BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Alex Townsend (Cornell University) DTSTART:20200427T200000Z DTEND:20200427T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/1 DESCRIPTION:Title: The ultraspherical spectral method\nby Alex Townsend (Corne ll University) as part of CRM Applied Math Seminar\n\nLecture held in Webi nar.\n\nAbstract\nPseudospectral methods\, based on high degree polynomial s\, have spectral accuracy when solving differential equations but typical ly lead to dense and ill-conditioned matrices. The ultraspherical spectral method is a numerical technique to solve ordinary and partial differentia l equations\, leading to almost banded well-conditioned linear systems whi le maintaining spectral accuracy. In this talk\, we introduce the ultrasph erical spectral method and develop it into a spectral element method using a modification to a hierarchical Poincaré-Steklov domain decomposition m ethod.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Thomas Bury (McGill University) DTSTART:20200511T200000Z DTEND:20200511T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/2 DESCRIPTION:Title: Detecting and distinguishing bifurcations from noisy time serie s data\nby Thomas Bury (McGill University) as part of CRM Applied Math Seminar\n\nLecture held in Webinar.\n\nAbstract\nNumerous systems in the natural sciences have the capacity to undergo an abrupt change in their dy namical behaviour as a threshold is crossed. Prominent examples include th e collapse of fisheries\, algal blooms and paleoclimatic transitions. Math ematical models reveal such transitions as the result of crossing a bifurc ation and help to elucidate the underlying mechanisms. However\, the numbe r of unknowns is often large\, making it difficult to infer where the bifu rcation occurs in the real system.\nIn this talk\, we will look at methods for detecting bifurcations using data-driven approaches. These methods ex ploit generic dynamical phenomena that occur prior to bifurcations\, such as critical slowing down\, in order to infer their approach. We will show how the power spectrum of noisy time series data provides information on t he type of bifurcation and validate this approach with empirical predator- prey experiment that undergoes a Hopf bifurcation. Finally\, we will explo re deep learning methods for detection of bifurcations and make comparison to the more traditional statistical methods in their ability to detect bi furcations.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Bamdad Hosseini (California Institute of Technology) DTSTART:20200622T200000Z DTEND:20200622T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/3 DESCRIPTION:Title: Data-driven supervised learning: Neural networks and uncertaint y quantification\nby Bamdad Hosseini (California Institute of Technolo gy) as part of CRM Applied Math Seminar\n\n\nAbstract\nIn this talk I will discuss some ideas at the intersection of machine learning and uncertaint y quantification with a particular focus on data-driven methods that do no t require explicit knowledge of processes that generate the data. In the first half of the talk I will discuss supervised learning on Banach spaces for emulation of PDE based models and outline a method that combines prin cipal component analysis with neural network regression for mesh-independe nt approximation of PDE solutions. In the second half I will take a diffe rent approach to supervised learning viewing it as a conditional sampling problem. I will then introduce a measure transport framework based on gen erative adversarial networks (GANs) for data-driven conditional sampling. ​\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Theodore Kolokolnikov (Dalhousie University) DTSTART:20200629T200000Z DTEND:20200629T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/4 DESCRIPTION:Title: Simple agent-based models and their continuum limit\nby The odore Kolokolnikov (Dalhousie University) as part of CRM Applied Math Semi nar\n\n\nAbstract\nWe discuss several very different ABM models and their continuum limits.\n\nFirst\, consider the following agent-based model of c oronavirus spread: people move randomly and infection occurs with some non zero probability when an infected individual comes within a certain ``infe ction radius'' of a susceptible individual. The question is how the infect ion radius affects the reproduction number. At low infection rates\, this model leads to the classical S-I-R ODE model as its continuum limit. Howev er higher infection rates lead to a saturation effect\, which we compute e xplicitly using basic probability theory. Its continuum limit It leads to an S-I-R type model with a specific saturation term. We also show that thi s modified model gives a much better fit to the real-world data than the c lassical SIR model.\n\nNext\, we will look at a very simple stochastic mod el of bacterial aggregation which leads to a novel fourth-order nonlinear PDE in its continuum limit. This PDE admits soliton-type solutions corresp onding to bacterial aggregation patterns\, which we explicitly construct. \n\nIf time allows\, we will consider a spatial model of wealth exchange w hich leads to novel integro-differential equations.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Stephen Becker (University of Colorado Boulder\, USA) DTSTART:20200921T183000Z DTEND:20200921T193000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/5 DESCRIPTION:Title: Algorithmic stability for generalization guarantees in machine learning\nby Stephen Becker (University of Colorado Boulder\, USA) as part of CRM Applied Math Seminar\n\n\nAbstract\nInspired by the practical success of deep learning\, the broader math community has been energized r ecently to find theoretical justification for these methods. There is a la rge amount of theory from the computer science community\, dating to the 1 980s and earlier\, but usually the quantitative guarantees are too loose t o be helpful in practice\, and it is rare that theory can predict somethin g useful (such as what iteration to perform early-stopping in order to pre vent over-fitting). \nMany of these theories are less well-known inside ap plied math\, so we briefly review essential results before focusing on the notion of algorithmic stability\, popularized in the early 2000s\, which is an alternative to the more mainstream VC dimension approach\, and is on e avenue that might give sharper theoretical guarantees. Algorithmic stabi lity is appealing to applied mathematicians\, and in particular analysts\, since a lot of the technical work is similar to analysis used for converg ence proofs. \nWe give an overview of the fundamental results of algorithm ic stability\, focusing on the stochastic gradient descent (SGD) method in the context of a nonconvex loss function\, and give the latest state-of-t he-art bounds\, including some of our own work (joint with L. Madden and E . Dall'Anese) which is one of the first results that suggests when to do e arly-stopping.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Yuji Nakatsukasa (Oxford University\, UK) DTSTART:20200928T200000Z DTEND:20200928T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/6 DESCRIPTION:Title: Fast and stable randomized low-rank matrix approximation\nb y Yuji Nakatsukasa (Oxford University\, UK) as part of CRM Applied Math Se minar\n\n\nAbstract\nRandomized SVD has become an extremely successful app roach for efficiently computing a low-rank approximation of matrices. In p articular the paper by Halko\, Martinsson\, and Tropp (SIREV 2011) contain s extensive analysis\, and has made it a very popular method. The typical complexity for a rank-r approximation of mxn matrices is O(mnlog n+(m+n)r^ 2) for dense matrices. The classical Nystrom method is much faster\, but o nly applicable to positive semidefinite matrices. This work studies a gene ralization of Nystrom's method applicable to general matrices\, and shows that (i) it has near-optimal approximation quality comparable to competing methods\, (ii) the computational cost is the near-optimal O(mnlog n+r^3) for dense matrices\, with small hidden constants\, and (iii) crucially\, i t can be implemented in a numerically stable fashion despite the presence of an ill-conditioned pseudoinverse. Numerical experiments illustrate that generalized Nystrom can significantly outperform state-of-the-art methods \, especially when r>>1\, achieving up to a 10-fold speedup. The method is also well suited to updating and downdating the matrix.