem: solution and verification (https://researchseminars.org/talk/BerekelyApplied/14/)
by Rolando Somma (Los Alamos National Laboratory) as part of Berkeley applied mathematics seminar

Abstract: TBA

Dong An (UC Berkeley)
2020-10-28 23:10 - 2020-10-29 00:00 | https://researchseminars.org/talk/BerekelyApplied/15/
Title: Quantum adiabatic evolution and applications in computational physics and quantum computing (https://researchseminars.org/talk/BerekelyApplied/15/)
by Dong An (UC Berkeley) as part of Berkeley applied mathematics seminar

Abstract
Originally discovered by Born and Fock (1928), a quantum mechanical system almost remains in its instantaneous eigenstates if the Hamiltonian varies sufficiently slowly and there is a gap between the eigenvalue and the rest of the Hamiltonian’s spectrum. Such a system is said to be a quantum adiabatic evolution, and has become a powerful tool for analyzing quantum dynamics and designing fast classical and quantum algorithms. In this talk, I will first discuss the mathematical formulation of quantum adiabatic evolutions, namely quantum adiabatic theorem. Several versions of the theorem will be discussed, with a focus on the factors that might significantly influence the adiabaticity. Then I will present two applications of the adiabatic evolutions and adiabatic theorems. One is accelerating numerical simulation of Schrodinger equations on classical computers, and the other is a quantum algorithm for solving linear system of equations with near optimal complexity on a quantum computer.

Xiaochuan Tian (UCSD)
2020-11-05 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/16/
Title: Reproducing kernel collocation methods for nonlocal models: asymptotic compatibility and numerical stability (https://researchseminars.org/talk/BerekelyApplied/16/)
by Xiaochuan Tian (UCSD) as part of Berkeley applied mathematics seminar

Abstract: TBA

Jiequn Han (Princeton University)
2020-11-12 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/17/
Title: Solving High-Dimensional PDEs, Controls, and Games with Deep Learning (https://researchseminars.org/talk/BerekelyApplied/17/)
by Jiequn Han (Princeton University) as part of Berkeley applied mathematics seminar

Abstract: TBA

Robert Webber (Courant Institute)
2020-12-03 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/18/
Title: Monte Carlo methods for the Hermitian eigenvalue problem (https://researchseminars.org/talk/BerekelyApplied/18/)
by Robert Webber (Courant Institute) as part of Berkeley applied mathematics seminar

Abstract
In quantum mechanics and the analysis of Markov processes, Monte Carlo methods are needed to identify low-lying eigenfunctions of dynamical generators. The standard Monte Carlo approaches for identifying eigenfunctions, however, can be inaccurate or slow to converge. What limits the efficiency of the currently available spectral estimation methods, and what is needed to build more efficient methods for the future? Through numerical analysis and computational examples, we begin to answer these questions. We present the first-ever convergence proof and error bounds for the variational approach to conformational dynamics (VAC), the dominant method for estimating eigenfunctions used in biochemistry. Additionally, we analyze and optimize variational Monte Carlo (VMC), which combines Monte Carlo with neural networks to accurately identify low-lying eigenstates of quantum systems.

Aditi Krishnapriyan (Lawrence Berkeley National Lab)
2020-10-07 23:10 - 2020-10-08 00:00 | https://researchseminars.org/talk/BerekelyApplied/19/
Title: Persistent Homology Advances Interpretable Machine Learning for Scientific Applications (https://researchseminars.org/talk/BerekelyApplied/19/)
by Aditi Krishnapriyan (Lawrence Berkeley National Lab) as part of Berkeley applied mathematics seminar

Abstract
Machine learning for scientific applications, ranging from physics and materials science to biology, has emerged as a promising alternative to more time-consuming experiments and simulations. The challenge with this approach is the selection of features that enable universal and interpretable system representations across multiple prediction tasks. We use persistent homology to construct holistic feature representations to describe the structure of scientific systems; for example, material and protein structures. We show that these representations can also be augmented with other generic features to capture further information. We demonstrate our approaches on multiple scientific datasets by predicting a variety of different targets across different conditions. Our results show considerable improvement in both accuracy and transferability across targets compared to models constructed from commonly used manually curated features. A key advantage of our approach is interpretability. For example, in material structures, our persistent homology features allow us to identify the location and size of pores in the structure that correlate best to different materials properties, contributing to understanding atomic level structure-property relationships for materials design.

James Stokes (Flatiron Institute)
2020-11-19 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/20/
Title: First-quantized neural networks for lattice fermions (https://researchseminars.org/talk/BerekelyApplied/20/)
by James Stokes (Flatiron Institute) as part of Berkeley applied mathematics seminar

Abstract
First-quantized deep neural network techniques are developed for analyzing strongly coupled fermionic systems on the lattice. Using a Slater-Jastrow inspired ansatz which exploits deep residual networks with convolutional residual blocks, we approximately determine the ground state of spinless fermions on a square lattice with nearest-neighbor interactions. The flexibility of the neural-network ansatz results in a high level of accuracy when compared to exact diagonalization results on small systems, both for energy and correlation functions. On large systems, we obtain accurate estimates of the boundaries between metallic and charge ordered phases as a function of the interaction strength and the particle density.

Dongbin Xiu (The Ohio State University)
2021-02-11 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/21/
by Dongbin Xiu (The Ohio State University) as part of Berkeley applied mathematics seminar

Abstract: TBA

Seung-Yeal Ha (Seoul National University)
2021-04-28 23:10 - 2021-04-29 00:00 | https://researchseminars.org/talk/BerekelyApplied/23/
Title: Emergent behaviors of Lohe tensor flocks (https://researchseminars.org/talk/BerekelyApplied/23/)
by Seung-Yeal Ha (Seoul National University) as part of Berkeley applied mathematics seminar

Abstract
In this talk, we present a new aggregation model on the space of rank-m tensors with the same size, and study emergent dynamics of the proposed model. Our proposed aggregation model is general enough to include Lohe type synchronization models such as the Kuramoto model, the Lohe sphere model and the Lohe matrix models for the ensemble of real rank-0, rank-1 and rank-2 tensors, respectively. In this regard, we call our proposed model as the Lohe tensor model for rank-m tensors with the same size. For the proposed model, we present several sufficient frameworks leading to the collective dynamics of the Lohe tensor model in terms of system parameters and initial data, and study existence of special solutions such as completely separable solutions and quadratically separable solutions. This is a joint work with Hansol Park (Seoul National Univ.) and Dohyun Kim (Sungshin Women’s Univ.)

Jianfeng Lu (Duke University)
2021-02-04 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/24/
Title: Towards solving high dimensional PDEs using neural networks (https://researchseminars.org/talk/BerekelyApplied/24/)
by Jianfeng Lu (Duke University) as part of Berkeley applied mathematics seminar

Abstract
Numerical solution to high dimensional PDEs has been one of the central challenges in scientific computing due to curse of dimension. In recent years, we have seen tremendous progress in applying neural networks to solve high dimensional PDEs, while the analysis for such methods is still lacking. In this talk, we will discuss some of these numerical methods for high dimensional PDEs and also some initial attempts in numerical analysis for high dimensional PDEs.

Doron Levy (University of Maryland)
2021-03-04 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/25/
Title: Fighting drug resistance with math (https://researchseminars.org/talk/BerekelyApplied/25/)
by Doron Levy (University of Maryland) as part of Berkeley applied mathematics seminar

Abstract
The emergence of drug-resistance is a major challenge in chemotherapy. In this talk we will overview some of our recent mathematical models for describing the dynamics of drug-resistance in solid tumors. These models follow the dynamics of the tumor, assuming that the cancer cell population depends on a phenotype variable that corresponds to the resistance level to a cytotoxic drug. Under certain conditions, our models predict that multiple resistant traits emerge at different locations within the tumor, corresponding to heterogeneous tumors. We show that a higher drug dosage may delay a relapse, yet, when this happens, a more resistant trait emerges. We will show how mathematics can be used to propose an efficient drug schedule aiming at minimizing the growth rate of the most resistant trait, and how such resistant cells can be eliminated.

Ruiwen Shu (University of Maryland, College Park)
2021-01-28 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/26/
Title: Linear interpolation convexity/concavity in the minimization of attractive-repulsive energy (https://researchseminars.org/talk/BerekelyApplied/26/)
by Ruiwen Shu (University of Maryland, College Park) as part of Berkeley applied mathematics seminar

Abstract
Energy minimization problems of attractive-repulsive pairwise interactions are very important in the study of pattern formation in biological and social sciences. In this talk, I will discuss some recent progress (joint work with Jose Carrillo) on the study of Wasserstein-$\infty$ local energy minimizers by using the method of linear interpolation convexity/concavity. In the first part, we prove the radial symmetry and uniqueness of local minimizers for interaction potentials satisfying the 'linear interpolation convexity', which generalizes the result of O. Lopes 17' for global minimizers. In the second part, we show that the failure of linear interpolation convexity could lead to the formation of small scales in the support of local minimizers, and construct interaction potentials whose local minimizers are supported on fractal sets. To our best knowledge, this is the first time people observe fractal sets as the support of local minimizers.

