BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Nataša Krejic (University of Novi Sad\, Serbia) DTSTART:20210518T140000Z DTEND:20210518T150000Z DTSTAMP:20260422T212553Z UID:SNAP/1 DESCRIPTION:Title: A s tochastic first-order trust-region method with inexact restoration for non convex optimization\nby Nataša Krejic (University of Novi Sad\, Serbi a) as part of Seminars on Numerics and Applications\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SNAP/1/ END:VEVENT BEGIN:VEVENT SUMMARY:David Ketcheson (King Abdullah University of Science and Technolog y\, Saudi Arabia) DTSTART:20210525T140000Z DTEND:20210525T150000Z DTSTAMP:20260422T212553Z UID:SNAP/2 DESCRIPTION:Title: Exp licit numerical integrators that conserve energy or dissipate entropy\ nby David Ketcheson (King Abdullah University of Science and Technology\, Saudi Arabia) as part of Seminars on Numerics and Applications\n\n\nAbstra ct\nMany mathematical models are equipped with an energy that is conserved or an entropy that is known to change monotonically in time. Integrators that preserve these properties discretely are usually expensive\, with the best-known examples being fully-implicit Runge-Kutta methods. I will pres ent a modification that can be applied to any integrator in order to prese rve such a structural property. The resulting method can be fully explicit \, or (depending on the functional) may require the solution of a scalar a lgebraic equation at each step. I will present examples to show the effect iveness of these “relaxation” methods\, and their advantages over full y implicit methods or orthogonal projection. Examples will include applica tions to compressible fluid dynamics\, dispersive nonlinear waves\, and Ha miltonian systems.\n LOCATION:https://researchseminars.org/talk/SNAP/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Luca Calatroni (Laboratoire d'Informatique\, Signaux et Systèmes de Sophia-Antipolis (I3S)\, France) DTSTART:20210601T140000Z DTEND:20210601T150000Z DTSTAMP:20260422T212553Z UID:SNAP/3 DESCRIPTION:Title: Sca led\, inexact and adaptive generalised FISTA for (strongly) convex imaging problems\nby Luca Calatroni (Laboratoire d'Informatique\, Signaux et Systèmes de Sophia-Antipolis (I3S)\, France) as part of Seminars on Numer ics and Applications\n\n\nAbstract\nWe consider an inexact\, scaled genera lised Fast Iterative Soft-Thresholding Algorithm (FISTA) for minimising th e sum of two (possibly strongly) convex functions\, which we name SAGE-FIS TA. Here\, the inexactness is explicitly taken into account so as to descr ibe standard situations where proximal operators cannot be evaluated in cl osed form. The idea of considering data-dependent scaling in forward-backw ard splitting methods has furthermore been shown to be effective in incorp orating Newton-type information along the optimisation via suitable variab le-metric updates. Finally\, in order to account for the adjustment of the algorithmic step-size along the iterations\, we propose a non-monotone ba cktracking strategy which improves the convergence speed compared to stand ard Armijoo-type analogs. Analytically\, linear convergence result for the function values is proved. The result depends on the strong convexity mod uli of the two functions\, the upper and lower bounds on the spectrum of t he variable metric operators and the inexactness/backtracking parameters. The performance of SAGE-FISTA is validated on convex and strongly-convex e xemplar image denoising\, deblurring and super-resolution problems where s parsity-promoting regularisation is combined with data-dependent Kullback- Leibler-type fidelity terms.
\nThis is joint work with S. Rebegold i (University of Florence).\n LOCATION:https://researchseminars.org/talk/SNAP/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Yannis Kevrekidis (Johns Hopkins University\, USA) DTSTART:20210615T140000Z DTEND:20210615T150000Z DTSTAMP:20260422T212553Z UID:SNAP/4 DESCRIPTION:Title: No equations\, no variables\, no space\, no time: Data and the modeling of co mplex systems\nby Yannis Kevrekidis (Johns Hopkins University\, USA) a s part of Seminars on Numerics and Applications\n\n\nAbstract\nObtaining p redictive dynamical equations from data lies at the heart of science and e ngineering modeling\, and is the linchpin of our technology. In mathematic al modeling one typically progresses from observations of the world (and s ome serious thinking!) first to equations for a model\, and then to the an alysis of the model to make predictions. Good mathematical models give goo d predictions (and inaccurate ones do not) - but the computational tools f or analyzing them are the same: algorithms that are typically based on clo sed form equations. While the skeleton of the process remains the same\, t oday we witness the development of mathematical techniques that operate di rectly on observations -data-\, and appear to circumvent the serious think ing that goes into selecting variables and parameters and deriving accurat e equations. The process then may appear to the user a little like making predictions by "looking in a crystal ball". Yet the "serious thinking" is still there and uses the same -and some new- mathematics: it goes into bui lding algorithms that jump directly from data to the analysis of the model (which is now not available in closed form) so as to make predictions. Ou r work here presents a couple of efforts that illustrate this "new” path from data to predictions. It really is the same old path\, but it is trav elled by new means.\n LOCATION:https://researchseminars.org/talk/SNAP/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Valeria Simoncini (Università di Bologna\, Italy) DTSTART:20210629T140000Z DTEND:20210629T150000Z DTSTAMP:20260422T212553Z UID:SNAP/5 DESCRIPTION:Title: Mat rix-oriented numerical methods for semilinear PDEs\nby Valeria Simonci ni (Università di Bologna\, Italy) as part of Seminars on Numerics and Ap plications\n\n\nAbstract\nThe numerical solution of time dependent semilin ear partial differential equations in two space dimensions typically leads to discretized problems of large size.