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/6/ END:VEVENT BEGIN:VEVENT SUMMARY:David Russell Luke (Universität Göttingen\, Germany) DTSTART:20201005T200000Z DTEND:20201005T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/7 DESCRIPTION:Title: Optimization on Spheres : Models and Proximal Algorithms with C omputational Performance Comparisons\nby David Russell Luke (Universit ät Göttingen\, Germany) as part of CRM Applied Math Seminar\n\n\nAbstrac t\nWe present a unified treatment of the abstract problem of finding the b est approximation between a cone and spheres in the image of affine transf ormations. Prominent instances of this problem are phase retrieval and sou rce localization. The common geometry binding these problems permits a gen eric application of algorithmic ideas and abstract convergence results for nonconvex optimization. We organize variational models for this problem i nto three different classes and derive the main algorithmic approaches wit hin these classes (13 in all). We identify the central ideas underlying th ese methods and provide thorough numerical benchmarks comparing their perf ormance on synthetic and laboratory data. The software and data of our exp eriments are all publicly accessible. We also introduce one new algorithm\ , a cyclic relaxed Douglas-Rachford algorithm\, which outperforms all othe r algorithms by every measure: speed\, stability and accuracy. The analysi s of this algorithm remains open.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Sheehan Olver (Imperial College London\, UK) DTSTART:20201019T200000Z DTEND:20201019T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/8 DESCRIPTION:Title: Sparse Spectral Methods for Power-Law Interactions\nby Shee han Olver (Imperial College London\, UK) as part of CRM Applied Math Semin ar\n\n\nAbstract\nAttractive-repulsive power law equilbriums are an import ant tool in modelling phenomena in collective behaviour: picture a flock o f birds which simultaneously group together\, but not too closely (i.e.\, they practice social distancing)\, until an equilibrium distribution is re ached. In this talk we show that orthogonal polynomials have sparse recurr ence relationships for power law (Riesz) kernels. This leads to highly str uctured and efficiently solvable linear systems for the attractive-repulsi ve case with two such kernels of opposite sign\, giving an effective numer ical method for computing such equilibrium distributions. This links to an d builds on related work in logarithmic potential theory\, singular integr al equations\, and fractional differential equations.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Johannes Royset (Naval Postgraduate School\, USA) DTSTART:20201026T200000Z DTEND:20201026T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/9 DESCRIPTION:Title: Variational Perspectives on Mathematical Optimization\nby J ohannes Royset (Naval Postgraduate School\, USA) as part of CRM Applied Ma th Seminar\n\n\nAbstract\nThe mathematical tools for building optimization models and algorithms grow out of linear algebra\, differential calculus and real analysis. However\, the needs of applications have led to a new a rea of mathematics that can handle systems of inequalities and functions t hat are neither smooth nor well-defined in a traditional sense. Variationa l analysis is the broad term for this area of mathematics. In this present ation\, we show its crucial role in the development of optimization models and algorithms in finite dimensions. First\, we examine variational geome try and definitions of normal and tangent vectors that extend the classica l notions for smooth manifolds. This in turn leads to subdifferentiability \, a wide range of calculus rules and optimality conditions for arbitrary functions. Second\, we develop an approximation theory for optimization pr oblems that leads to consistent approximations\, error bounds and rates of convergence even in the nonconvex and nonsmooth setting.\n\nDr. Johannes O. Royset is Professor of Operations Research at the Naval Postgraduate Sc hool. Dr. Royset's research focuses on formulating and solving stochastic and deterministic optimization problems arising in data analytics\, sensor management\, and reliability engineering. He was awarded a National Resea rch Council postdoctoral fellowship in 2003\, a Young Investigator Award f rom the Air Force Office of Scientific Research in 2007\, and the Barchi P rize as well as the MOR Journal Award from the Military Operations Researc h Society in 2009. He received the Carl E. and Jessie W. Menneken Faculty Award for Excellence in Scientific Research in 2010 and the Goodeve Medal from the Operational Research Society in 2019. Dr. Royset was a plenary sp eaker at the International Conference on Stochastic Programming in 2016 an d at the SIAM Conference on Uncertainty Quantification in 2018. He has a D octor of Philosophy degree from the University of California at Berkeley ( 2002). Dr. Royset has been an associate or guest editor of Operations Rese arch\, Mathematical Programming\, Journal of Optimization Theory and Appli cations\, Journal of Convex Analysis\, Set-Valued and Variational Analysis \, Naval Research Logistics\, and Computational Optimization and Applicati ons.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Heather Harrington (Oxford University\, UK) DTSTART:20201116T210000Z DTEND:20201116T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/10 DESCRIPTION:Title: Algebraic Systems Biology\nby Heather Harrington (Oxford U niversity\, UK) as part of CRM Applied Math Seminar\n\n\nAbstract\nSignall ing pathways in molecular biology can be modelled by polynomial dynamical systems. I will present models describing two biological systems involved in development and cancer. I will overview approaches to analyse these mod els with data using computational algebraic geometry\, differential algebr a and statistics. Finally\, I will present how topological data analysis c an provide additional information to distinguish wild-type and mutant mole cules in one pathway. These case studies showcase how computational geomet ry\, topology and dynamics can provide new insights in the biological syst ems\, specifically how changes at the molecular scale (e.g. molecular muta tions) result in kinetic differences that are observed as phenotypic chang es (e.g.mutations in fruit fly wings).\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Thomas Surowiec (Philipps-Universität Marburg\, Germany) DTSTART:20201123T210000Z DTEND:20201123T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/11 DESCRIPTION:Title: A Primal-Dual Algorithm for Risk Minimization in PDE-Constrain ed Optimization\nby Thomas Surowiec (Philipps-Universität Marburg\, G ermany) as part of CRM Applied Math Seminar\n\n\nAbstract\nWe present an a lgorithm for the solution of risk-averse optimization problems. The settin g is sufficiently general so as to encompass both finite-dimensional and P DE-constrained stochastic optimization problems. Due to a lack of smoothne ss of many popular risk measures and non-convexity of the objective functi ons\, both the numerical approximation and numerical solution is a major c omputational challenge. The proposed algorithm addresses these issues in p art by making use of the favorable dual properties of coherent risk measur es. The algorithm itself is motivated by the classical method of multiplie rs and exploits recent results on epigraphical regularization of risk meas ures. Consequently\, the algorithm requires the solution of a sequence of smooth problems using derivative-based methods. We prove convergence of th e algorithm in the fully continuous setting and conclude with several nume rical examples. The algorithm is seen to outperform a popular bundle-trust method and a direct smoothing-plus-continuation approach.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Damek Davis (Cornell University\, USA) DTSTART:20201130T210000Z DTEND:20201130T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/12 DESCRIPTION:Title: Nonconvex Optimization for Estimation and Learning: Dynamics\, Conditioning\, and Nonsmoothness\nby Damek Davis (Cornell University\ , USA) as part of CRM Applied Math Seminar\n\n\nAbstract\nNonconvex optimi zation algorithms play a major role in solving statistical estimation and learning problems. Indeed\, simple nonconvex heuristics\, such as the stoc hastic gradient method\, often provide satisfactory solutions in practice\ , despite such problems being NP hard in the worst case. Key examples incl ude deep neural network training and signal estimation from nonlinear meas urements. While practical success stories are common\, strong theoretical guarantees\, are rarer. The purpose of this talk is to overview a few (hig hly non exhaustive!) settings where rigorous performance guarantees can be established for nonconvex optimization\, focusing on the interplay of alg orithm dynamics\, problem conditioning\, and nonsmoothness.\n\nBio: Damek Davis received his Ph.D. in mathematics from the University of California\ , Los Angeles in 2015. In July 2016 he joined Cornell University's School of Operations Research and Information Engineering as an Assistant Profess or. Damek is broadly interested in the mathematics of data science\, parti cularly the interplay of optimization\, signal processing\, statistics\, a nd machine learning. He is the recipient of several awards\, including the INFORMS Optimization Society Young Researchers Prize in (2019) and a Sloa n Research Fellowship in Mathematics (2020).