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:20260404T095336Z UID:SNAP/1 DESCRIPTION:Title: A stochastic first-order trust-region method with inexact restoration for nonconvex optimization\nby Nataša Krejic (University of Novi Sad\ , Serbia) as part of Seminars on Numerics and Applications\n\nAbstract: TB A\n LOCATION:https://stable.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:20260404T095336Z UID:SNAP/2 DESCRIPTION:Title: Explicit numerical integrators that conserve energy or dissipate entro py\nby David Ketcheson (King Abdullah University of Science and Techno logy\, Saudi Arabia) as part of Seminars on Numerics and Applications\n\n\ nAbstract\nMany mathematical models are equipped with an energy that is co nserved or an entropy that is known to change monotonically in time. Integ rators that preserve these properties discretely are usually expensive\, w ith the best-known examples being fully-implicit Runge-Kutta methods. I wi ll present a modification that can be applied to any integrator in order t o preserve such a structural property. The resulting method can be fully e xplicit\, or (depending on the functional) may require the solution of a s calar algebraic equation at each step. I will present examples to show the effectiveness of these “relaxation” methods\, and their advantages ov er fully implicit methods or orthogonal projection. Examples will include applications to compressible fluid dynamics\, dispersive nonlinear waves\, and Hamiltonian systems.\n LOCATION:https://stable.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:20260404T095336Z UID:SNAP/3 DESCRIPTION:Title: Scaled\, inexact and adaptive generalised FISTA for (strongly) convex imaging problems\nby Luca Calatroni (Laboratoire d'Informatique\, Sign aux et Systèmes de Sophia-Antipolis (I3S)\, France) as part of Seminars o n Numerics and Applications\n\n\nAbstract\nWe consider an inexact\, scaled generalised Fast Iterative Soft-Thresholding Algorithm (FISTA) for minimi sing the sum of two (possibly strongly) convex functions\, which we name S AGE-FISTA. Here\, the inexactness is explicitly taken into account so as t o describe standard situations where proximal operators cannot be evaluate d in closed form. The idea of considering data-dependent scaling in forwar d-backward splitting methods has furthermore been shown to be effective in incorporating Newton-type information along the optimisation via suitable variable-metric updates. Finally\, in order to account for the adjustment of the algorithmic step-size along the iterations\, we propose a non-mono tone backtracking strategy which improves the convergence speed compared t o standard Armijoo-type analogs. Analytically\, linear convergence result for the function values is proved. The result depends on the strong convex ity moduli of the two functions\, the upper and lower bounds on the spectr um of the variable metric operators and the inexactness/backtracking param eters. The performance of SAGE-FISTA is validated on convex and strongly-c onvex exemplar image denoising\, deblurring and super-resolution problems where sparsity-promoting regularisation is combined with data-dependent Ku llback-Leibler-type fidelity terms.
\nThis is joint work with S. R ebegoldi (University of Florence).\n LOCATION:https://stable.researchseminars.org/talk/SNAP/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Yannis Kevrekidis (Johns Hopkins University\, USA) DTSTART:20210615T140000Z DTEND:20210615T150000Z DTSTAMP:20260404T095336Z UID:SNAP/4 DESCRIPTION:Title: No equations\, no variables\, no space\, no time: Data and the modelin g of complex systems\nby Yannis Kevrekidis (Johns Hopkins University\, USA) as part of Seminars on Numerics and Applications\n\n\nAbstract\nObta ining predictive dynamical equations from data lies at the heart of scienc e and engineering modeling\, and is the linchpin of our technology. In mat hematical modeling one typically progresses from observations of the world (and some serious thinking!) first to equations for a model\, and then to the analysis of the model to make predictions. Good mathematical models g ive good predictions (and inaccurate ones do not) - but the computational tools for analyzing them are the same: algorithms that are typically based on closed form equations. While the skeleton of the process remains the s ame\, today we witness the development of mathematical techniques that ope rate directly on observations -data-\, and appear to circumvent the seriou s thinking that goes into selecting variables and parameters and deriving accurate equations. The process then may appear to the user a little like making predictions by "looking in a crystal ball". Yet the "serious thinki ng" is still there and uses the same -and some new- mathematics: it goes i nto building algorithms that jump directly from data to the analysis of th e model (which is now not available in closed form) so as to make predicti ons. Our 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 travelled by new means.\n LOCATION:https://stable.researchseminars.org/talk/SNAP/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Valeria Simoncini (Università di Bologna\, Italy) DTSTART:20210629T140000Z DTEND:20210629T150000Z DTSTAMP:20260404T095336Z UID:SNAP/5 DESCRIPTION:Title: Matrix-oriented numerical methods for semilinear PDEs\nby Valeria Simoncini (Università di Bologna\, Italy) as part of Seminars on Numerics and Applications\n\n\nAbstract\nThe numerical solution of time dependent semilinear partial differential equations in two space dimensions typicall y leads to discretized problems of large size.