Mitchell Luskin (University of Minnesota)
2021-03-31 23:10 - 2021-04-01 00:00 | https://researchseminars.org/talk/BerekelyApplied/27/
Title: Mathematics and Physics at the Moiré Scale (https://researchseminars.org/talk/BerekelyApplied/27/)
by Mitchell Luskin (University of Minnesota) as part of Berkeley applied mathematics seminar

Abstract
Placing a two-dimensional lattice on another with a small rotation gives rise to periodic “moire” patterns on a superlattice scale much larger than the original lattice. This effective large-scale fundamental domain allows phenomena such as the fractal Hofstadter butterfly in the spectrum of Harper’s equation to be observed in real crystals. Experimentalists have more recently observed new correlated phases at the “magic” twist angles predicted by theorists.

We will give mathematical and computational models to predict and gain insight into new physical phenomena at the moiré scale including our recent mathematical and experimental results for twisted trilayer graphene.

Andrew Horning (Cornell University)
2021-01-21 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/28/
Title: Computing spectral properties of infinite-dimensional operators (https://researchseminars.org/talk/BerekelyApplied/28/)
by Andrew Horning (Cornell University) as part of Berkeley applied mathematics seminar

Abstract
Computing the spectrum of a differential or integral operator is usually done in two steps: (1) discretize the operator to obtain a matrix eigenvalue problem and (2) compute eigenvalues of the matrix with numerical linear algebra. This “discretize-then-solve” paradigm is flexible and powerful, but tension between spectral properties of the operator and the matrix discretizations can lead to numerical artifacts that pollute computed spectra and degrade accuracy. Moreover, it is unclear how to robustly capture infinite-dimensional phenomena, like continuous spectra, with “discretize-then-solve.” In this talk, we introduce a new computational framework that extracts discrete and continuous spectral properties of a broad class of operators by strategically sampling the resolvent operator in the complex plane. The resulting algorithms respect key structure from the operator, regardless of the underlying matrix discretizations used for computation. We illustrate the approach through a range of examples, including a Dirac operator and a magnetic tight-binding model of graphene.

Zhewei Yao (University of California, Berkeley)
2021-02-18 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/29/
Title: Second Order Methods for Neural Network Analysis, Training, and Inference (https://researchseminars.org/talk/BerekelyApplied/29/)
by Zhewei Yao (University of California, Berkeley) as part of Berkeley applied mathematics seminar

Abstract: TBA

Zhenning Cai (National University of Singapore)
2021-02-25 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/30/
Title: On the method of complex Langevin (https://researchseminars.org/talk/BerekelyApplied/30/)
by Zhenning Cai (National University of Singapore) as part of Berkeley applied mathematics seminar

Abstract
The complex Langevin (CL) method is a numerical approach to alleviate the numerical sign problem in the computation of path integrals in lattice field theories. Mathematically, it is a simple numerical tool to compute a wide class of high-dimensional and oscillatory integrals with the form of an ensemble average. The method was developed in 1980s. However, after it was proposed, it had very few applications due to its subtle nature. It is often observed that the CL method converges but the limiting result is incorrect. Less than one decade ago, the CL method was improved by gauge cooling method and dynamical stabilization, after which the CL method acquired much more attention and was later successfully applied to a number of fields including finite density quantum chromodynamics, superstring theory, and the spin-orbit coupling. In this talk, I will take the mathematical perspective to explain the basic idea of the CL method and the reason of its failure. The current limitation of the method and the possible remedies will also be discussed.

Fernando Casas (Universitat Jaume I, Spain)
2021-03-11 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/31/
Title: Symmetric-conjugate composition methods in the numerical integration of differential equations (https://researchseminars.org/talk/BerekelyApplied/31/)
by Fernando Casas (Universitat Jaume I, Spain) as part of Berkeley applied mathematics seminar

Abstract
In this talk I will analyze composition methods with complex coefficients exhibiting the so-called “symmetry-conjugate” pattern in their distribution. In particular, I will study their behavior with respect to preservation of qualitative properties when projected on the real axis and how they compare with the usual left-right palindromic compositions. New schemes within this family up to order 8 will be proposed and illustrated on several examples. Some of the special features of this class of methods will also be reviewed.

Yuan Su (Caltech)
2021-03-19 17:10 - 18:00 | https://researchseminars.org/talk/BerekelyApplied/32/
Title: Nearly tight Trotterization of interacting electrons (https://researchseminars.org/talk/BerekelyApplied/32/)
by Yuan Su (Caltech) as part of Berkeley applied mathematics seminar

Abstract
We consider simulating quantum systems on digital quantum computers. We show that the performance of quantum simulation can be improved by simultaneously exploiting the commutativity of Hamiltonian, the sparsity of interactions, and the prior knowledge of initial state. We achieve this using Trotterization for a class of interacting electrons that encompasses various physical systems, including the plane-wave-basis electronic structure and the Fermi-Hubbard model. We estimate the simulation error by taking the transition amplitude of nested commutators of Hamiltonian terms within the $\eta$-electron manifold. We develop multiple techniques for bounding the transition amplitude and expectation of general fermionic operators, which may be of independent interest. We show that it suffices to use $\left(\frac{n^{5/3}}{\eta^{2/3}}+n^{4/3}\eta^{2/3}\right)n^{o(1)}$ gates to simulate electronic structure in the plane-wave basis with $n$ spin orbitals and $\eta$ electrons, improving the best previous result in second quantization while outperforming the first-quantized simulation when $n=\eta^{2-o(1)}$. We also obtain an improvement for simulating the Fermi-Hubbard model. We construct concrete examples for which our bounds are almost saturated, giving a nearly tight Trotterization of interacting electrons.

Jimmy Xia (University of California, Berkeley)
2021-04-07 23:10 - 2021-04-08 00:00 | https://researchseminars.org/talk/BerekelyApplied/33/
Title: Mathematical modeling of human learning and decision making (https://researchseminars.org/talk/BerekelyApplied/33/)
by Jimmy Xia (University of California, Berkeley) as part of Berkeley applied mathematics seminar

Abstract
Reinforcement learning (RL) has been widely used to study and model human, animal and artificial intelligence. In this talk, we focus on modeling human learning and decision making, and exemplify two ways that mathematical RL modeling adds to our existing knowledge of human cognition: (1) as a powerful quantitative tool for parametrizing and compressing individual differences in human behavior, and (2) as an important theoretical framework for complex human cognition. In the first study, we use RL modeling to capture trial-by-trial learning dynamics in a probabilistic task and to understand how learning changes during puberty. In the second study, we augment existing RL models to explain transfer and generalization effects in multi-step learning and decision making tasks.

Zhennan Zhou (Peking University)
2021-04-14 23:10 - 2021-04-15 00:00 | https://researchseminars.org/talk/BerekelyApplied/34/
Title: Efficient Sampling of Thermal Averages of Interacting Quantum Particle Systems: preconditioning and simulation with random batches (https://researchseminars.org/talk/BerekelyApplied/34/)
by Zhennan Zhou (Peking University) as part of Berkeley applied mathematics seminar

Abstract
We investigate the continuum limit that the number of beads goes to infinity in the ring polymer representation of thermal averages. Studying the continuum limit of the trajectory sampling equation sheds light on possible preconditioning techniques for sampling ring polymer configurations with large number of beads. In the case where the potential is quadratic, we show that the continuum limit of the preconditioned mass modified Langevin dynamics converges to its equilibrium exponentially fast, which suggests that the finite-dimensional counterpart has a dimension-independent convergence rate. In the second part of the talk, an efficient sampling method, the pmmLang+RBM, is proposed to compute the quantum thermal average in the interacting quantum particle system. Benefiting from the random batch method (RBM), the pmmLang+RBM reduces the complexity due to the interaction forces per timestep from O(NP^2) to O(NP), where N is the number of beads and P is the number of particles. We also propose an extension of the pmmLang+RBM, which is based on the splitting Monte Carlo method and is applicable when the interacting potential contains a singular part.

Phillip Colella (Lawrence Berkeley National Laboratory and UC Berkeley)
2021-04-21 23:10 - 2021-04-22 00:00 | https://researchseminars.org/talk/BerekelyApplied/35/
Title: Numerical Analysis of Particle-in-Cell Methods for Advection-Type Partial Differential Equations (https://researchseminars.org/talk/BerekelyApplied/35/)
by Phillip Colella (Lawrence Berkeley National Laboratory and UC Berkeley) as part of Berkeley applied mathematics seminar

Abstract
Particle-in-cell (PIC) methods for advection equations use particles that move along integral curves of the advection velocity to represent the primary dependent variables, while using a structured grid to which the particle state has been interpolated to compute the dependence of the velocities, and forcing terms on the solution. PIC is one of the oldest methods in numerical PDE, dating back to the 1950s for fluid dynamics, and the 1960s for plasma physics, and are still used extensively today. Nonetheless, there appears to be no mathematically-systematic numerical analysis framework for understanding the error in PIC methods. This is in contrast to traditional finite-difference, finite element, and grid-free particle methods, for which such framework exist and are used very successfully to design methods for complex problems. In this talk, we will present such a numerical analysis framework for both advection and for kinetics problems. One of the principal results is that PIC methods, as they are currently used in scientific applications, have an O(1) contribution to the error, relative to the number of particles, that grows exponentially in time. We will describe the source of this error, and strategies for controlling it.

Joint work with Henry Boateng, Bhavna Singh, Erick Velez, and Colin Wahl.