\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Tim Hoheisel (McGill University) DTSTART:20210111T210000Z DTEND:20210111T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/13 DESCRIPTION:Title: Halting Time is Predictable for Large Models: A Universality P roperty and Average-case Analysis\nby Tim Hoheisel (McGill University) as part of CRM Applied Math Seminar\n\n\nAbstract\nAverage-case analysis computes the complexity of an algorithm averaged over all possible inputs. Compared to worst-case analysis\, it is more representative of the typica l behavior of an algorithm\,but remains largely unexplored in optimization . One difficulty is that the analysis can depend on the probability distri bution of the inputs to the model. However\, we show that this is not the case for a class of large-scale problems trained with first-order methods including random least squares and one-hidden layer neural networks with r andom weights. In fact\, the halting time exhibits a universality propert y: it is independent of the probability distribution. With this barrier fo r average-case analysis removed\, we provide the first explicit average-ca se convergence rates showing a tighter complexity not captured by traditio nal worst-case analysis. Finally\, numerical simulations suggest this univ ersality property holds for a more general class of algorithms and problem s.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael P. Friedlander (University of British Columbia) DTSTART:20210118T210000Z DTEND:20210118T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/14 DESCRIPTION:Title: Polar deconvolution of mixed signals\nby Michael P. Friedl ander (University of British Columbia) as part of CRM Applied Math Seminar \n\n\nAbstract\nThe signal demixing problem seeks to separate the superpos ition of multiple signals into its constituent components. We model the s uperposition process as the polar convolution of atomic sets\, which allow s us to use the duality of convex cones to develop an efficient two-stage algorithm with sublinear iteration complexity and linear storage. If the signal measurements are random\, the polar deconvolution approach stably r ecovers low-complexity and mutually-incoherent signals with high probabili ty and with optimal sample complexity. Numerical experiments on both real and synthetic data confirm the theory and efficiency of the proposed appr oach. Joint work with Zhenan Fan\, Halyun Jeong\, and Babhru Joshi at the University of British Columbia.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Nathan Kutz (University of Washington) DTSTART:20210125T210000Z DTEND:20210125T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/15 DESCRIPTION:Title: Targeted use of deep learning for physics and engineering\ nby Nathan Kutz (University of Washington) as part of CRM Applied Math Sem inar\n\n\nAbstract\nMachine learning and artificial intelligence algorithm s are now being used to automate the discovery of governing physical equat ions and coordinate systems from measurement data alone. However\, positi ng a universal physical law from data is challenging: (i) An appropriate c oordinate system must also be advocated and (ii) simultaneously proposing an accompanying discrepancy model to account for the inevitable mismatch b etween theory and measurements must be considered. Using a combination of deep learning and sparse regression\, specifically the sparse identificat ion of nonlinear dynamics (SINDy) algorithm\, we show how a robust mathema tical infrastructure can be formulated for simultaneously learning physics models and their coordinate systems. This can be done with limited data and sensors. We demonstrate the methods on a diverse number of examples\, showing how data can maximally be exploited for scientific and engineerin g applications.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Zhaojun Bai (UC Davis) DTSTART:20210201T210000Z DTEND:20210201T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/16 DESCRIPTION:Title: Rayleigh quotient optimizations and eigenvalue problems\nb y Zhaojun Bai (UC Davis) as part of CRM Applied Math Seminar\n\n\nAbstract \nMany computational science and data analysis techniques lead to optimizi ng Rayleigh quotient (RQ) and RQ-type objective functions\, such as comput ing excitation states (energies) of electronic structures\, robust classif ication to handle uncertainty and constrained data clustering to incorpora te domain knowledge. We will discuss emerging RQ optimization problems\, variational principles\, and reformulations to algebraic linear and nonlin ear eigenvalue problems. We will show how to exploit underlying propertie s of these eigenvalue problems for designing fast solvers\, and illustrate the efficacy of these solvers in applications.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Marwa El Halabi (MILA) DTSTART:20210208T210000Z DTEND:20210208T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/17 DESCRIPTION:Title: Optimal approximation for unconstrained non-submodular minimiz ation\nby Marwa El Halabi (MILA) as part of CRM Applied Math Seminar\n \n\nAbstract\nSubmodular function minimization is well studied\, and exist ing algorithms solve it exactly or up to arbitrary accuracy. However\, in many applications\, such as structured sparse learning or batch Bayesian optimization\, the objective function is not exactly submodular\, but clos e. In this case\, no theoretical guarantees exist. Indeed\, submodular m inimization algorithms rely on intricate connections between submodularity and convexity. We show how these relations can be extended to obtain app roximation guarantees for minimizing non-submodular functions\, characteri zed by how close the function is to submodular. We also extend this resul t to noisy function evaluations. Our approximation results are the first for minimizing non-submodular functions\, and are optimal\, as established by our matching lower bound.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Patrick Combettes (NC State) DTSTART:20210215T210000Z DTEND:20210215T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/18 DESCRIPTION:Title: Perspective Functions and Applications\nby Patrick Combett es (NC State) as part of CRM Applied Math Seminar\n\n\nAbstract\nIn this t alk I will discuss mathematical and computational issues pertaining to per spective functions\, a powerful concept that permits to extend a convex fu nction to a jointly convex one in terms of an additional scale variable. A pplications in inverse problems and statistics will be presented.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Heinz Bauschke (UBC) DTSTART:20210222T210000Z DTEND:20210222T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/19 DESCRIPTION:Title: Compositions of projection mappings: fixed point sets and diff erence vectors\nby Heinz Bauschke (UBC) as part of CRM Applied Math Se minar\n\n\nAbstract\nProjection operators and associated projection algori thms are fundamental building blocks in fixed point theory and optimizatio n. In this talk\, I will survey recent results on the displacement mappin g of the right-shift operator and sketch a new application deepening our u nderstanding of the geometry of the fixed point set of the composition of projection operators in Hilbert space. Based on joint works with Salha Al wadani\, Julian Revalski\, and Shawn Wang.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Paul E. Hand (Northeastern University) DTSTART:20210308T210000Z DTEND:20210308T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/20 DESCRIPTION:Title: Signal Recovery with Generative Priors\nby Paul E. Hand (N ortheastern University) as part of CRM Applied Math Seminar\n\n\nAbstract\ nRecovering images from very few measurements is an important task in imag ing problems. Doing so requires assuming a model of what makes some image s natural. Such a model is called an image prior. Classical priors such as sparsity have led to the speedup of Magnetic Resonance Imaging in certa in cases. With the recent developments in machine learning\, neural netwo rks have been shown to provide efficient and effective priors for inverse problems arising in imaging. In this talk\, we will discuss the use of ne ural network generative models for inverse problems in imaging. We will p resent a rigorous recovery guarantee at optimal sample complexity for comp ressed sensing and other inverse problems under a suitable random model. We will see that generative models enable an efficient algorithm for phase retrieval from generic measurements with optimal sample complexity. In c ontrast\, no efficient algorithm is known for this problem in the case of sparsity priors. We will discuss strengths\, weaknesses\, and future oppo rtunities of neural networks and generative models as image priors. These works are in collaboration with Vladislav Voroninski\, Reinhard Heckel\, Ali Ahmed\, Wen Huang\, Oscar Leong\, Jorio Cocola\, Muhammad Asim\, and M ax Daniels.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Clarice Poon (University of Bath) DTSTART:20210315T200000Z DTEND:20210315T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/21 DESCRIPTION:Title: Off-the-grid sparse estimation\nby Clarice Poon (Universit y of Bath) as part of CRM Applied Math Seminar\n\n\nAbstract\nThe behaviou r of sparse regularization using the Lasso method is well understood when dealing with discretized linear models. However\, the behaviour of Lasso is poor when dealing with models with very large parameter spaces and in recent years\, there has been much interest in the use of "off-the-grid" approaches\, using a continuous parameter space in conjunction with convex optimization problem over measures. In my talk\, I will present some re cent results which explain the behaviour of this method in arbitrary dimen sions. Some highlights include the use of the Fisher metric to study the performance of Blasso over general domains and the application of this fo r quantitative MRI.