\nUnder certain hypotheses o n the physical domain\, the space-discretized problem can be formulated as a matrix differential equation\, with significant advantages in the compu tational costs\, memory requirements and structure preservation. Moreover\ , time integrators can conveniently exploit this matrix framework.
\ nTo mitigate the difficulties associated with fine discretizations\, prope r orthogonal decompositions (POD) methodologies and discrete empirical int erpolation (DEIM) strategies are commonly employed to reduce the problem d imensions. We propose a novel matrix-oriented POD/DEIM approach that allow s us to apply matrix time integrators to the reduced differential problem.
\nThese are joint works with Maria Chiara D'Autilia and Ivonne Sg ura (Università del Salento)\, and Gerhard Kirsten (Università di Bologn a).\n LOCATION:https://researchseminars.org/talk/SNAP/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Raffaele D’Ambrosio (Università dell'Aquila\, Italy) DTSTART:20210706T140000Z DTEND:20210706T150000Z DTSTAMP:20260422T212553Z UID:SNAP/6 DESCRIPTION:Title: Pri nciples of stochastic geometric numerical integration\nby Raffaele D ’Ambrosio (Università dell'Aquila\, Italy) as part of Seminars on Numer ics and Applications\n\n\nAbstract\nThis talk is devoting to sharing recen t advances in the numerical preservation of invariant\nlaws characterizing the underlying dynamics of stochastic problems\, following the spirit\nof the so-called stochastic geometric numerical integration. We first addres s stochastic\nHamiltonian problems\, in order to obtain long-term energy c onservation. Specifically\,\nwe study the behaviour of stochastic Runge-Ku tta methods arising as stochastic perturbation\nof symplectic Runge-Kutta methods. The analysis is provided through epsilon-expansions\nof the solut ions (where epsilon is the amplitude of the stochastic fluctuation) and sh ows\nthe presence of secular terms destroying the long-term preservation o f the expected Hamiltonian.\nThen\, an energy-preserving scheme is develop ed and analyzed to fill this gap in.\nWe finally consider the nonlinear st ability properties of stochastic theta-methods with\nrespect to mean-squar e dissipative nonlinear test problems\, generating a mean-square\ncontract ive behaviour. The pursued aim is that of making the same property visible also along\nthe numerical discretization via stochastic theta–methods: this issue is translated into\nsharp stepsize restrictions depending on so me parameters of the problem\, accurately estimated.\nA selection of numer ical tests confirming the effectiveness of the analysis and its sharpness\ nis also provided.\n
\nReferences\n
\n[1] C. Chen\, D. Co hen\, R. D’Ambrosio\, A. Lang\, "Drift-preserving numerical inte-grat ors for stochastic Hamiltonian systems"\, Adv. Comput. Math. 46\, arti cle number 27 (2020).\n
\n[2] R. D’Ambrosio\, "Numerical approxi mation of differential problems"\, Springer (toappear).\n
\n[3] R . D’Ambrosio\, S. Di Giovacchino\, "Mean-square contractivity of stochastic theta-methods"\, Comm. Nonlin. Sci. Numer. Simul. 96\, art icle number 105671 (2021).\n
\n[4] R. D’Ambrosio\, S. Di Giovacch ino\, "Nonlinear stability issues for stochastic Runge-Kutta method s"\, Comm. Nonlin. Sci. Numer. Simul. 94\, article number 105549 (2021 ).\n
\n[5] R. D’Ambrosio\, G. Giordano\, B. Paternoster\, A. Ventol a\, "Perturbative analysis of stochastic Hamiltonian problems under tim e discretizations"\, Appl. Math. Lett. 120\, article number 107223 (20 21).\n LOCATION:https://researchseminars.org/talk/SNAP/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Jens Starke (University of Rostock\, Germany) DTSTART:20210713T140000Z DTEND:20210713T150000Z DTSTAMP:20260422T212553Z UID:SNAP/7 DESCRIPTION:by Jens Starke (University of Rostock\, Germany) as part of Se minars on Numerics and Applications\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SNAP/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Nataša Krklec Jerinkic (University of Novi Sad\, Serbia) DTSTART:20210727T140000Z DTEND:20210727T150000Z DTSTAMP:20260422T212553Z UID:SNAP/8 DESCRIPTION:Title: EFI X: Exact Fixed Point Methods for Distributed Optimization\nby Nataša Krklec