\nUnder certain hypot heses on the physical domain\, the space-discretized problem can be formul ated as a matrix differential equation\, with significant advantages in th e computational costs\, memory requirements and structure preservation. Mo reover\, time integrators can conveniently exploit this matrix framework.
\nTo mitigate the difficulties associated with fine discretizations\ , proper orthogonal decompositions (POD) methodologies and discrete empiri cal interpolation (DEIM) strategies are commonly employed to reduce the pr oblem dimensions. We propose a novel matrix-oriented POD/DEIM approach tha t allows us to apply matrix time integrators to the reduced differential p roblem.
\nThese are joint works with Maria Chiara D'Autilia and Iv onne Sgura (Università del Salento)\, and Gerhard Kirsten (Università di Bologna).\n LOCATION:https://stable.researchseminars.org/talk/SNAP/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Raffaele D’Ambrosio (Università dell'Aquila\, Italy) DTSTART:20210706T140000Z DTEND:20210706T150000Z DTSTAMP:20260404T095336Z UID:SNAP/6 DESCRIPTION:Title: Principles of stochastic geometric numerical integration\nby Raffa ele D’Ambrosio (Università dell'Aquila\, Italy) as part of Seminars on Numerics and Applications\n\n\nAbstract\nThis talk is devoting to sharing recent advances in the numerical preservation of invariant\nlaws character izing the underlying dynamics of stochastic problems\, following the spiri t\nof the so-called stochastic geometric numerical integration. We first a ddress stochastic\nHamiltonian problems\, in order to obtain long-term ene rgy conservation. Specifically\,\nwe study the behaviour of stochastic Run ge-Kutta methods arising as stochastic perturbation\nof symplectic Runge-K utta methods. The analysis is provided through epsilon-expansions\nof the solutions (where epsilon is the amplitude of the stochastic fluctuation) a nd shows\nthe presence of secular terms destroying the long-term preservat ion of the expected Hamiltonian.\nThen\, an energy-preserving scheme is de veloped and analyzed to fill this gap in.\nWe finally consider the nonline ar stability properties of stochastic theta-methods with\nrespect to mean- square dissipative nonlinear test problems\, generating a mean-square\ncon tractive behaviour. The pursued aim is that of making the same property vi sible also along\nthe numerical discretization via stochastic theta–meth ods: this issue is translated into\nsharp stepsize restrictions depending on some parameters of the problem\, accurately estimated.\nA selection of numerical tests confirming the effectiveness of the analysis and its sharp ness\nis also provided.\n
\nReferences\n
\n[1] C. Chen\, D. Cohen\, R. D’Ambrosio\, A. Lang\, "Drift-preserving numerical inte -grators for stochastic Hamiltonian systems"\, Adv. Comput. Math. 46\, article number 27 (2020).\n
\n[2] R. D’Ambrosio\, "Numerical ap proximation 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\ , article number 105671 (2021).\n
\n[4] R. D’Ambrosio\, S. Di Gio vacchino\, "Nonlinear stability issues for stochastic Runge-Kutta m ethods"\, Comm. Nonlin. Sci. Numer. Simul. 94\, article number 105549 (2021).\n
\n[5] R. D’Ambrosio\, G. Giordano\, B. Paternoster\, A. V entola\, "Perturbative analysis of stochastic Hamiltonian problems unde r time discretizations"\, Appl. Math. Lett. 120\, article number 10722 3 (2021).\n LOCATION:https://stable.researchseminars.org/talk/SNAP/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Jens Starke (University of Rostock\, Germany) DTSTART:20210713T140000Z DTEND:20210713T150000Z DTSTAMP:20260404T095336Z 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://stable.researchseminars.org/talk/SNAP/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Nataša Krklec Jerinkic (University of Novi Sad\, Serbia) DTSTART:20210727T140000Z DTEND:20210727T150000Z DTSTAMP:20260404T095336Z UID:SNAP/8 DESCRIPTION:Title: EFIX: Exact Fixed Point Methods for Distributed Optimization\nby N ataša Krklec Jerinkic (University of Novi Sad\, Serbia) as