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Olga Mula (Paris Dauphine) DTSTART:20210322T200000Z DTEND:20210322T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/22 DESCRIPTION:Title: Depth-Adaptive Neural Networks from the Optimal Control viewpo int\nby Olga Mula (Paris Dauphine) as part of CRM Applied Math Seminar \n\n\nAbstract\nIn recent years\, deep learning has been connected with op timal control as a way to define a notion of a continuous underlying learn ing problem. In this view\, neural networks can be interpreted as a disc retization of a parametric Ordinary Differential Equation which\, in the l imit\, defines a continuous-depth neural network. The learning task th en consists in finding the best ODE parameters for the problem under consi deration\, and their number increases with the accuracy of the time discr etization. Although important steps have been taken to realize the advant ages of such continuous formulations\, most current learning techniques f ix a discretization (i.e.~the number of layers is fixed). In this work\, we propose an iterative adaptive algorithm where we progressively refine the time discretization (i.e.~we increase the number of layers). Provided that certain tolerances are met across the iterations\, we prove that th e strategy converges to the underlying continuous problem. One salient ad vantage of such a shallow-to-deep approach is that it helps to benefit in practice from the higher approximation properties of deep networks by mit igating over-parametrization issues. The performance of the approach is i llustrated in several numerical examples.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Sasha Aravkin (University of Washington) DTSTART:20210412T200000Z DTEND:20210412T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/23 DESCRIPTION:Title: A tale of two models for Covid-19 scenarios\nby Sasha Arav kin (University of Washington) as part of CRM Applied Math Seminar\n\n\nAb stract\nCovid-19 Pandemic is a defining global health event in the 21st ce ntury. Forecasting the evolution of the pandemic is a key problem for any one trying to plan ahead. Since March 2020\, IHME has been generating Cov id-19 scenarios\, first for US states and then for all Admin-1 locations a round the world. These scenarios have been intensively used\; results are uploaded weekly to an interactive website: https://covid19.healthdata.org / \nIn this talk\, we describe two core mathematical models underlying the IHME scenarios. The first model\, dubbed CurveFit\, used strong assumpti ons to get useful predictions using extremely limited data\, and was used during March and April of 2020. The second model\, a data-driven SEIIR mo del\, was put in play in June 2020\, and provides a flexible way to incorp orate relationships with key drivers such as mobility\, mask use\, and pne umonia seasonality. We describe the mathematics underlying both models\, and discuss the interplay between stability\, scalability\, and complexity in mathematical modeling.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/23/ END:VEVENT BEGIN:VEVENT SUMMARY:test DTSTART:20210614T200000Z DTEND:20210614T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/24 DESCRIPTION:Title: test\nby test as part of CRM Applied Math Seminar\n\n\nAbs tract\nWe show that intertwining operators for the discrete Fourier transf orm form a cubic algebra $C_q$ with $q$ a root of unity. This algebra is i ntimately related to the two other well-known\nrealizations of the cubic a lgebra: the Askey-Wilson algebra and the Askey-Wilson-Heun algebra.\nThis is joint work with Mesuma Atakishiyeva (Universidad Autónoma del Estado d e Morelos\,\nCentro de Investigación en Ciencias\, Cuernavaca\, 62250\, M orelos\, México) and Alexei Zhedanov (School of Mathematics\, Renmin Univ ersity of China\, Beijing 100872\, China) curious theorem on S-integrables D$\\Delta$Es and its consequences\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Kuhn (EPFL) DTSTART:20210913T183000Z DTEND:20210913T193000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/26 DESCRIPTION:Title: Mathematical Foundations of Robust and Distributionally Robust Optimization\nby Daniel Kuhn (EPFL) as part of CRM Applied Math Semin ar\n\n\nAbstract\nRobust and distributionally robust optimization are mode ling paradigms for decision-making under uncertainty where the uncertain p arameters are only known to reside in an uncertainty set or are governed b y any probability distribution from within an ambiguity set\, respectively \, and a decision is sought that minimizes a cost function under the most adverse outcome of the uncertainty. In this paper\, we develop a rigorous and general theory of robust and distributionally robust nonlinear optimi zation using the language of convex analysis. Our framework is based on a generalized `primal-worst-equals-dual-best' principle that establishes st rong duality between a semi-infinite primal worst and a non-convex dual be st formulation\, both of which admit finite convex reformulations. This p rinciple offers an alternative formulation for robust optimization problem s that may be computationally advantageous\, and it obviates the need to m obilize the machinery of abstract semi-infinite duality theory to prove st rong duality in distributionally robust optimization. We illustrate the m odeling power of our approach through convex reformulations for distributi onally robust optimization problems whose ambiguity sets are defined throu gh general optimal transport distances\, which generalize earlier results for Wasserstein ambiguity sets.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Diane Guignard (University of Ottawa) DTSTART:20210920T200000Z DTEND:20210920T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/27 DESCRIPTION:Title: Nonlinear reduced models for parametric PDEs\nby Diane Gui gnard (University of Ottawa) as part of CRM Applied Math Seminar\n\n\nAbst ract\nWe consider model reduction methods for parametric partial different ial equations. The usual approach to model reduction is to construct a lo w dimensional linear space which accurately approximates the parameter-to- solution map\, and use it to build an efficient forward solver. However\, the construction of a suitable linear space is not always feasible numeri cally. It is well-known that nonlinear methods may provide improved effic iency. In a so-called library approximation\, the idea is to replace the linear space by a collection of linear (or affine) spaces of smaller dimen sion. In this talk\, we first review standard linear methods for model re duction. Then\, we present a strategy which can be used to generate a non linear reduced model\, namely a library based on piecewise (Taylor) polyno mials. We provide an analysis of the method\, in particular the derivatio n of an upper bound on the size of the library\, and illustrate its perfor mance through several numerical experiments.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Gitta Kutyniok (LMU Munich) DTSTART:20210927T183000Z DTEND:20210927T193000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/28 DESCRIPTION:Title: The Modern Mathematics of Deep Learning\nby Gitta Kutyniok (LMU Munich) as part of CRM Applied Math Seminar\n\n\nAbstract\nDespite t he outstanding success of deep neural networks in real-world applications\ , ranging from science to public life\, most of the related research is em pirically driven and a comprehensive mathematical foundation is still miss ing. At the same time\, these methods have already shown their impressive potential in mathematical research areas such as imaging sciences\, inver se problems\, or numerical analysis of partial differential equations\, so metimes by far outperforming classical mathematical approaches for particu lar problem classes. The goal of this lecture is to first provide an intr oduction into this new vibrant research area. We will then survey recent advances in two directions\, namely the development of a mathematical foun dation of deep learning and the introduction of novel deep learning-based approaches to mathematical problem settings.