Thomas Hou (Caltech)
2021-09-01 23:10 - 2021-09-02 00:00 | https://researchseminars.org/talk/BerekelyApplied/36/
Title: Potential singularity of 3D incompressible Euler equations and the nearly singular behavior of 3D Navier-Stokes equations (https://researchseminars.org/talk/BerekelyApplied/36/)
by Thomas Hou (Caltech) as part of Berkeley applied mathematics seminar

Abstract
Whether the 3D incompressible Euler and Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In an effort to provide a rigorous proof of the potential Euler singularity revealed by Luo-Hou's computation, we develop a novel method of analysis and prove that the original De Gregorio model and the Hou-Lou model develop a finite time singularity from smooth initial data. Using this framework and some techniques from Elgindi's recent work on the Euler singularity, we prove the finite time blowup of the 2D Boussinesq and 3D Euler equations with $C^{1,\alpha}$ initial velocity and boundary. Further, we present some new numerical evidence that the 3D incompressible Euler equations with smooth initial data develop a potential finite time singularity at the origin, which is quite different from the Luo-Hou scenario. Our study also shows that the 3D Navier-Stokes equations develop nearly singular solutions with maximum vorticity increasing by a factor of $10^7$. However, the viscous effect eventually dominates vortex stretching and the 3D Navier-Stokes equations narrowly escape finite time blowup. Finally, we present strong numerical evidence that the 3D Navier-Stokes equations with slowly decaying viscosity develop a finite time singularity.

Jeffrey Kuan (UC Berkeley)
2021-09-08 23:10 - 2021-09-09 00:00 | https://researchseminars.org/talk/BerekelyApplied/37/
Title: A stochastic fluid-structure interaction problem describing Stokes flow interacting with a membrane (https://researchseminars.org/talk/BerekelyApplied/37/)
by Jeffrey Kuan (UC Berkeley) as part of Berkeley applied mathematics seminar

Abstract
In this talk, we present a well-posedness result for a stochastic fluid-structure interaction model. We study a fully coupled stochastic fluid-structure interaction problem, with linear coupling between Stokes flow and an elastic structure modeled by the wave equation, and stochastic noise in time acting on the structure. Such stochasticity is of interest in applications of fluid-structure interaction, in which there is random noise present which may affect the dynamics and statistics of the full system. We construct a solution by using a new splitting method for stochastic fluid-structure interaction, and probabilistic methods. To the best of our knowledge, this is the first result on well-posedness for fully coupled stochastic fluid-structure interaction. This is joint work with Sunčica Čanić (UC Berkeley).

Natalia Komarova (UC Irvine)
2021-09-15 23:10 - 2021-09-16 00:00 | https://researchseminars.org/talk/BerekelyApplied/38/
Title: Mathematical methods in cancer dynamics (https://researchseminars.org/talk/BerekelyApplied/38/)
by Natalia Komarova (UC Irvine) as part of Berkeley applied mathematics seminar

Abstract
Evolutionary dynamics are at the core of carcinogenesis. Mathematical methods can be used to study evolutionary processes, such as selection and mutation, and to shed light onto cancer origins, progression, and mechanisms of treatment. I will present two broad approaches to cancer modeling that we have developed. One is concerned with near-equilibrium dynamics of stem cells, with the goal of figuring out how tissue cell turnover is orchestrated, and how control networks prevent “selfish” cell growth. The other direction is studying evolutionary dynamics of drug resistance in cancer.

Michael Lindsey (Courant Institute)
2021-09-22 23:10 - 2021-09-23 00:00 | https://researchseminars.org/talk/BerekelyApplied/39/
Title: Embedding approaches for classical and quantum statistical mechanics (https://researchseminars.org/talk/BerekelyApplied/39/)
by Michael Lindsey (Courant Institute) as part of Berkeley applied mathematics seminar

Abstract
We show how a synthesis of ideas from graphical models, tensor networks, optimal transport, and semidefinite programming can be brought to bear on problems from classical and quantum statistical mechanics, broadly construed. Specifically, we discuss applications including classical and quantum spin systems on the lattice, continuous global optimization, and electronic structure.

Simon Becker (University of Cambridge)
2021-09-29 23:10 - 2021-09-30 00:00 | https://researchseminars.org/talk/BerekelyApplied/40/
Title: Mathematical properties of twisted bilayer graphene (https://researchseminars.org/talk/BerekelyApplied/40/)
by Simon Becker (University of Cambridge) as part of Berkeley applied mathematics seminar

Abstract
Twistronics is the study of how the angle (the twist) between layers of two-dimensional materials can change their electronic structure. When two sheets of graphene are twisted by those angles the resulting material exhibits flat bands which, as argued in the physics literature, is related to superconductivity, ferromagnetism, and Mott-insulators. I will start with a very simple operator whose spectral properties are supposed to determine which angles are magical and describe some of the mathematical challenges and results. Then, I will introduce a method to study the response of this material to an external magnetic field in a regime of large magnetic fields and explain some of the phenomena. Finally, I will move on to even simpler one-dimensional models, that naturally appear when strain is applied in one direction of the van der Waals material to make it periodic in one spatial direction, which allow for a more refined mathematical analysis (Cantor spectrum and metal/insulator transitions). If time permits, I will briefly touch upon the connection between such materials and topological insulators.

This is joint work with M Embree, R Kim, J Wittsten, X Zhu, M Zworski.

Emanuel Gull (University of Michigan)
2021-10-20 23:10 - 2021-10-21 00:00 | https://researchseminars.org/talk/BerekelyApplied/41/
Title: Nevanlinna Analytical Continuation (https://researchseminars.org/talk/BerekelyApplied/41/)
by Emanuel Gull (University of Michigan) as part of Berkeley applied mathematics seminar

Abstract
Simulations of finite temperature quantum systems provide imaginary frequency Green’s functions that correspond one-to-one to experimentally measurable real-frequency spectra. However, due to the bad conditioning of the continuation transform from imaginary to real frequencies, established methods tend to either wash out spectral features at high frequencies or produce spectral functions with unphysical negative parts. Here, we show that explicitly respecting the analytic ‘Nevanlinna' structure of the Green’s function leads to intrinsically positive and normalized spectral functions and we present a continued fraction expansion that yields all possible functions consistent with the analytic structure. Application to synthetic trial data shows that sharp, smooth, and multi-peak data is resolved accurately. Application to the band structure of silicon demonstrates that high energy features are resolved precisely. Continuations in a realistic correlated setup reveal additional features that were previously unresolved. By substantially increasing the resolution of the real frequency calculations, our work overcomes one of the main limitations of finite-temperature quantum simulations.

Song Mei (UC Berkeley)
2021-10-27 18:10 - 19:00 | https://researchseminars.org/talk/BerekelyApplied/42/
Title: The efficiency of kernel methods on structured datasets (https://researchseminars.org/talk/BerekelyApplied/42/)
by Song Mei (UC Berkeley) as part of Berkeley applied mathematics seminar

Abstract
Inspired by the proposal of tangent kernels of neural networks (NNs), a recent research line aims to design kernels with a better generalization performance on standard datasets. Indeed, a few recent works showed that certain kernel machines perform as well as NNs on certain datasets, despite their separations in specific cases implied by theoretical results. Furthermore, it was shown that the induced kernels of convolutional neural networks perform much better than any former handcrafted kernels. These empirical results pose a theoretical challenge to understanding the performance gaps in kernel machines and NNs in different scenarios.

In this talk, we show that data structures play an essential role in inducing these performance gaps. We consider a few natural data structures, and study their effects on the performance of these learning methods. Based on a fine-grained high dimensional asymptotics framework of analyzing random features models and kernel machines, we show the following: 1) If the feature vectors are nearly isotropic, kernel methods suffer from the curse of dimensionality, while NNs can overcome it by learning the best low-dimensional representation; 2) If the feature vectors display the same low-dimensional structure as the target function (the spiked covariates model), this curse of dimensionality becomes milder, and the performance gap between kernel methods and NNs become smaller; 3) On datasets that display some invariance structure (e.g., image dataset), there is a quantitative performance gain of using invariant kernels (e.g., convolutional kernels) over inner product kernels. Beyond explaining the performance gaps, these theoretical results can further provide some intuitions towards designing kernel methods with better performance.

Jason Kaye (Flatiron institute)
2021-11-11 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/43/
Title: Efficient numerical algorithms for simulating quantum dynamics (https://researchseminars.org/talk/BerekelyApplied/43/)
by Jason Kaye (Flatiron institute) as part of Berkeley applied mathematics seminar

Abstract
I will describe a few new algorithms which reduce computational bottlenecks in simulations of quantum many-body dynamics.

In time-dependent density functional theory (TDDFT), the many-body wavefunction is approximated using a collection of single-particle wavefunctions, which independently satisfy the Schrodinger equation and are coupled through an effective potential. I will introduce a high-order, FFT-based solver for free space (nonperiodic) problems in TDDFT which sidesteps the usual requirement of imposing artificial boundary conditions.

Many-body Green's functions, which describe correlations between quantum observables, enable practical simulations beyond the effective one-body picture of TDDFT. The Green's functions satisfy history dependent Volterra integro-differential equations with kernel nonlinearities. I will outline efficient history integration algorithms which significantly extend feasible propagation times in both equilibrium and nonequilibrium calculations.