Jerinkic (University of Novi Sad\, Serbia) as part of Seminars on N umerics and Applications\n\n\nAbstract\nWe consider strongly convex distri buted consensus optimization over connected networks. EFIX\, the proposed method\, is derived using quadratic penalty approach. In more detail\, we use the standard reformulation − transforming the original problem into a constrained problem in a higher dimensional space − to define a sequen ce of suitable quadratic penalty subproblems with increasing penalty param eters. For quadratic objectives\, the corresponding sequence consists of q uadratic penalty subproblems. For the generic strongly convex case\, the o bjective function is approximated with a quadratic model and hence the seq uence of the resulting penalty subproblems is again quadratic. EFIX is the n derived by solving each of the quadratic penalty subproblems via a fixed point (R)-linear solver\, e.g.\, Jacobi Over-Relaxation method. The exact convergence is proved as well as the worst case complexity of order for t he quadratic case. In the case of strongly convex generic functions\, the standard result for penalty methods is obtained. Numerical results indicat e that the method is highly competitive with state-of-the-art exact first order methods\, requires smaller computational and communication effort\, and is robust to the choice of algorithm parameters.\n LOCATION:https://researchseminars.org/talk/SNAP/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Maurizio Falcone (Sapienza Università di Roma\, Italy) DTSTART:20210928T140000Z DTEND:20210928T150000Z DTSTAMP:20260422T212553Z UID:SNAP/9 DESCRIPTION:Title: Dyn amic Programming on a tree for the approximation of finite horizon optimal control problems\nby Maurizio Falcone (Sapienza Università di Roma\, Italy) as part of Seminars on Numerics and Applications\n\n\nAbstract\nTh e classical Dynamic Programming (DP) approach to optimal control problems is based on the characterization of the value function as the unique visco sity solution of a Hamilton-Jacobi-Bellman (HJB) equation [2]. The DP sche me for the numerical approximation of viscosity solutions of those equatio ns is typically based on a time discretization coupled with a projection o n a fixed space triangulation of the numerical domain [3]. The time discre tization is obtained by a one-step scheme for the dynamics and the project ion is based on a polynomial interpolation. This approach allows to get a synthesis of optimal controls in feedback form and is very powerful for no nlinear optimal control problems in low dimension although general converg ence results are valid in any dimension. The computational cost is severe in high dimension and several methods have been proposed to mitigate the " curse of dimensionality" of DP schemes\, e.g. static and dynamic domain de composition\, fast-marching and fast-sweeping methods\, discrete represent ation formulas (when available)\, see [3] and the references therein.
\nWe present a new approach for finite horizon optimal control problems [ 1\, 4] where we compute the value function on a tree structure generated b y the time discrete dynamics avoiding the construction of a space grid/tri angulation to solve the HJB equation. This drops the computational cost of space interpolation although the tree mantains a perfect matching with th e discrete dynamics. We prove first order convergence to the value functio n for a first order discretization of the dynamics. We will also discuss e xtensions to high-order schemes and to problems with state constraints als o showing some numerical tests.
\nWorks in collaboration with A. A lla (PUC\, Rio de Janeiro) and L. Saluzzi (Sapienza\, Roma).
\nReferences
\n[1] A. Alla\, M. Falcone and L. Saluzzi. An efficie nt DP algorithm on a tree-structure for finite horizon optimal control pro blems\, SIAM Journal on Scientific Computing\, (41) 4\, 2019\, A2384-A2406
\n[2] M. Bardi\, I. Capuzzo-Dolcetta\, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations\, Birkhäuser\, Basel\, 19 97.
\n[3] M. Falcone\, R. Ferretti\, Semi-Lagrangian Approximation Sc hemes for Linear and Hamilton-Jacobi Equations\, Society for Industrial an d Applied Mathematics\, Philadelphia\, 2013.