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Jason Bramburger (George Mason University) DTSTART:20211004T200000Z DTEND:20211004T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/29 DESCRIPTION:Title: Deep learning of conjugate mappings\nby Jason Bramburger ( George Mason University) as part of CRM Applied Math Seminar\n\n\nAbstract \nDespite many of the most common chaotic dynamical systems being continuo us in time\, it is through discrete time mappings that much of the underst anding of chaos is formed. Henri Poincaré first made this connection by t racking consecutive iterations of the continuous flow with a lower-dimensi onal\, transverse subspace. The mapping that iterates the dynamics through consecutive intersections of the flow with the subspace is now referred t o as a Poincaré map\, and it is the primary method available for interpre ting and classifying chaotic dynamics. Unfortunately\, in all but the simp lest systems\, an explicit form for such a mapping remains outstanding. In this talk I present a method of discovering explicit Poincaré mappings u sing deep learning to construct an invertible coordinate transformation in to a conjugate representation where the dynamics are governed by a relativ ely simple chaotic mapping. The invertible change of variable is based on an autoencoder\, which allows for dimensionality reduction\, and has the a dvantage of classifying chaotic systems using the equivalence relation of topological conjugacies. We illustrate with low-dimensional systems such a s the Rössler and Lorenz systems\, while also demonstrating the utility o f the method on the infinite-dimensional Kuramoto--Sivashinsky equation.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Terry Rockafellar (University of Washington) DTSTART:20211018T200000Z DTEND:20211018T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/30 DESCRIPTION:Title: Hidden convexity in nonconvex optimization\nby Terry Rocka fellar (University of Washington) as part of CRM Applied Math Seminar\n\n\ nAbstract\nIn nonconvex optimization\, not only the objective but even the feasible set may lack convexity. It may seem therefore that the concepts and methodology of convex optimization can no longer have a fundamental r ole\, but this is actually wrong. Standard sufficient conditions for loca l optimality in nonlinear programming and its extensions turn out to corre spond to characterizing optimality in terms of a local convex-concave-type saddle point of an augmented Lagrangian function. Algorithms that effect ively in both primal and dual elements are thereby revealed as working jus t as they would in the convex case.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Aaron Berk (University of British Columbia) DTSTART:20211025T200000Z DTEND:20211025T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/31 DESCRIPTION:Title: On LASSO parameter sensitivity\nby Aaron Berk (University of British Columbia) as part of CRM Applied Math Seminar\n\n\nAbstract\nCo mpressed sensing theory explains why LASSO programs recover structured hig h-dimensional signals with minimax order-optimal error. Yet\, the optimal choice of the program's governing parameter is often unknown in practice. It is still unclear how variation of the governing parameter impacts rec overy error in compressed sensing\, which is otherwise provably stable and robust. We provide an overview of parameter sensitivity in LASSO program s in the setting of proximal denoising\; and of compressed sensing with su bgaussian measurement matrices and gaussian noise. We demonstrate how two popular ell-1 minimization programs exhibit sensitivity with respect to t heir parameter choice and illustrate the theory with numerical simulations . For example\, a 1% error in the estimate of a parameter can cause the e rror to increase by a factor of 10^9\, while choosing a different LASSO pr ogram avoids such sensitivity issues. We hope that revealing parameter se nsitivity regimes of LASSO programs helps to inform a practitioner's choic e.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Frank E. Curtis (Lehigh University) DTSTART:20211101T200000Z DTEND:20211101T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/32 DESCRIPTION:Title: Algorithms for Deterministically Constrained Stochastic Opt imization\nby Frank E. Curtis (Lehigh University) as part of CRM Appli ed Math Seminar\n\n\nAbstract\nI will present the recent work by my resear ch group on the design\, analysis\, and implementation of algorithms for s olving nonlinear optimization problems that involve a stochastic objecti ve function and deterministic constraints.  The talk will focus on our s equential quadratic optimization (commonly known as SQP) methods for cases when the constraints are defined by nonlinear systems of equations\, wh ich arise in various applications including optimal control\, PDE-constrai ned optimization\, and network optimization problems.  One might also co nsider our techniques for training machine learning (e.g.\, deep learning) models with constraints.  I will also discuss the various extensions th at my group is exploring along with other related open questions.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Rainer Groh (Bristol Composites Institute (ACCIS)) DTSTART:20211108T210000Z DTEND:20211108T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/33 DESCRIPTION:Title: Experimental continuation of nonlinear load-bearing structures \nby Rainer Groh (Bristol Composites Institute (ACCIS)) as part of CRM Applied Math Seminar\n\n\nAbstract\nThe drive for lightweighting in struc tural engineering leads to ever thinner structures that deform in nonlinea r ways and that are prone to sudden instabilities. Simultaneously\, a ren ewed interest in structural instability revolves around purposefully embed ding instabilities in structures to add functionality beyond structural lo ad-carrying capability (e.g. dynamic shape adaptivity). To date\, the de sign of nonlinear structures is guided almost entirely by computational mo delling\, in particular the use of numerical continuation tools. Advances in experimental testing of nonlinear structures\, on the other hand\, are significantly lagging behind numerical methods. While numerical continua tion principles such as path-following\, calculation of bifurcations\, bra nch-switching\, and bifurcation tracking are now well established\, nonlin ear experimental methods of structures have not advanced beyond simple dis placement and force control. This means that the nonlinear response of ev en simple nonlinear structures cannot be fully characterised\, as establis hed techniques induce dynamic snaps at limit points and subcritical bifurc ations. There is thus huge potential for devising novel and non-destructi ve ways of testing nonlinear structures by applying concepts from the fiel d of continuation to experimental mechanics. At the University of Bristol \, we have developed a testing method based on adding control points with auxiliary sensors and actuators to a structure to: (i) stabilise otherwise unstable equilibria\; (ii) control the shape of the structure to transiti on between different stable equilibria\; and (iii) compute an experimental “tangential” stiffness matrix (the Jacobian)\, which ultimate ly allows Newton's root-finding algorithm to be implemented experimentally . With this approach all the features of the numerical techniques mention ed above can (theoretically) be replicated. The testing method has been a pplied to laboratory scale test specimens such as the snap-through of a sh allow arch\, and this seminar will provide an overview of the mathematical background to experimental continuation\, its application\, and outlook t o future experiments.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Wolfgang Dahmen (University of South Carolina) DTSTART:20211115T210000Z DTEND:20211115T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/34 DESCRIPTION:Title: Some Thoughts on Physics Informed Neural Networks\nby Wolf gang Dahmen (University of South Carolina) as part of CRM Applied Math Sem inar\n\n\nAbstract\nEmploying Deep Learning concepts to "learn" physical l aws\, has been recently attracting significant attention. In particular\, so called "Physics Informed Neural Networks” (PINN) refers to a paradigm where the training of model surrogates is based on empirical risks that r equire only point-wise evaluation of residuals. This avoids the expensive computation of a sufficiently large number of training data\, typically gi ven in terms of high fidelity approximations of model states. \n\nThe core issue addressed in this talk is the prediction capability of such methods for models given in terms of parameter-dependent families of partial diff erential equations. Related specific questions concern\, for instance\, th e choice of "variationally correct" training risks\, that convey certifiab le information about the achieved accuracy in problem relevant metrics\, t he role of a priori versus a posteriori error bounds\, connections with Ge nerative Adversarial Networks\, as well as related implications on trainin g strategies and network adaptation.