Alexandre Chorin (UC Berkeley and LBNL)
2021-12-09 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/44/
by Alexandre Chorin (UC Berkeley and LBNL) as part of Berkeley applied mathematics seminar

Abstract: TBA

Yu Tong (UC Berkeley)
2021-11-03 23:10 - 2021-11-04 00:00 | https://researchseminars.org/talk/BerekelyApplied/45/
Title: Quantum eigenstate filtering and its applications (https://researchseminars.org/talk/BerekelyApplied/45/)
by Yu Tong (UC Berkeley) as part of Berkeley applied mathematics seminar

Abstract
In this talk I will introduce a quantum algorithmic technique called quantum eigenstate filtering, which is based on approximation theory results and the quantum singular value transformation. I will discuss its applications in preparing eigenstates, solving quantum linear systems, and estimating the ground state energy. For these tasks this technique leads to significantly better query complexities, fewer ancilla qubits, and does so without requiring complex subroutines that may not be realistically implementable. Besides these algorithmic applications, the essential idea also leads to a useful proof technique for studying the ground state property of quantum many-body systems.

Di Fang (UC Berkeley)
2021-10-06 23:10 - 2021-10-07 00:00 | https://researchseminars.org/talk/BerekelyApplied/46/
Title: Time-dependent unbounded Hamiltonian simulation with vector norm scaling (https://researchseminars.org/talk/BerekelyApplied/46/)
by Di Fang (UC Berkeley) as part of Berkeley applied mathematics seminar

Abstract
Hamiltonian simulation is a basic task in quantum computation. The accuracy of such simulation is usually measured by the error of the unitary evolution operator in the operator norm, which in turn depends on certain norm of the Hamiltonian. For unbounded operators, after suitable discretization, the norm of the Hamiltonian can be very large, which significantly increases the simulation cost. However, the operator norm measures the worst-case error of the quantum simulation, while practical simulation concerns the error with respect to a given initial vector at hand. We demonstrate that under suitable assumptions of the Hamiltonian and the initial vector, if the error is measured in terms of the vector norm, the computational cost may not increase at all as the norm of the Hamiltonian increases using Trotter type methods. In this sense, our result outperforms all previous error bounds in the quantum simulation literature. We also clarify the existence and the importance of commutator scalings of Trotter and generalized Trotter methods for time-dependent Hamiltonian simulations.

Franziska Weber (Carnegie Mellon University)
2021-10-13 23:10 - 2021-10-14 00:00 | https://researchseminars.org/talk/BerekelyApplied/47/
Title: A Convergent Numerical Method for a Model of Liquid Crystal Director Coupled to An Electric Field (https://researchseminars.org/talk/BerekelyApplied/47/)
by Franziska Weber (Carnegie Mellon University) as part of Berkeley applied mathematics seminar

Abstract
Starting from the Oseen-Frank theory, we derive a simple model for the dynamics of a nematic liquid crystal director field under the influence of an electric field. The resulting nonlinear system of partial differential equations consists of the electrostatics equations for the electric field coupled with the damped wave map equation for the evolution of the liquid crystal director field, which is a normal vector pointing in the direction of the main orientation of the liquid crystal molecules. The liquid crystal director field enters the electrostatics equations in the constitutive relations while the electric field enters the wave map equation in the form of a nonlinear source term. Since it is a normal vector, the variable for the liquid crystal director field has to satisfy the constraint that it takes values in the unit sphere. We derive an energy-stable and constraint preserving numerical method for this system and prove convergence of a subsequence of approximations to a weak solution of the system of partial differential equations. In particular, this implies the existence of weak solutions for this model.

Aditi Krishnapriyan (Lawrence Berkeley National Lab)
2021-12-02 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/48/
Title: Integrating machine learning with physics-based spatial and temporal modeling (https://researchseminars.org/talk/BerekelyApplied/48/)
by Aditi Krishnapriyan (Lawrence Berkeley National Lab) as part of Berkeley applied mathematics seminar

Abstract
Deep learning has achieved great success in numerous areas, and is also seeing increasing interest in scientific applications. However, challenges still remain: scientific phenomena are difficult to model, and can also be limited by a lack of training data. As a result, scientific machine learning approaches are being developed by incorporating domain knowledge into the machine learning process to enable more accurate and general predictions. One such popular approach, colloquially known as physics-informed neural networks (PINNs), incorporates domain knowledge as soft constraints on an empirical loss function. I will discuss the challenges associated with such an approach, and show that by changing the learning paradigm to curriculum regularization or sequence-to-sequence learning, we can achieve significantly lower error. Another approach, colloquially known as ODE-Nets, aims to couple dynamical systems/numerical methods with neural networks. I will discuss how exploiting techniques from numerical analysis for these systems can enable learning continuous dynamics for scientific problems. This method will be illustrated by showing that it can: resolve fine-scale features in a temporal solution despite training on coarse data, successfully resolve fine-scale features in the temporal solution even when the training data is irregularly spaced with non-uniform time intervals, and learn dynamics from image snapshots by generating super-resolution videos at higher frame rates of the much finer solution.

No seminar
2022-01-27 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/49/
by No seminar as part of Berkeley applied mathematics seminar

Abstract: TBA

Alina Chertock (North Carolina State University)
2022-02-03 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/50/
Title: Structure Preserving Numerical Methods for Hyperbolic Systems of Conservation and Balance Laws (https://researchseminars.org/talk/BerekelyApplied/50/)
by Alina Chertock (North Carolina State University) as part of Berkeley applied mathematics seminar

Abstract
Many physical models, while quite different in nature, can be described by nonlinear hyperbolic systems of conservation and balance laws. The main source of difficulties one comes across when numerically solving these systems is lack of smoothness as solutions of hyperbolic conservation/balance laws may develop very complicated nonlinear wave structures including shocks, rarefaction waves and contact discontinuities. The level of complexity may increase even further when solutions of the hyperbolic system reveal a multiscale character and/or the system includes additional terms such as friction terms, geometrical terms, nonconservative products, etc., which are needed to be taken into account in order to achieve a proper description of the studied physical phenomena. In such cases, it is extremely important to design a numerical method that is not only consistent with the given PDEs, but also preserves certain structural and asymptotic properties of the underlying problem at the discrete level. While a variety of numerical methods for such models have been successfully developed, there are still many open problems, for which the derivation of reliable high-resolution numerical methods still remains to be an extremely challenging task.

In this talk, I will discuss recent advances in the development of two classes of structure preserving numerical methods for nonlinear hyperbolic systems of conservation and balance laws. In particular, I will present (i) well-balanced and positivity preserving numerical schemes, that is, the methods which are capable of exactly preserving some steady-state solutions as well as maintaining the positivity of the numerical quantities when it is required by the physical application, and (ii) asymptotic preserving schemes, which provide accurate and efficient numerical solutions in certain stiff and/or asymptotic regimes of physical interest.

Yunan Yang (ETH Zurich)
2022-02-09 18:10 - 19:00 | https://researchseminars.org/talk/BerekelyApplied/51/
Title: Adjoint DSMC Method for Boltzmann-Constrained Optimization Problems (https://researchseminars.org/talk/BerekelyApplied/51/)
by Yunan Yang (ETH Zurich) as part of Berkeley applied mathematics seminar

Abstract
Applications for kinetic equations such as optimal design and inverse problems often involve finding unknown parameters through gradient-based optimization algorithms. Based on the adjoint-state method, we derive two different frameworks for approximating the gradient of an objective functional constrained by the nonlinear Boltzmann equation. While the forward problem can be solved by the Direct Simulation Monte Carlo (DSMC) method, it is difficult to efficiently solve the high-dimensional continuous adjoint equation obtained by the "optimize-then-discretize" approach. This challenge motivates us to propose an adjoint DSMC method following the "discretize-then-optimize" approach for Boltzmann-constrained optimization. We also analyze the properties of the two frameworks and their connections. Several numerical examples are presented to demonstrate their accuracy and efficiency.

Andre Laestadius (Hylleraas Centre for Quantum Molecular Sciences)
2022-02-24 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/52/
Title: Energy error estimate for coupled-cluster excited states (https://researchseminars.org/talk/BerekelyApplied/52/)
by Andre Laestadius (Hylleraas Centre for Quantum Molecular Sciences) as part of Berkeley applied mathematics seminar

Abstract
In our recent work, the nonlinear equations of the single-reference Coupled-Cluster method have been analyzed using topological degree theory. This generalizes previous work based on (local) strong monotonicity. We have established existence results and qualitative information about the solutions of these equations that also sheds light on some of the numerically observed behavior. In particular, to investigate truncation schemes within the Coupled-Cluster method, we have utilized the Kowalski-Piecuch homotopy. In this setting, we have derived an energy error bound for approximate eigenstates of the Schrödinger equation, i.e., for both ground and excited states.