\n[4] L. Saluzzi\, A. Al la and M. Falcone. Error estimates for a tree structure algorithm for dyna mic programming equations\, submitted\, 2018 https://arxiv.org/abs/1812.11 194\n LOCATION:https://researchseminars.org/talk/SNAP/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Houman Owhadi (California Institute of Technology\, USA) DTSTART:20210921T140000Z DTEND:20210921T150000Z DTSTAMP:20260422T212553Z UID:SNAP/10 DESCRIPTION:Title: On solving/learning differential equations with kernels\nby Houman Owhad i (California Institute of Technology\, USA) as part of Seminars on Numeri cs and Applications\n\n\nAbstract\nWe present a simple\, rigorous\, and un ified framework for solving and learning (possibly nonlinear) differential equations (PDEs and ODEs) using the framework of Gaussian processes/kerne l methods.\nFor PDEs the proposed approach:
\n(1) provides a natural generalization of collocation kernel methods to nonlinear PDEs and Inverse Problems\;
\n(2) has guaranteed convergence for a very general class of PDEs\, and comes equipped with a path to compute error bounds for spec ific PDE approximations\;
\n(3) inherits the state-of-the-art computa tional complexity of linear solvers for dense kernel matrices.
\nFor ODEs\, we illustrate the efficacy of the proposed approach by extrapolatin g weather/climate time series obtained from satellite data and illustrate the importance of using adapted/learned kernels.
\nParts of this t alk are joint work with Yifan Chen\, Boumediene Hamzi\, Bamdad Hosseini\, Romit Maulik\, Florian Schäfer\, Clint Scovel and Andrew Stuart.\n LOCATION:https://researchseminars.org/talk/SNAP/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Giancarlo Sangalli (Università di Pavia\, Italy) DTSTART:20210914T140000Z DTEND:20210914T150000Z DTSTAMP:20260422T212553Z UID:SNAP/11 DESCRIPTION:Title: Is ogeometric Analysis: high-order numerical solution of PDEs and computation al challenges\nby Giancarlo Sangalli (Università di Pavia\, Italy) as part of Seminars on Numerics and Applications\n\n\nAbstract\nThe concept of $k$-refinement was proposed as one of the key features of isogeometric analysis\,\n"a new\, more efficient\, higher-order concept"\, in the semin al work [1]. The idea of using\nhigh-degree and continuity splines (or NUR BS\, etc.) as a basis for a new high-order method\nappeared very promising from the beginning\, and received confirmations from the next development s.\nThe $k$-refinement leads to several advantages: higher accuracy per de gree-of-freedom\,\nimproved spectral accuracy\, the possibility of structu re-preserving smooth discretizations are\nthe most interesting features th at have been studied actively in the community. At the same\ntime\, the $k $-refinement brings significant challenges at the computational level: usi ng standard\nfinite element routines\, its computational cost grows with r espect to the degree\, making\ndegree raising computationally expensive. H owever\, recent ideas allow a computationally efficient\n$k$-refinement.\n
\nReferences\n
\n[1] T.J.R. Hughes\, J.A. Cottrell\, and Y. Bazilevs\, "Isogeometric analysis: CAD\, finite elements\,\nNURBS\, exact geometry and mesh refinement"\, Comput. Methods Appl. Mech. Eng rg.\, Vol. 194\,\npp. 4135-4195 (2005).\n LOCATION:https://researchseminars.org/talk/SNAP/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Elisabeth Larsson (Uppsala Universitet\, Sweden) DTSTART:20211005T140000Z DTEND:20211005T150000Z DTSTAMP:20260422T212553Z UID:SNAP/12 DESCRIPTION:Title: Lo calized least-squares radial basis function methods for PDEs\nby Elisa beth Larsson (Uppsala Universitet\, Sweden) as part of Seminars on Numeric s and Applications\n\n\nAbstract\nRadial basis function (RBF) approximatio n methods are attractive\nfor solving PDEs due to their flexibility with r espect to geometry\,\nthe potential for high-order accuracy\, and their ea se of use.\nSince global approximations come with a high computational cos t\,\nthe trend has been towards localized approximations.\nThe two main cl asses are stencil-based methods (RBF-FD) and partition\nof unity methods ( RBF-PUM). These are cost efficient and work well.\nHowever\, it has been d ifficult to derive complete convergence\nproofs for the collocation method s. Recently\, several authors have\ninvestigated how to introduce oversam pling into the PDE solution procedures.\nThis improves the approximation s tability\, and the least-square versions\nof the methods can be computatio nally competitive compared with their\ncollocation counterparts. Furthermo re\, for the least-squares methods\nwe are able to derive convergence proo fs using approaches based on the\ncontinuous approximation problem. In thi s talk\, we present recent algorithmic\nand theoretical developments as we ll as numerical results for a variety of PDE problems.\n LOCATION:https://researchseminars.org/talk/SNAP/12/ END:VEVENT END:VCALENDAR