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Soizic Terrien (CNRS - Université du Mans) DTSTART:20211122T210000Z DTEND:20211122T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/35 DESCRIPTION:Title: Equidistant and non equidistant pulsing patterns in an excitab le microlaser with delayed feedback\nby Soizic Terrien (CNRS - Univers ité du Mans) as part of CRM Applied Math Seminar\n\n\nAbstract\nExcitabil ity is observed in many natural and artificial systems\, from spiking neur ons to cardiac cells and semiconductor lasers. It corresponds to the all- or-none pulse-shaped response of a system to an external perturbation\, de pending whether or not the perturbation amplitude exceeds the so-called ex citable threshold. When subject to delayed feedback\, an excitable system can regenerate its own excitable response when it is reinjected after a d elay time τ. As the process repeats\, this results in sustained pulsing regimes\, which can be of interest for many applications\, from data trans mission to all-optical signal processing or neuromorphic photonic networks . \n\nHere we investigate the short-term and long-term dynamics of an exc itable microlaser subject to delayed optical feedback. This is done both experimentally and numerically through a bifurcation analysis of a suitabl e model written in the form of three delay-differential equations (DDEs) w ith one fast and two slow variables. We show that almost any pulse sequen ce can be excited and regenerated by the system over short periods of time . In the long-term\, on the other hand\, the system settles down to one o f the coexisting\, slowly-attracting periodic orbits\, which correspond to different numbers of pulses in the feedback cavity. We show that\, depen ding on the internal timescales of the excitable system\, these pulses app ear to be either equidistant (i.e. with equalized pulse intervals) or non -equidistant in the feedback cavity. A bifurcation analysis demonstrates that non-equidistant pulsing patterns originate in resonance phenomena. T he mechanism for the emergence of very large locking regions in the parame ter space in investigated. \n\nJoint work with Bernd Krauskopf (Universit y of Auckland)\, Neil Broderick (University of Auckland) and Sylvain Barba y (C2N\, CNRS / Univ. Paris Saclay)\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrus Giraldo (The University of Auckland) DTSTART:20211129T210000Z DTEND:20211129T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/36 DESCRIPTION:Title: Degenerate singular cycles and chaotic switching in the two-si te open Bose--Hubbard model\nby Andrus Giraldo (The University of Auck land) as part of CRM Applied Math Seminar\n\n\nAbstract\nThe two-site open Bose--Hubbard dimer model is a celebrated fundamental quantum optical mod el that accounts for the dynamics of bosons at two lossy interacting sites . Recently\, two coupled\, driven\, and lossy photonic crystal nanocavitie s ---which are optical devices that operate with only a few hundred photon s due to their extremely small size--- have been shown to realise this mod el experimentally. Thus\, there is much interest in understanding the diff erent behaviours that such model exhibits for theoretical and practical pu rposes.\nThis talk will show the different dynamics in the semiclassical a pproximation of this quantum optical system by presenting a comprehensive bifurcation analysis. We characterised different transitions of chaotic at tractors in parameter plane by numerically computing tangency bifurcations between stable and unstable manifolds of saddle equilibria and periodic o rbits. By doing so\, we identify codimension-two degenerate singular cycle s\, and their generalisations\, as responsible for the organisations of di fferent tangency and heteroclinic bifurcations between saddle equilibria p eriodic orbits in parameter plane. Thus\, we provide a roadmap for observa ble chaotic dynamics in the semiclassical approximation of the two-site Bo se--Hubbard dimer model\, which connects novel results in bifurcation theo ry with novel applications through numerical continuation techniques.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/36/ END:VEVENT BEGIN:VEVENT SUMMARY:Haesun Park (Georgia Institute of Technology) DTSTART:20211206T210000Z DTEND:20211206T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/37 DESCRIPTION:Title: Multi-view Unsupervised and Semi-Supervised Clustering based o n Content and Connection Information\nby Haesun Park (Georgia Institut e of Technology) as part of CRM Applied Math Seminar\n\n\nAbstract\nConstr ained Low Rank Approximation (CLRA) is a powerful framework for a variety of important tasks in large scale data analytics such as topic discovery i n text data and community detection in social network data. In this talk\, a hybrid method called Joint Nonnegative Matrix Factorization (JointNMF) is introduced for latent information discovery from multi-view data sets t hat contain both text content and connection structure information. The me thod jointly optimizes an integrated objective function\, which is a combi nation of the Nonnegative Matrix Factorization (NMF) objective function fo r handling text content/attribute information and the Symmetric NMF (SymNM F) objective function for handling relation/connection information. An eff ective algorithm for the joint NMF objective function is proposed utilizin g the block coordinate descent (BCD) method.\nThe proposed hybrid method s imultaneously discovers content associations and related latent connection s without any need for post-processing or additional clustering. In additi on\, known partial label information can be incorporated into a JointNMF f or semi-supervised clustering framework. The experimental results from sev eral real-life application problems illustrate the advantages of the propo sed approaches.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/37/ END:VEVENT BEGIN:VEVENT SUMMARY:Alex Bihlo (Memorial University) DTSTART:20220110T210000Z DTEND:20220110T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/38 DESCRIPTION:Title: Deep neural networks for solving differential equations on gen eral orientable surfaces\nby Alex Bihlo (Memorial University) as part of CRM Applied Math Seminar\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/38/ END:VEVENT BEGIN:VEVENT SUMMARY:Matus Benko (Johannes Kepler University Linz) DTSTART:20220117T210000Z DTEND:20220117T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/39 DESCRIPTION:Title: Variational Analysis: Basics\, Calculus\, and Semismoothness*< /a>\nby Matus Benko (Johannes Kepler University Linz) as part of CRM Appli ed Math Seminar\n\n\nAbstract\nThe purpose of this talk is to offer a brie f introduction into set-valued and variational analysis and to try to moti vate the study of this area. To this end\, we first discuss some basic not ions and ideas. Namely\, we try to explain why set-valued mappings should be analyzed\, what properties of such mappings seems to be useful and are typically studied\, as well as how one can analyze them\, i.e.\, what are the available tools. It should not be very surprising that\, just like in the standard analysis of functions\, derivatives play a crucial role. Thus \, we clarify how to differentiate set-valued mappings using the machinery of variational geometry (tangent and normal cones). Then we discuss in mo re depth the topic of calculus rules that enable one to properly manipulat e with generalized derivatives and apply them to practically relevant prob lems. We conclude with some remarks about the new property of semismoothne ss* for set-valued mappings and the related Newton method for solving gene ralized equations (inclusions).\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/39/ END:VEVENT BEGIN:VEVENT SUMMARY:David Rolnick (McGill University) DTSTART:20220124T210000Z DTEND:20220124T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/40 DESCRIPTION:Title: Expressivity and learnability in deep neural networks\nby David Rolnick (McGill University) as part of CRM Applied Math Seminar\n\n\ nAbstract\nIn this talk\, we show that there is a large gap between the ma ximum complexity of the functions that a neural network can express and th e expected complexity of the functions that it learns in practice. Deep R eLU networks are piecewise linear functions\, and the number of distinct l inear regions is a natural measure of their expressivity. It is well-know n that the maximum number of linear regions grows exponentially with the d epth of the network\, and this has often been used to explain the success of deeper networks. We show that the expected number of linear regions in fact grows polynomially in the size of the network\, far below the expone ntial upper bound and independent of the depth of the network. This state ment holds true both at initialization and after training\, under natural assumptions for gradient-based learning algorithms. We also show that the linear regions of a ReLU network reveal information about the network's p arameters. In particular\, it is possible to reverse-engineer the weights and architecture of an unknown deep ReLU network merely by querying it.