Samuel Olivier (University of California, Berkeley)
2022-03-03 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/53/
Title: High Order Finite Element Discretizations of the Variable Eddington Factor Equations for Accelerating Radiation Transport Calculations on Curved Meshes (https://researchseminars.org/talk/BerekelyApplied/53/)
by Samuel Olivier (University of California, Berkeley) as part of Berkeley applied mathematics seminar

Abstract
The Variable Eddington Factor (VEF) method is one of the oldest techniques for solving the radiation transport equation. In VEF, the kinetic equation is iteratively coupled to the moment equations through discrete closures. This moment-based approach enables significant algorithmic flexibility and more efficient multiphysics coupling. However, despite considerable attention in the literature, VEF is rarely used in practice due to the lack of scalable iterative preconditioners for the discretized moment equations. In this talk, I present three classes of VEF methods with high-order accuracy on curved meshes that can be efficiently and scalably solved. Discretization and preconditioning techniques known to be effective on simpler model elliptic problems are extended to the VEF moment equations to derive Discontinuous Galerkin, continuous finite element, and mixed finite element VEF methods. These methods are demonstrated to be effective on a proxy problem from thermal radiative transfer in both outer transport iterations and inner preconditioned linear solver iterations and to scale out to 1152 processors and over 10 million scalar flux unknowns.

Michael I Weinstein (Columbia University)
2022-04-06 23:10 - 2022-04-07 00:00 | https://researchseminars.org/talk/BerekelyApplied/54/
Title: Discrete honeycombs, rational edges and edge states (https://researchseminars.org/talk/BerekelyApplied/54/)
by Michael I Weinstein (Columbia University) as part of Berkeley applied mathematics seminar

Abstract
We first discuss the derivation of tight binding (discrete) Hamiltonians from an underlying continuum Schroedinger Hamiltonians in both non-magnetic and strongly magnetic systems (joint works with with CL Fefferman and J Shapiro).

We then present very recent work (with CL Fefferman and S Fliss) on the tight binding model of graphene, sharply terminated along a rational edge, a line I parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges into those of "zigzag type" and those of "armchair type", generalizing the classical zigzag and armchair edges. Edge states are eigenstates which are plane wave like in directions parallel to the edge and are localized in directions transverse to the edge. We prove that zero energy/flat band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulas for flat band edge states when they exist. Finally, we produce strong evidence for the existence of dispersive (non flat) edge state curves of nonzero energy for most rational edges.

Houman Owhadi (California Institute of Technology)
2022-04-27 23:10 - 2022-04-28 00:00 | https://researchseminars.org/talk/BerekelyApplied/55/
Title: Computational Graph Completion (https://researchseminars.org/talk/BerekelyApplied/55/)
by Houman Owhadi (California Institute of Technology) as part of Berkeley applied mathematics seminar

Abstract
We present a framework for generating, organizing, and reasoning with computational knowledge. It is motivated by the observation that most problems in Computational Sciences and Engineering (CSE) can be formulated as that of completing (from data) a computational graph (or hypergraph) representing dependencies between functions and variables. Nodes represent variables, and edges represent functions. Functions and variables may be known, unknown, or random. Data comes in the form of observations of distinct values of a finite number of subsets of the variables of the graph (satisfying its functional dependencies). The underlying problem combines a regression problem (approximating unknown functions) with a matrix completion problem (recovering unobserved variables in the data). Replacing unknown functions by Gaussian Processes (GPs) and conditioning on observed data provides a simple but efficient approach to completing such graphs. Since this completion process can be reduced to an algorithm, as one solves $\sqrt{2}$ on a pocket calculator without thinking about it, one could, with the automation of the proposed framework, solve a complex CSE problem by drawing a diagram.

Jingwei Hu (University of Washington)
2022-02-17 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/56/
Title: An efficient dynamical low-rank algorithm for the Boltzmann-BGK equation close to the compressible viscous flow regime (https://researchseminars.org/talk/BerekelyApplied/56/)
by Jingwei Hu (University of Washington) as part of Berkeley applied mathematics seminar

Abstract
It has recently been demonstrated that dynamical low-rank algorithms can provide robust and efficient approximations to a range of kinetic equations. This is true especially if the solution is close to some asymptotic limit where it is known that the solution is low-rank. A particularly interesting case is the fluid dynamic limit that is commonly obtained in the limit of small Knudsen number. However, in this case the Maxwellian which describes the corresponding equilibrium distribution is not necessarily low-rank; because of this, the methods known in the literature are only applicable to the weakly compressible case. In this paper, we propose an efficient dynamical low-rank integrator that can capture the fluid limit—the Navier–Stokes equations—of the Boltzmann-BGK model even in the compressible regime. This is accomplished by writing the solution as f = Mg, where M is the Maxwellian and the low-rank approximation is only applied to g. To efficiently implement this decomposition within a low-rank framework requires, in the isothermal case, that certain coefficients are evaluated using convolutions, for which fast algorithms are known. Using the proposed decomposition also has the advantage that the rank required to obtain accurate results is significantly reduced compared to the previous state of the art. We demonstrate this by performing a number of numerical experiments and also show that our method is able to capture sharp gradients/shock waves. This is joint work with Lukas Einkemmer (Innsbruck) and Lexing Ying (Stanford).

Guillaume Bal (University of Chicago)
2022-03-30 23:10 - 2022-03-31 00:00 | https://researchseminars.org/talk/BerekelyApplied/57/
Title: Asymmetric transport and topological invariants (https://researchseminars.org/talk/BerekelyApplied/57/)
by Guillaume Bal (University of Chicago) as part of Berkeley applied mathematics seminar

Abstract
Robust asymmetric transport at the interface between two-dimensional insulating bulks has been observed in many areas of (geo)physical and materials sciences. The main practical appeal of this asymmetry is its immunity to large classes of perturbations. This stability is explained by topological considerations.

A physical observable, a one-dimensional conductivity, is assigned to the asymmetric transport. Interface Hamiltonians modeling the transition between the bulk phases are next introduced and classified by a topological charge, the index of an appropriate Fredholm operator. A general principle, the bulk-edge correspondence, then states that the conductivity is quantized and equal to the topological charge, which may be interpreted as a difference of bulk topologies.

While ubiquitous in the physical and engineering literatures, the bulk-edge correspondence remains difficult to establish mathematically or in fact even heuristically. This talk presents recent results on the derivation of the correspondence for reasonably large algebras of (pseudo-)differential operators that appear generically as low-energy large-wavelength models in the applications. We use the correspondence to compute the asymmetry in several settings where a direct estimation seems hopeless, with applications, e.g., in graphene-based Floquet topological insulators and topological properties of twisted bilayer graphene.

Time permitting, we will contrast the above spectral properties with the practically more relevant temporal picture and, for instance, the propagation of semi-classical wavepackets along curved interfaces.

Eitan Tadmor (University of Maryland)
2022-03-16 23:10 - 2022-03-17 00:00 | https://researchseminars.org/talk/BerekelyApplied/58/
Title: Hierarchical decomposition of images and the problem of Bourgain-Brezis (https://researchseminars.org/talk/BerekelyApplied/58/)
by Eitan Tadmor (University of Maryland) as part of Berkeley applied mathematics seminar

Abstract
Edges are noticeable features in images which can be extracted from noisy data using different variational models. The analysis of such models leads to the question of expressing general L^2-data, f, as the divergence of uniformly bounded vector fields, div(U). We present a multi-scale approach to construct uniformly bounded solutions of div(U)=f for general f’s in the critical regularity space L^d(T^d). The study of this equation and related problems was motivated by results of Bourgain & Brezis. The intriguing critical aspect here is that although the problems are linear, construction of their solution is not. Our constructive solution for such problems is a special case of a rather general framework for solving linear equations, formulated as inverse problems in critical regularity spaces. The solutions are realized in terms of nonlinear hierarchical decomposition, U=image001.png, which we introduced earlier in the context of image processing, and yield a multi-scale decomposition of “objects” U.

Dejan Slepcev (Carnegie Mellon University)
2022-04-13 23:10 - 2022-04-14 00:00 | https://researchseminars.org/talk/BerekelyApplied/59/
Title: Proper regularizers for semi-supervised learning (https://researchseminars.org/talk/BerekelyApplied/59/)
by Dejan Slepcev (Carnegie Mellon University) as part of Berkeley applied mathematics seminar

Abstract
We will discuss a standard problem of semi-supersised learning: given a data set (considered as a point cloud in a euclidean space) with a small number of labeled points the task is to extrapolate the label values to the whole data set. In order to utilize the geometry of the dataset one creates a graph by connecting the nodes which are sufficiently close. Many standard approaches rely on minimizing graph-based functionals, which reward the agreement with the labels and the regularity of the estimator. Choosing a good regularization leads to questions about the relations between discrete functionals in random setting and continuum nonlocal and differential functionals. We will discuss how insights about this relation provide ways to properly choose the functionals for semi-supervised learning and appropriately set the weights of the graph so that the information is propagated in a desirable way from the labeled points. Theoretical results, numerical illustrations and performance on standard test data sets will be provided.

Minh Tran (MIT)
2022-04-20 23:10 - 2022-04-21 00:00 | https://researchseminars.org/talk/BerekelyApplied/60/
Title: The propagation of information in power-law interacting systems (https://researchseminars.org/talk/BerekelyApplied/60/)
by Minh Tran (MIT) as part of Berkeley applied mathematics seminar

Abstract
Most physical many-body quantum systems are geometrically local; it takes time to propagate quantum information in the systems. Such locality imposes fundamental limits on many quantum information processing tasks. In this talk, we will review the state-of-the-art speed limits for the propagation of information in quantum systems with power-law interactions. We discuss applications of the speed limits and, in particular, use them to constrain the propagation of error and improve the performance of quantum simulation algorithms. Inversely, we also prove new speed limits using quantum simulation algorithms, suggesting a deep connection between the propagation of information and digital quantum simulation.