part of Semina rs on Numerics and Applications\n\n\nAbstract\nWe consider strongly convex distributed consensus optimization over connected networks. EFIX\, the pr oposed method\, is derived using quadratic penalty approach. In more detai l\, we use the standard reformulation − transforming the original proble m into a constrained problem in a higher dimensional space − to define a sequence of suitable quadratic penalty subproblems with increasing penalt y parameters. For quadratic objectives\, the corresponding sequence consis ts of quadratic penalty subproblems. For the generic strongly convex case\ , the objective function is approximated with a quadratic model and hence the sequence of the resulting penalty subproblems is again quadratic. EFIX is then derived by solving each of the quadratic penalty subproblems via a fixed point (R)-linear solver\, e.g.\, Jacobi Over-Relaxation method. Th e exact convergence is proved as well as the worst case complexity of orde r for the quadratic case. In the case of strongly convex generic functions \, the standard result for penalty methods is obtained. Numerical results indicate that the method is highly competitive with state-of-the-art exact first order methods\, requires smaller computational and communication ef fort\, and is robust to the choice of algorithm parameters.\n LOCATION:https://stable.researchseminars.org/talk/SNAP/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Maurizio Falcone (Sapienza Università di Roma\, Italy) DTSTART:20210928T140000Z DTEND:20210928T150000Z DTSTAMP:20260404T095336Z UID:SNAP/9 DESCRIPTION:Title: Dynamic 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\nAbstr act\nThe classical Dynamic Programming (DP) approach to optimal control pr oblems is based on the characterization of the value function as the uniqu e viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation [2]. The DP scheme for the numerical approximation of viscosity solutions of those equations is typically based on a time discretization coupled with a proje ction on a fixed space triangulation of the numerical domain [3]. The time discretization is obtained by a one-step scheme for the dynamics and the projection is based on a polynomial interpolation. This approach allows to get a synthesis of optimal controls in feedback form and is very powerful for nonlinear optimal control problems in low dimension although general convergence results are valid in any dimension. The computational cost is severe in high dimension and several methods have been proposed to mitigat e the "curse of dimensionality" of DP schemes\, e.g. static and dynamic do main decomposition\, fast-marching and fast-sweeping methods\, discrete re presentation formulas (when available)\, see [3] and the references therei n.
\nWe present a new approach for finite horizon optimal control pro blems [1\, 4] where we compute the value function on a tree structure gene rated by the time discrete dynamics avoiding the construction of a space g rid/triangulation to solve the HJB equation. This drops the computational cost of space interpolation although the tree mantains a perfect matching with the discrete dynamics. We prove first order convergence to the value function for a first order discretization of the dynamics. We will also di scuss extensions to high-order schemes and to problems with state constrai nts also showing some numerical tests.
\nWorks in collaboration wi th A. Alla (PUC\, Rio de Janeiro) and L. Saluzzi (Sapienza\, Roma).
\nReferences
\n[1] A. Alla\, M. Falcone and L. Saluzzi. An efficient DP algorithm on a tree-structure for finite horizon optimal cont rol problems\, SIAM Journal on Scientific Computing\, (41) 4\, 2019\, A238 4-A2406
\n[2] M. Bardi\, I. Capuzzo-Dolcetta\, Optimal Control and Vi scosity Solutions of Hamilton-Jacobi-Bellman Equations\, Birkhäuser\, Bas el\, 1997.
\n[3] M. Falcone\, R. Ferretti\, Semi-Lagrangian Approxima tion Schemes for Linear and Hamilton-Jacobi Equations\, Society for Indust rial and Applied Mathematics\, Philadelphia\, 2013.