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/40/ END:VEVENT BEGIN:VEVENT SUMMARY:Sebastien Le Digabel (Polytechnique Montreal) DTSTART:20220214T210000Z DTEND:20220214T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/41 DESCRIPTION:by Sebastien Le Digabel (Polytechnique Montreal) as part of CR M Applied Math Seminar\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/41/ END:VEVENT BEGIN:VEVENT SUMMARY:Tom Trogdon (U of Washington) DTSTART:20220221T210000Z DTEND:20220221T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/42 DESCRIPTION:by Tom Trogdon (U of Washington) as part of CRM Applied Math S eminar\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/42/ END:VEVENT BEGIN:VEVENT SUMMARY:Guy Wolf (UdeM) DTSTART:20220307T210000Z DTEND:20220307T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/43 DESCRIPTION:by Guy Wolf (UdeM) as part of CRM Applied Math Seminar\n\nAbst ract: TBA\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/43/ END:VEVENT BEGIN:VEVENT SUMMARY:Margarida Carvalho (UdeM) DTSTART:20220321T200000Z DTEND:20220321T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/44 DESCRIPTION:Title: Mathematical Programming Games: pushing the limits of equilibr ia computation\nby Margarida Carvalho (UdeM) as part of CRM Applied Ma th Seminar\n\n\nAbstract\nMathematical programming games (MPGs) encompass flexible problem modeling when decision makers interact. Through them\, w e can reflect each player's goal in the game through a parametric optimiza tion problem. In this talk\, we will first provide examples of such games and their MPG formulation. Then\, we will focus on MPGs where decisions can take integer values\, the so-called integer programming games. We wil l also discuss Nash games among Stackelberg leaders. The theoretical intr actability of these games will be presented as well as algorithmic schemes to solve them in practice.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/44/ END:VEVENT BEGIN:VEVENT SUMMARY:Soledad Villar (Johns Hopkins) DTSTART:20220404T200000Z DTEND:20220404T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/45 DESCRIPTION:Title: Units-equivariant machine learning\nby Soledad Villar (Joh ns Hopkins) as part of CRM Applied Math Seminar\n\n\nAbstract\nWe combine ideas from dimensional analysis and from equivariant machine learning to p rovide an approach for units-equivariant machine learning. Units equivaria nce is the exact symmetry that follows from the requirement that relations hips among measured quantities must obey self-consistent dimensional scali ngs. Our approach is to construct a dimensionless version of the learning task\, using classic results from dimensional analysis\, and then perform the learning task in the dimensionless space. This approach can be used to impose units equivariance on almost any contemporary machine-learning met hods\, including those that are equivariant to rotations and other groups. Units equivariance is expected to be particularly valuable in the context s of symbolic regression and emulation. We discuss the in-sample and out-o f-sample prediction accuracy gains one can obtain if exact units equivaria nce is imposed\; the symmetry is extremely powerful in some contexts. We i llustrate these methods with simple numerical examples involving dynamical systems in physics and ecology.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/45/ END:VEVENT BEGIN:VEVENT SUMMARY:Giang Tran (University of Waterloo) DTSTART:20220411T200000Z DTEND:20220411T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/46 DESCRIPTION:Title: Sparse Random Feature Models: Theoretical Guarantees and Appli cations\nby Giang Tran (University of Waterloo) as part of CRM Applied Math Seminar\n\n\nAbstract\nRandom feature methods have been successful i n various machine learning tasks\, are easy to compute\, and come with the oretical accuracy bounds. They serve as an alternative approach to standa rd neural networks since they can represent similar function spaces withou t a costly training phase. However\, for accuracy\, random feature method s require more measurements than trainable parameters\, limiting their use for data-scarce applications or problems in scientific machine learning. In this talk\, we will introduce the sparse random feature expansion to o btain parsimonious random feature models. Specifically\, we leverage idea s from compressive sensing to generate random feature expansions with theo retical guarantees even in the data-scarce setting. We also present a ran dom feature model for approximating high-dimensional sparse additive funct ions and a sparse random mode decomposition to extract intrinsic modes fro m challenging time-series data. Comparisons show that our proposed approa ches perform better or are comparable to other state-of-the-art or popular methods. Applications of our methods on identifying important variables in high-dimensional settings as well as on decomposing music pieces and vi sualizing black-hole mergers will be addressed.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/46/ END:VEVENT BEGIN:VEVENT SUMMARY:Fabian Pedregosa (Google) DTSTART:20220207T210000Z DTEND:20220207T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/47 DESCRIPTION:Title: Efficient and Modular Implicit Differentiation\nby Fabian Pedregosa (Google) as part of CRM Applied Math Seminar\n\n\nAbstract\nAuto matic differentiation (autodiff) has revolutionized machine learning. It allows expressing complex computations by composing elementary ones in cre ative ways and removes the burden of computing their derivatives by hand. More recently\, differentiation of optimization problem solutions has att racted widespread attention with applications such as optimization layers\ , and in bi-level problems such as hyper-parameter optimization and meta-l earning. However\, so far\, implicit differentiation remained difficult t o use for practitioners\, as it often required case-by-case tedious mathem atical derivations and implementations. In this paper\, we propose a unif ied\, efficient and modular approach for implicit differentiation of optim ization problems. In our approach\, the user defines directly in Python a function F capturing the optimality conditions of the problem to be diffe rentiated. Once this is done\, we leverage autodiff of F and implicit dif ferentiation to automatically differentiate the optimization problem. Our approach thus combines the benefits of implicit differentiation and autod iff. It is efficient as it can be added on top of any state-of-the-art so lver and modular as the optimality condition specification is decoupled fr om the implicit differentiation mechanism. We show that seemingly simple principles allow to recover many exiting implicit differentiation methods and create new ones easily. We demonstrate the ease of formulating and so lving bi-level optimization problems using our framework. We also showcas e an application to the sensitivity analysis of molecular dynamics.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/47/ END:VEVENT BEGIN:VEVENT SUMMARY:Anna Ma (University of California\, Irvine) DTSTART:20220314T200000Z DTEND:20220314T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/48 DESCRIPTION:Title: The Kaczmarz Algorithm: Greed\, Randomness\, and Tensors\n by Anna Ma (University of California\, Irvine) as part of CRM Applied Math Seminar\n\n\nAbstract\nIn settings where data sets become extremely large -scale\, stochastic iterative methods such as the Kaczmarz algorithm and R andomized Coordinate Descent become advantageous due to their low memory f ootprint. The Randomized Kaczmarz algorithm in particular has garnered at tention owing to its applicability in large-scale settings and its elegant geometric interpretation. In this talk\, we will discuss the Randomized Kaczmarz algorithm\, it's connection to the popular Stochastic Gradient De scent algorithm and it's greedy counter-part: Motzkin's Method. This pres entation contains joint work with Jamie Haddock and Denali Molitor.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/48/ END:VEVENT BEGIN:VEVENT SUMMARY:Tom Trogdon (U of Washington) DTSTART:20220328T200000Z DTEND:20220328T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/49 DESCRIPTION:Title: Perturbations of orthogonal polynomials: Riemann-Hilbert probl ems\, random matrices and numerical linear algebra\nby Tom Trogdon (U of Washington) as part of CRM Applied Math Seminar\n\n\nAbstract\nWe consi der the perturbation of orthogonal polynomials (OPs) with respect to chang es in the orthogonality measure. While the transformation from a measure to its orthogonal polynomials is typically ill-conditioned as the degree o f the polynomial grows\, using the Fokas-Its-Kitaev Riemann--Hilbert probl em\, we show that in certain settings this mapping is well-conditioned. A usable perturbation theory can then be obtained. The results are strengt hened when the asymptotics of the OPs with respect to a limiting measure a re known. Our main applications are to random matrices and to numerical al gorithms and dynamical systems applied to these random matrices. This is joint work with Percy Deift\, Xuicai Ding and Elliot Paquette.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/49/ END:VEVENT BEGIN:VEVENT SUMMARY:Bruno Després (Jacques-Louis Lions Laboratory) DTSTART:20220519T200000Z DTEND:20220519T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/50 DESCRIPTION:Title: Neural Networks from the viewpoint of Numerical Analysis\n by Bruno Després (Jacques-Louis Lions Laboratory) as part of CRM Applied Math Seminar\n\n\nAbstract\nThe presentation will focus on the interplay b etween\, on the one hand Neural Networks and Machine Learning which are em erging hot topics\, and on the other hand Numerical Analysis which is now a classical topic. The Yarotsky theorem will be discussed together with a recent alternative to the polarisation formula (D.-Ancellin 19'). The stab ility of the Adam algorithm will be shown with a particular Lyapunov funct ion. Discretization of transport equations for CFD will serve as an applic ative illustration.