Spring break. No seminar
2022-03-23 23:10 - 2022-03-24 00:00 | https://researchseminars.org/talk/BerekelyApplied/61/
by Spring break. No seminar as part of Berkeley applied mathematics seminar

Abstract: TBA

Michael Franco (University of California, Berkeley)
2022-03-10 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/62/
Title: Relating high-order fluid flow problems to simpler subproblems to create efficient preconditioners (https://researchseminars.org/talk/BerekelyApplied/62/)
by Michael Franco (University of California, Berkeley) as part of Berkeley applied mathematics seminar

Abstract
This talk will focus on two solvers for high-order methods, with the common thread being that their efficiency derives from relating the original problem to a simpler subproblem. First, a matrix-free flow solver for high-order finite element discretizations of the incompressible Navier-Stokes and Stokes equations with GPU acceleration will be presented. For high polynomial degrees, assembling the matrix for the linear systems resulting from the finite element discretization can be prohibitively expensive, both in terms of computational complexity and memory. For this reason, it is necessary to develop matrix-free operators and preconditioners, which can be used to efficiently solve these linear systems without access to the matrix entries themselves. Particular attention will be given to the matrix-free operator evaluations that utilize GPU-accelerated sum-factorization techniques to minimize memory movement and maximize throughput. I will also briefly introduce novel preconditioners based on a low-order refined methodology with parallel subspace corrections. Second, I will introduce a novel class of iterative subregion correction preconditioners for solving flow problems with geometrically localized stiffness. Just as multigrid methods spend more effort on cheaper grids to apply a correction that improves convergence on lower frequency components, our subregion correction preconditioners spend more effort on a subregion of the domain demonstrating slow convergence to improve overall convergence rates. Convergence theory and numerical results validating this theory will be presented.

Di Fang (University of California, Berkeley)
2022-02-25 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/63/
Title: Mathematics Department Colloquium: Quantum algorithms for Hamiltonian simulation with unbounded operators (https://researchseminars.org/talk/BerekelyApplied/63/)
by Di Fang (University of California, Berkeley) as part of Berkeley applied mathematics seminar

Lecture held in 60 Evans Hall.

Abstract
Recent years have witnessed tremendous progress in developing and analyzing quantum algorithms for quantum dynamics simulation of bounded operators (Hamiltonian simulation). However, many scientific and engineering problems require the efficient treatment of unbounded operators, which frequently arise due to the discretization of differential operators. Such applications include molecular dynamics, electronic structure theory, quantum control and quantum differential equations solver. We will introduce some recent advances in quantum algorithms for efficient unbounded Hamiltonian simulation, including Trotter-type splitting and the quantum highly oscillatory protocol (qHOP) in the interaction picture. The latter yields a surprising superconvergence result for regular potentials.

Mo Zhou (Duke University)
2022-09-07 23:10 - 2022-09-08 00:00 | https://researchseminars.org/talk/BerekelyApplied/64/
Title: Neural network approaches for high dimensional problems (https://researchseminars.org/talk/BerekelyApplied/64/)
by Mo Zhou (Duke University) as part of Berkeley applied mathematics seminar

Abstract
Neural networks are effective tools for solving high dimensional problems. In this talk, I will summarize the popular methods to solve high dimensional problems with neural networks. Then I will briefly introduce two of my works based on the DeepBSDE method. In the first work, we solve the eigenvalue problem by transforming it into a fixed-point formulation, which is a diffusion Monte Carlo like approach. In the second work, we leverage the actor-critic framework from reinforcement learning to solve the static Hamilton—Jacobi—Bellman equations. We propose a variance reduced temporal difference method for the critic and apply an adaptive step size algorithm for the actor to improve accuracy.

Li Wang (University of Minnesota)
2022-09-21 23:10 - 2022-09-22 00:00 | https://researchseminars.org/talk/BerekelyApplied/65/
Title: Variational methods for gradient flow (https://researchseminars.org/talk/BerekelyApplied/65/)
by Li Wang (University of Minnesota) as part of Berkeley applied mathematics seminar

Abstract
In this talk, I will introduce a general variational framework for nonlinear evolution equations with a gradient flow structure, which arise in material science, animal swarms, chemotaxis, and deep learning, among many others. Building upon this framework, we develop numerical methods that have built-in properties such as positivity preserving and entropy decreasing, and resolve stability issues due to the strong nonlinearity. Two specific applications will be discussed. One is the Wasserstein gradient flow, where the major challenge is to compute the Wasserstein distance and resulting optimization problem. I will show techniques to overcome these difficulties. The other is to simulate crystal surface evolution, which suffers from significant stiffness and therefore prevents simulation with traditional methods on fine spatial grids. On the contrary, our method resolves this issue and is proved to converge at a rate independent of the grid size.

Felix Leditzky (UIUC)
2022-09-28 23:10 - 2022-09-29 00:00 | https://researchseminars.org/talk/BerekelyApplied/66/
Title: The platypus of the quantum channel zoo (https://researchseminars.org/talk/BerekelyApplied/66/)
by Felix Leditzky (UIUC) as part of Berkeley applied mathematics seminar

Abstract
Understanding quantum channels and the strange behavior of their capacities is a key driver of quantum information theory. Despite having rigorous coding theorems, quantum capacities are poorly understood due to super-additivity effects. We will talk about a remarkably simple, low-dimensional, single-parameter family of quantum channels with exotic quantum information-theoretic features. As the simplest example from this family, we focus on a qutrit-to-qutrit channel that is intuitively obtained by hybridizing together a simple degradable channel and a completely useless qubit channel. Such hybridizing makes this channel's capacities behave in a variety of interesting ways. For instance, the private and classical capacity of this channel coincide and can be explicitly calculated, even though the channel does not belong to any class for which the underlying information quantities are known to be additive. Moreover, the quantum capacity of the channel can be computed explicitly, given a clear and compelling conjecture is true. This "spin alignment conjecture," which may be of independent interest, is proved in certain special cases and additional numerical evidence for its validity is provided. We further show that this qutrit channel demonstrates superadditivity when transmitting quantum information jointly with a variety of assisting channels, in a manner unknown before. A higher-dimensional variant of this qutrit channel displays super-additivity of quantum capacity together with an erasure channel. Subject to the spin-alignment conjecture, our results on super-additivity of quantum capacity extend to lower-dimensional channels as well as larger parameter ranges. In particular, super-additivity occurs between two weakly additive channels each with large capacity on their own, in stark contrast to previous results. Remarkably, a single, novel transmission strategy achieves super-additivity in all examples. Our results show that super-additivity is much more prevalent than previously thought. It can occur across a wide variety of channels, even when both participating channels have large quantum capacity.

This is joint work with Debbie Leung, Vikesh Siddhu, Graeme Smith, and John Smolin, and based on the papers https://arxiv.org/abs/2202.08380 and https://arxiv.org/abs/2202.08377.

Chao Ma (Stanford University)
2022-10-05 23:10 - 2022-10-06 00:00 | https://researchseminars.org/talk/BerekelyApplied/67/
Title: Implicit bias of optimization algorithms for neural networks and their effects on generalization (https://researchseminars.org/talk/BerekelyApplied/67/)
by Chao Ma (Stanford University) as part of Berkeley applied mathematics seminar

Abstract
Modern neural networks are usually over-parameterized—the number of parameters exceeds the number of training data. In this case the loss functions tend to have many (or even infinite) global minima, which imposes an additional challenge of minima selection on optimization algorithms besides the convergence. Specifically, when training a neural network, the algorithm not only has to find a global minimum, but also needs to select minima with good generalization among many other bad ones. In this talk, I will share a series of works studying the mechanisms that facilitate global minima selection of optimization algorithms. First, with a linear stability theory, we show that stochastic gradient descent (SGD) favors flat and uniform global minima. Then, we build a theoretical connection of flatness and generalization performance based on a special structure of neural networks. Next, we study the global minima selection dynamics—the process that an optimizer leaves bad minima for good ones—in two settings. For a manifold of minima around which the loss function grows quadratically, we derive effective exploration dynamics on the manifold for SGD and Adam, using a quasistatic approach. For a manifold of minima around which the loss function grows subquadratically, we study the behavior and effective dynamics for GD, which also explains the edge of stability phenomenon.

Mark Fornace (Caltech)
2022-10-12 23:10 - 2022-10-13 00:00 | https://researchseminars.org/talk/BerekelyApplied/68/
Title: Theoretical methods for nucleic acid secondary structure thermodynamics and kinetics (https://researchseminars.org/talk/BerekelyApplied/68/)
by Mark Fornace (Caltech) as part of Berkeley applied mathematics seminar

Abstract
Nucleic acid secondary structure models offer a simplified but powerful lens through which to view, analyze, and design nucleic acid chemistry. Computational approaches based on such models are central to current research directions across molecular programming and the life sciences more broadly. Considering only structures involving noncrossing partitions of nucleotides, dynamic programming algorithms can exactly compute equilibrium quantities (with respect to an approximate free energy model) in cubic complexity. I first show how such algorithms may be improved in speed, augmented in accuracy, and unified across a variety of physical quantities.