\n[4] L. Saluzzi\ , A. Alla and M. Falcone. Error estimates for a tree structure algorithm f or dynamic programming equations\, submitted\, 2018 https://arxiv.org/abs/ 1812.11194\n LOCATION:https://stable.researchseminars.org/talk/SNAP/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Houman Owhadi (California Institute of Technology\, USA) DTSTART:20210921T140000Z DTEND:20210921T150000Z DTSTAMP:20260404T095336Z UID:SNAP/10 DESCRIPTION:Title: On solving/learning differential equations with kernels\nby Houma n Owhadi (California Institute of Technology\, USA) as part of Seminars on Numerics and Applications\n\n\nAbstract\nWe present a simple\, rigorous\, and unified framework for solving and learning (possibly nonlinear) diffe rential equations (PDEs and ODEs) using the framework of Gaussian processe s/kernel methods.\nFor PDEs the proposed approach:
\n(1) provides a n atural generalization of collocation kernel methods to nonlinear PDEs and Inverse Problems\;
\n(2) has guaranteed convergence for a very genera l class of PDEs\, and comes equipped with a path to compute error bounds f or specific PDE approximations\;
\n(3) inherits the state-of-the-art computational complexity of linear solvers for dense kernel matrices.
\nFor ODEs\, we illustrate the efficacy of the proposed approach by extra polating weather/climate time series obtained from satellite data and illu strate the importance of using adapted/learned kernels.
\nParts of this talk are joint work with Yifan Chen\, Boumediene Hamzi\, Bamdad Hoss eini\, Romit Maulik\, Florian Schäfer\, Clint Scovel and Andrew Stuart.\n LOCATION:https://stable.researchseminars.org/talk/SNAP/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Giancarlo Sangalli (Università di Pavia\, Italy) DTSTART:20210914T140000Z DTEND:20210914T150000Z DTSTAMP:20260404T095336Z UID:SNAP/11 DESCRIPTION:Title: Isogeometric Analysis: high-order numerical solution of PDEs and comp utational challenges\nby Giancarlo Sangalli (Università di Pavia\, It aly) as part of Seminars on Numerics and Applications\n\n\nAbstract\nThe c oncept of $k$-refinement was proposed as one of the key features of isogeo metric analysis\,\n"a new\, more efficient\, higher-order concept"\, in th e seminal work [1]. The idea of using\nhigh-degree and continuity splines (or NURBS\, etc.) as a basis for a new high-order method\nappeared very pr omising from the beginning\, and received confirmations from the next deve lopments.\nThe $k$-refinement leads to several advantages: higher accuracy per degree-of-freedom\,\nimproved spectral accuracy\, the possibility of structure-preserving smooth discretizations are\nthe most interesting feat ures that have been studied actively in the community. At the same\ntime\, the $k$-refinement brings significant challenges at the computational lev el: using standard\nfinite element routines\, its computational cost grows with respect to the degree\, making\ndegree raising computationally expen sive. However\, recent ideas allow a computationally efficient\n$k$-refine ment.\n
\nReferences\n
\n[1] T.J.R. Hughes\, J.A. Cottrel l\, and Y. Bazilevs\, "Isogeometric analysis: CAD\, finite elements\,\n NURBS\, exact geometry and mesh refinement"\, Comput. Methods Appl. Me ch. Engrg.\, Vol. 194\,\npp. 4135-4195 (2005).\n LOCATION:https://stable.researchseminars.org/talk/SNAP/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Elisabeth Larsson (Uppsala Universitet\, Sweden) DTSTART:20211005T140000Z DTEND:20211005T150000Z DTSTAMP:20260404T095336Z UID:SNAP/12 DESCRIPTION:Title: Localized least-squares radial basis function methods for PDEs\nb y Elisabeth Larsson (Uppsala Universitet\, Sweden) as part of Seminars on Numerics and Applications\n\n\nAbstract\nRadial basis function (RBF) appro ximation methods are attractive\nfor solving PDEs due to their flexibility with respect to geometry\,\nthe potential for high-order accuracy\, and t heir ease of use.\nSince global approximations come with a high computatio nal cost\,\nthe trend has been towards localized approximations.\nThe two main classes are stencil-based methods (RBF-FD) and partition\nof unity me thods (RBF-PUM). These are cost efficient and work well.\nHowever\, it has been difficult to derive complete convergence\nproofs for the collocation methods. Recently\, several authors have\ninvestigated how to introduce oversampling into the PDE solution procedures.\nThis improves the approxim ation stability\, and the least-square versions\nof the methods can be com putationally competitive compared with their\ncollocation counterparts. Fu rthermore\, for the least-squares methods\nwe are able to derive convergen ce proofs using approaches based on the\ncontinuous approximation problem. In this talk\, we present recent algorithmic\nand theoretical development s as well as numerical results for a variety of PDE problems.\n LOCATION:https://stable.researchseminars.org/talk/SNAP/12/ END:VEVENT END:VCALENDAR