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/50/ END:VEVENT BEGIN:VEVENT SUMMARY:James Forbes (McGill) DTSTART:20220912T200000Z DTEND:20220912T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/51 DESCRIPTION:Title: Regularization Techniques in Koopman-based System Identificati on\nby James Forbes (McGill) as part of CRM Applied Math Seminar\n\n\n Abstract\nUsing the Koopman operator\, nonlinear systems can be expressed as infinite-dimensional linear systems. Data-driven methods can then be us ed to approximate a finite-dimensional Koopman operator\, which is particu larly useful for system identification\, control\, and state estimation ta sks. However\, approximating large Koopman operators is numerically challe nging\, leading to unstable Koopman operators being identified for otherwi se stable systems.\nThis talk will present a selection of techniques to re gularize the Koopman regression problem\, including a novel H-infinity nor m regularizer. In particular\, how to re-pose the system identification pr oblem in order to leverage numerically efficient optimization tools\, such as linear matrix inequalities\, will be presented. This talk is based on the pre-print arxiv\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/51/ END:VEVENT BEGIN:VEVENT SUMMARY:Quentin Bertrand (Mila) DTSTART:20220919T200000Z DTEND:20220919T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/52 DESCRIPTION:Title: Implicit Differentiation in Non-Smooth Convex Learning\nby Quentin Bertrand (Mila) as part of CRM Applied Math Seminar\n\n\nAbstract \nFinding the optimal hyperparameters of a model can be cast as a bilevel optimization problem\, typically solved zero-order techniques. In this wor k we study first-order methods when the inner optimization problem is conv ex but non-smooth. We show that the forward-mode differentiation of proxim al gradient descent and proximal coordinate descent yield sequences of Jac obians converging toward the exact Jacobian. Using implicit differentiatio n\, we show it is possible to leverage the non-smoothness of the inner pro blem to speed up the computation. Finally\, we provide a bound on the erro r made on the hypergradient when the inner optimization problem is solved approximately. Results on regression and classification problems reveal co mputational benefits for hyperparameter optimization\, especially when mul tiple hyperparameters are required.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/52/ END:VEVENT BEGIN:VEVENT SUMMARY:Akitoshi Takayasu (University of Tsukuba) DTSTART:20220926T200000Z DTEND:20220926T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/53 DESCRIPTION:Title: A general approach for rigorously integrating PDEs using semig roup theory\nby Akitoshi Takayasu (University of Tsukuba) as part of C RM Applied Math Seminar\n\n\nAbstract\nIn this talk we introduce a general rigorous PDE integrator that proves the existence of a solution to the Ca uchy problem of time-dependent PDEs. We derive a fixed-point formulation t o prove the existence of a solution locally in time\, which is based on th e solution map of a linearized problem called evolution operator. Using ri gorous numerics we validate the contraction of the fixed-point form on a n eighborhood of a numerically computed approximate solution. Then we extend the time interval to exist the solution via time stepping. The main advan tage of our approach is that the rigorous integrator can be applied to a g eneral class of PDEs\, even performed for higher spatial dimensional PDEs. \nThis is joint work with Jean-Philippe Lessard and Gabriel Duchesne.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/53/ END:VEVENT BEGIN:VEVENT SUMMARY:David Goluskin (U. of Victoria) DTSTART:20221003T200000Z DTEND:20221003T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/54 DESCRIPTION:Title: Verifying global stability of fluid flows despite transient gr owth of energy\nby David Goluskin (U. of Victoria) as part of CRM Appl ied Math Seminar\n\n\nAbstract\nVerifying nonlinear stability of a laminar fluid flow against all perturbations is a classic challenge in fluid dyna mics. All past results rely on monotonic decrease of a perturbation energy or a similar quadratic generalized energy. This "energy method" cannot sh ow global stability of any flow in which perturbation energy may grow tran siently. For the many flows that allow transient energy growth but seem to be globally stable (e.g. pipe flow and other parallel shear flows at cert ain Reynolds numbers) there has been no way to mathematically verify globa l stability. After explaining why the energy method was the only way to ve rify global stability of fluid flows for over 100 years\, I will describe a different approach that is broadly applicable but more technical. This a pproach\, proposed in 2012 by Goulart and Chernyshenko\, uses sum-of-squar es polynomials to computationally construct non-quadratic Lyapunov functio ns that decrease monotonically for all flow perturbations. I will present a computational implementation of this approach for the example of 2D plan e Couette flow\, where we have verified global stability at Reynolds numbe rs above the energy stability threshold. This energy stability result for 2D Couette flow had not been improved upon since being found by Orr in 190 7. The results I will present are the first verification of global stabili ty – for any fluid flow – that surpasses the energy method. This is joint work with Federico Fuentes (Universidad Católica de Chile ) and Sergei Chernyshenko (Imperial College London).\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/54/ END:VEVENT BEGIN:VEVENT SUMMARY:Nicola Guglielmi (Gran Sasso Science Institute) DTSTART:20221017T200000Z DTEND:20221017T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/55 DESCRIPTION:by Nicola Guglielmi (Gran Sasso Science Institute) as part of CRM Applied Math Seminar\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/55/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Baraldi (Sandia National Labs) DTSTART:20221024T200000Z DTEND:20221024T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/56 DESCRIPTION:by Robert Baraldi (Sandia National Labs) as part of CRM Applie d Math Seminar\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/56/ END:VEVENT BEGIN:VEVENT SUMMARY:Serge Prudhomme (Polytechnique Montreal) DTSTART:20221107T210000Z DTEND:20221107T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/57 DESCRIPTION:by Serge Prudhomme (Polytechnique Montreal) as part of CRM App lied Math Seminar\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/57/ END:VEVENT BEGIN:VEVENT SUMMARY:Kimon Fountoulakis (U. of Waterloo) DTSTART:20221128T210000Z DTEND:20221128T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/58 DESCRIPTION:by Kimon Fountoulakis (U. of Waterloo) as part of CRM Applied Math Seminar\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/58/ END:VEVENT BEGIN:VEVENT SUMMARY:Stephanie Dodson (Colby College) DTSTART:20221205T210000Z DTEND:20221205T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/59 DESCRIPTION:by Stephanie Dodson (Colby College) as part of CRM Applied Mat h Seminar\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/59/ END:VEVENT BEGIN:VEVENT SUMMARY:Tatiana Bubba (University of Bath) DTSTART:20221114T200000Z DTEND:20221114T210000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/60 DESCRIPTION:by Tatiana Bubba (University of Bath) as part of CRM Applied M ath Seminar\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/60/ END:VEVENT BEGIN:VEVENT SUMMARY:Ashwin Pananjady (Georgia Tech) DTSTART:20221121T210000Z DTEND:20221121T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/61 DESCRIPTION:by Ashwin Pananjady (Georgia Tech) as part of CRM Applied Math Seminar\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/61/ END:VEVENT BEGIN:VEVENT SUMMARY:Test (test) DTSTART:20221212T210000Z DTEND:20221212T220000Z DTSTAMP:20260422T225700Z UID:AppliedMathematics/62 DESCRIPTION:Title: test\nby Test (test) as part of CRM Applied Math Seminar\n \n\nAbstract\nA tensor T(x_1\, ...\, x_n) is a multilinear function of the input vectors x_j in F_q^n\, F_q a finite field. T has a small analytic rank if its output distribution is far from uniform. It has partition rank `r' if we can write T = f_1 * g_1 + ... + f_r * g_r\, where f_r and g_r a re tensors in fewer variables. Analytic rank measures the amount of random ness\, and partition rank measures the amount of structure. It is known th at if `f' has small partition rank\, it must have small analytic rank. Gre en and Tao proved an inverse theorem stating that if `f' has small analyti c rank then it has small partition rank. Their bound was qualitative\, how ever\, and several authors gave quantitative improvements. Janzer and Mili cevic independently proved a polynomial dependence. We prove an optimal in verse theorem: the analytic rank and partition rank are equivalent up to c onstant factors over large enough fields. Our techniques are very differen t from the usual methods in this area\, we rely on algebraic geometry rath er than additive combinatorics. This is joint work with Guy Moshkovitz.\n LOCATION:https://researchseminars.org/talk/AppliedMathematics/62/ END:VEVENT END:VCALENDAR