While analysis and design paradigms for nucleic acid thermodynamics are long-established in essence, nucleic acid kinetics have proved vexing for accurate and principled estimation algorithms. Past approaches have thus generally relied on stochastic simulation of the respective continuous time Markov chains (an asymptotically correct but computationally costly approach). In contrast, I show how a principled Galerkin-type approach to the kinetics proves remarkably amenable to deterministic estimation by dynamic programming algorithms. While inexact, the approach proves empirically accurate and is theoretically extensible to treatments of mass-action kinetics, macrostate models, and sequence design.

Li Gao (University of Houston)
2022-10-19 23:10 - 2022-10-20 00:00 | https://researchseminars.org/talk/BerekelyApplied/69/
Title: Logarithmic Sobolev inequalities for matrices and matrix-valued functions (https://researchseminars.org/talk/BerekelyApplied/69/)
by Li Gao (University of Houston) as part of Berkeley applied mathematics seminar

Abstract
Logarithmic Sobolev inequalities, first introduced by Gross in 70s, have rich connections to probability, geometry, as well as information theory. In recent years, logarithmic Sobolev inequalities for quantum Markov semigroups attracted a lot of attentions for its applications in quantum information theory and quantum many-body systems. In this talk, I'll present a simple, information-theoretic approach to modified logarithmic Sobolev inequalities for both quantum Markov semigroup on matrices, and classical Markov semigroup on matrix-valued functions. In the classical setting, our results implies every sub-Laplacian of a Hörmander system admits a uniform modified logarithmic Sobolev constant for all its matrix valued functions. For quantum Markov semigroups, we improve a previous result of Gao and Rouzé by replacing the dimension constant by its logarithm. This talk is based on a joint work with Marius Junge, Nicholas, LaRacunte, and Haojian Li.

Matthew Colbrook (University of Cambridge)
2022-10-24 23:10 - 2022-10-25 00:00 | https://researchseminars.org/talk/BerekelyApplied/70/
Title: Residual Dynamic Mode Decomposition: Rigorous Data-Driven Computation of Spectral Properties of Koopman Operators for Dynamical Systems (https://researchseminars.org/talk/BerekelyApplied/70/)
by Matthew Colbrook (University of Cambridge) as part of Berkeley applied mathematics seminar

Abstract
Koopman operators are infinite-dimensional operators that globally linearize
nonlinear dynamical systems, making their spectral information valuable for
understanding dynamics. However, Koopman operators can have continuous
spectra, can lack finite-dimensional invariant subspaces, and approximations can
suffer from spectral pollution (spurious modes). These issues make computing
the spectral properties of Koopman operators a considerable challenge. This two-
part talk will detail the first scheme (ResDMD) with convergence guarantees for
computing the spectra and pseudospectra of general Koopman operators from
snapshot data. Furthermore, we use the resolvent operator and ResDMD to
compute smoothed approximations of spectral measures (including continuous
spectra), with explicit high-order convergence. ResDMD is similar to extended
DMD, except it rigorously concurrently computes a residual from the same
snapshot data, allowing practitioners to gain confidence in the computed results.
Kernelized variants of our algorithms allow for dynamical systems with a high-
dimensional state-space, and the error control provided by ResDMD allows a
posteriori verification of learnt dictionaries. We apply ResDMD to compute the
spectral measure associated with the dynamics of a protein molecule (20,046-dimensional state-space) and demonstrate several problems in fluid dynamics
(with state-space dimensions > 100,000). For example, we compare ResDMD
and DMD for particle image velocimetry data from turbulent wall-jet flow, the
acoustic signature of laser-induced plasma, and turbulent flow past a cascade of
aerofoils.

Samuel Lanthaler (Caltech)
2022-11-10 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/71/
Title: Supervised learning in function space (https://researchseminars.org/talk/BerekelyApplied/71/)
by Samuel Lanthaler (Caltech) as part of Berkeley applied mathematics seminar

Abstract
Neural networks have proven to be effective approximators of high dimensional functions in a wide variety of applications. In scientific applications the goal is often to approximate an underlying operator, which defines a mapping between infinite-dimensional spaces of input and output functions. Extensions of neural networks to this infinite-dimensional setting have been proposed in recent years, giving rise to the rapidly emerging field of operator learning. Despite their practical success, our theoretical understanding of these approaches is still in its infancy. In this talk, I will review some of the proposed operator learning architectures (deep operator networks/neural operators), and present recent results on their approximation theory and sample complexity. This work identifies basic mechanisms by which neural operators can avoid a curse of dimensionality in the underlying (very high- or even infinite-dimensional) approximation task, thus providing a first rationale for their practical success for concrete operators of interest. The analysis also reveals fundamental limitations of some of these approaches.

Lucas Bouck (University of Maryland)
2022-11-17 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/72/
Title: Finite Element Approximation of a Membrane Model for Liquid Crystal Polymeric Networks (https://researchseminars.org/talk/BerekelyApplied/72/)
by Lucas Bouck (University of Maryland) as part of Berkeley applied mathematics seminar

Abstract
Liquid crystal polymeric networks are materials where a nematic liquid crystal is coupled with a rubbery material. When actuated with heat or light, the interaction of the liquid crystal with the rubber creates complex shapes. Starting from the classical 3D trace formula energy of Bladon, Warner and Terentjev (1994), we derive a 2D membrane energy as the formal asymptotic limit of the 3D energy. The derivation is similar to derivations in Ozenda, Sonnet, and Virga (2020) and Cirak et. al. (2014). We characterize the zero energy deformations and prove that the energy lacks certain convexity properties. We propose a finite element method to discretize the problem. To address the lack of convexity of the membrane energy, we regularize with a term that mimics a higher order bending energy. We prove that minimizers of the discrete energy converge to minimizers of the continuous energy. For minimizing the discrete problem, we employ a nonlinear gradient flow scheme, which is energy stable. Additionally, we present computations showing the geometric effects that arise from liquid crystal defects. Computations of configurations from nonisometric origami are also presented.

No seminar. Happy thanksgiving.
2022-11-24 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/73/
by No seminar. Happy thanksgiving. as part of Berkeley applied mathematics seminar

Abstract: TBA

Sui Tang (UCSB)
2022-12-01 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/74/
Title: Bridging the interacting particle models and data science via Gaussian process (https://researchseminars.org/talk/BerekelyApplied/74/)
by Sui Tang (UCSB) as part of Berkeley applied mathematics seminar

Abstract
System of interacting particles that display a wide variety of collective behaviors are ubiquitous in science and engineering, such as self-propelled particles, flocking of birds, milling of fish. Modeling interacting particle systems by a system of differential equations plays an essential role in exploring how individual behavior engenders collective behaviors, which is one of the most fundamental and important problems in various disciplines. Although the recent theoretical and numerical study bring a flood of models that can reproduce many macroscopical qualitative collective patterns of the observed dynamics, the quantitative study towards matching the well-developed models to observational data is scarce.

We consider the data-driven discovery of macroscopic particle models with latent interactions. We propose a learning approach that models the latent interactions as Gaussian processes, which provides an uncertainty-aware modeling of interacting particle systems. We introduce an operator-theoretic framework to provide a detailed analysis of recoverability conditions, and establish statistical optimality of the proposed approach. Numerical results on prototype systems and real data demonstrate the effectiveness of the proposed approach.

Krutika Tawri (University of California, Berkeley)
2022-09-14 23:10 - 2022-09-15 00:00 | https://researchseminars.org/talk/BerekelyApplied/75/
Title: On stochastic partial differential equations with a Ladyzenskaya-Smagorinsky type nonlinearity (https://researchseminars.org/talk/BerekelyApplied/75/)
by Krutika Tawri (University of California, Berkeley) as part of Berkeley applied mathematics seminar

Lecture held in 939 Evans Hall.

Abstract
The theory of monotone operators plays a central role in many areas of nonlinear analysis. Monotone operators often appear in fluid dynamics, for example the p-Laplacian appears in a non-Newtonian variant of the Navier-Stokes equations modeled by Ladyzenskaya or in the Smagorinsky model of turbulence. In this talk, we will discuss global existence results of both martingale and pathwise solutions of stochastic equations with a monotone operator, of the Ladyzenskaya-Smagorinsky type, driven by a general Levy noise. The classical approach based on using directly the Galerkin approximation is not valid. In this talk we will discuss how one can approximate a monotone operator by a family of monotone operators acting in a Hilbert space, so as to recover certain useful properties of the orthogonal projectors and overcome the challenges faced while applying the Galerkin scheme.

Yonah Borns-Weil (University of California, Berkeley)
2022-11-02 23:10 - 2022-11-03 00:00 | https://researchseminars.org/talk/BerekelyApplied/76/
Title: Observable Trotter error bounds in the semiclassical regime (https://researchseminars.org/talk/BerekelyApplied/76/)
by Yonah Borns-Weil (University of California, Berkeley) as part of Berkeley applied mathematics seminar

Abstract
The Trotter product formula is perhaps the oldest and most well-known method for computing Schrödinger propagators. We consider its application to the semiclassical Schrodinger equation where the parameter $h$ is taken to be very small. If one wishes to do practical computations in such a regime, they must take at least $O(h^{-1})$ spatial grid points, which gives the Hamiltonian terms and their nested commutators to be of norm O(h^{-1}). This would appear to cause serious trouble for both Trotter and post-Trotter methods, as their time complexity depends on such norms.

The issue resolves itself when we consider approximating the propagator in observable norm, which measures how much an observable propagated via the actual Hamiltonian differs from one propagated by our Trotter approximation. By a simple argument using Egorov's theorem from semiclasscial analysis, we show the error in this norm to be uniform in the semiclassical parameter $h$. In addition, we consider the discretized space case of interest in quantum computing, and use discrete microlocal analysis on the quantized torus to extend our results to this case without added error.

Govind Menon (Brown University)
2023-01-21 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/77/
Title: Stochastic Nash evolution (https://researchseminars.org/talk/BerekelyApplied/77/)
by Govind Menon (Brown University) as part of Berkeley applied mathematics seminar

Abstract
About ten years ago, De Lellis and Szekelyhidi made the surprising discovery that Nash’s results on the isometric embedding problem for Riemannian manifoldscould be adapted to construct counterintuitive solutions to the Euler equations for incompressible flow. Their work shed new light on turbulence and nonlinear PDE. We use this link in the other direction, transferring ideas from turbulence to geometry.

A thermodynamic framework is introduced that connects two problems previously thought to be distinct: the isometric embedding problem for Riemannian manifolds and the construction of Brownian motion on Riemannian manifolds. This link is used to introduce a geometric stochastic flow that we term stochastic Nash evolution.

These ideas will be explained in a (hopefully) elementary manner. My main goal is to present the stochastic flows in a manner that is suited to implementations by modifications of level set methods. The absence of numerical computations of isometric embeddings is an important gap in our understanding.

This is joint work with Dominik Inauen (University of Leipzig).

Anuj Kumar (UC Santa Cruz)
2023-01-26 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/78/
Title: Application of branching flows to optimal scalar transport and a result concerning the nonuniqueness of trajectories (https://researchseminars.org/talk/BerekelyApplied/78/)
by Anuj Kumar (UC Santa Cruz) as part of Berkeley applied mathematics seminar

Abstract: TBA

Michael Albergo (New York University)
2023-02-23 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/79/
by Michael Albergo (New York University) as part of Berkeley applied mathematics seminar

Abstract: TBA

Xiantao Li (Penn State University)
2023-03-02 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/80/
Title: Efficient algorithms for quantum optimal control problems (https://researchseminars.org/talk/BerekelyApplied/80/)
by Xiantao Li (Penn State University) as part of Berkeley applied mathematics seminar

Abstract
Many recent developments in chemistry and material science utilize systems that exhibit unique quantum properties. A quantum optimal control strategy can maximize the performance of electronic devices that rely on quantum properties, e.g., minimizing the currents in molecular junctions. Computationally, solving the control problem requires visiting the time-dependent Schrödinger equation frequently, and the corresponding solutions will be combined with an optimization method. In this talk, we examine the overall approximation error and estimate the complexity given the error tolerance. We focus on various methods for integrating optimization algorithms and Hamiltonian simulations to achieve provable accuracy.

Fanhui Xu (Harvard University)
2023-03-15 23:10 - 2023-03-16 00:00 | https://researchseminars.org/talk/BerekelyApplied/81/
by Fanhui Xu (Harvard University) as part of Berkeley applied mathematics seminar

Abstract: TBA

Chi-Fang Chen (Caltech)
2023-03-22 23:10 - 2023-03-23 00:00 | https://researchseminars.org/talk/BerekelyApplied/82/
Title: Sparse random Hamiltonians are quantumly easy (https://researchseminars.org/talk/BerekelyApplied/82/)
by Chi-Fang Chen (Caltech) as part of Berkeley applied mathematics seminar

Abstract
A candidate application for quantum computers is to simulate the low-temperature properties of quantum systems. For this task, there is a well-studied quantum algorithm that performs quantum phase estimation on an initial trial state that has a nonnegligible overlap with a low-energy state. However, it is notoriously hard to give theoretical guarantees that such a trial state can be prepared efficiently. Moreover, the heuristic proposals that are currently available, such as with adiabatic state preparation, appear insufficient in practical cases. This paper shows that, for most random sparse Hamiltonians, the maximally mixed state is a sufficiently good trial state, and phase estimation efficiently prepares states with energy arbitrarily close to the ground energy. Furthermore, any low-energy state must have nonnegligible quantum circuit complexity, suggesting that low-energy states are classically nontrivial and phase estimation is the optimal method for preparing such states (up to polynomial factors). These statements hold for two models of random Hamiltonians: (i) a sum of random signed Pauli strings and (ii) a random signed d-sparse Hamiltonian. The main technical argument is based on some new results in nonasymptotic random matrix theory. In particular, a refined concentration bound for the spectral density is required to obtain complexity guarantees for these random Hamiltonians.

no seminar. spring break
2023-03-29 23:10 - 2023-03-30 00:00 | https://researchseminars.org/talk/BerekelyApplied/83/
by no seminar. spring break as part of Berkeley applied mathematics seminar

Abstract: TBA

Yangwen Zhang (Carnegie Mellon University)
2023-04-05 23:10 - 2023-04-06 00:00 | https://researchseminars.org/talk/BerekelyApplied/84/
by Yangwen Zhang (Carnegie Mellon University) as part of Berkeley applied mathematics seminar

Abstract: TBA

Sandeep Sharma (University of Colorado Boulder)
2023-04-12 23:10 - 2023-04-13 00:00 | https://researchseminars.org/talk/BerekelyApplied/85/
by Sandeep Sharma (University of Colorado Boulder) as part of Berkeley applied mathematics seminar

Abstract: TBA

Uros Seljak (UC Berkeley)
2023-04-19 23:10 - 2023-04-20 00:00 | https://researchseminars.org/talk/BerekelyApplied/86/
by Uros Seljak (UC Berkeley) as part of Berkeley applied mathematics seminar

Abstract: TBA

Aleksandar Donev (Courant Institute)
2023-03-17 23:10 - 2023-03-18 00:00 | https://researchseminars.org/talk/BerekelyApplied/87/
Title: Hydrodynamics and rheology of fluctuating, semiflexible, inextensible, and slender filaments in Stokes flow (https://researchseminars.org/talk/BerekelyApplied/87/)
by Aleksandar Donev (Courant Institute) as part of Berkeley applied mathematics seminar

Abstract: TBA

Zhiyan Ding (UC Berkeley)
2023-02-09 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/88/
Title: Mean-field analysis of interacting particle system and overparameterization of neural network (https://researchseminars.org/talk/BerekelyApplied/88/)
by Zhiyan Ding (UC Berkeley) as part of Berkeley applied mathematics seminar

Abstract
The interacting particle system is a dynamic system that contains a lot of interacting particles. Because of the interaction between different particles, the direct analysis and simulation of the system are very difficult. The mean-field analysis is a framework for analyzing these large interacting particle systems. In this framework, instead of directly studying the coupled system, one approximates the system by a partial differential equation whose solution characterizes the distribution of the particles. This strategy largely simplifies the problem and provides a more efficient way to study the evolution of the particle system. In this talk, I will mainly focus on the derivation of the mean-field analysis and its application in analyzing overparameterization of neural network.

Uros Seljak (UC Berkeley)
2023-02-16 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/89/

Julian Urban (MIT)
2023-03-09 00:10 - 01:00 | https://researchseminars.org/talk/BerekelyApplied/90/
by Julian Urban (MIT) as part of Berkeley applied mathematics seminar

Abstract: TBA

Yuehaw Khoo (University of Chicago)
2023-04-26 23:10 - 2023-04-27 00:00 | https://researchseminars.org/talk/BerekelyApplied/91/
by Yuehaw Khoo (University of Chicago) as part of Berkeley applied mathematics seminar

Abstract: TBA

Eun-Jae Park (Yonsei University)
2023-06-15 17:30 - 18:30 | https://researchseminars.org/talk/BerekelyApplied/92/
Title: Polygonal Staggered Galerkin Methods (https://researchseminars.org/talk/BerekelyApplied/92/)
by Eun-Jae Park (Yonsei University) as part of Berkeley applied mathematics seminar

Abstract
In this talk, we first present the staggered discontinuous Galerkin method on general meshes for the Poisson equation. Adaptive mesh refinement is an attractive tool for general meshes due to their flexibility and simplicity in handling hanging nodes. We derive a simple residual-type error estimator. Numerical results indicate that optimal convergence can be achieved for both the potential and vector variables, and the singularity can be well-captured by the proposed error estimator. Then, some applications to interface problems are considered such as coupling of Darcy-Forchheimer and Stokes equations, and a single-phase flow in porous media with a fracture. In the case of s fractured porous media, the bulk variables are solved using staggered DG method and an interface variable is solved using the continuous Galerkin method. We derive optimal convergence for both pressure and velocity fields. Numerical experiments suggest that our method is more accurate when polygonal meshes are used among various mesh configurations; moreover, our method is robust to mesh distortion. These observations allow us to consider unfitted methods without any special treatment. With background meshes generated independent of fracture, numerical solutions converge in optimal order.

This is joint work with Eric Chung, Dohyun Kim, and Lina Zhao.
