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A posteriori error analysis for random scalar conservation laws using the stochastic galerkin method

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arXiv 2017年

作者： Meyery, Fabian Rohde, Christian Giesselmann, Jan Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 United Kingdom MathCCES RWTH Aachen University Schinkelstraße 2 Aachen52062 United Kingdom

In this article we present an a posteriori error estimator for the spatial-stochastic error of a Galerkin-type discretization of an initial value problem for a random hyperbolic conservation law. For the stochastic discretization we use the Stochastic Galerkin method and for the spatial-temporal discretization of the Stochastic Galerkin system a Runge-Kutta Discontinuous Galerkin method. The estimator is obtained using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework of Dafermos (2016), this leads to computable error bounds for the space-stochastic discretization error. Moreover, it turns out that the error estimator admits a splitting into one part representing the spatial error, and a remaining term, which can be interpreted as the stochastic error. This decomposition allows us to balance the errors arising from spatial and stochastic discretization. We conclude with some numerical examples confirming the theoretical findings.35L65, 35R60 (Primary) 65M15, 65M60 (Secondary) Copyright © 2017, The Authors. All rights reserved.

关键词： Galerkin methods

Kernel methods for center manifold approximation and a data-based version of the center manifold theorem

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arXiv 2020年

作者： Haasdonk, Bernard Hamzi, Boumediene Santin, Gabriele Wittwar, Dominik Institute for Applied Analysis and Numerical Simulation University of Stuttgart Germany Department of Mathematics Imperial College London United Kingdom Center for Information and Communication Technology Bruno Kessler Foundation Trento Italy

For dynamical systems with a non hyperbolic equilibrium, it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to the equilibrium point and to obtain meaningful predictions of its behavior by analyzing a reduced order system on the so-called center manifold. Since the center manifold is usually not known, good approximation methods are important as the center manifold theorem states that the stability properties of the origin of the reduced order system are the same as those of the origin of the full order system. In this work, we establish a data-based version of the center manifold theorem that works by considering an approximation in place of an exact manifold. Also the error between the approximated and the original reduced dynamics are quantified. We then use an apposite data-based kernel method to construct a suitable approximation of the manifold close to the equilibrium, which is compatible with our general error theory. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used as a training set. The method is tested on different examples which show promising performance and good accuracy. Copyright © 2020, The Authors. All rights reserved.

关键词： Dynamical systems

Consistent and Asymptotic-Preserving Finite-Volume Robin Transmission Conditions for Singularly Perturbed Elliptic Equations 26th

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Consistent and Asymptotic-Preserving Finite-Volume Robin Tra...

26th International Conference on Domain Decomposition Methods, 2020

作者： Gander, Martin J. Lunowa, Stephan B. Rohde, Christian Section de Mathématiques Université de Genève Rue du Conseil-Général 7-9 Genève SuisseCP 64 1205 Switzerland Computational Mathematics UHasselt – Hasselt University Agoralaan D 3590 Diepenbeek Hasselt Belgium Institute for Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

ISBN: (纸本)9783030950248

Adaptive Dirichlet–Neumann and Robin–Neumann algorithms for singularlyperturbed advection-diffusion equations were introduced in [2], accounting for transport along characteristics, see also [6] for the discrete setting and damped versions using a modified quadrature rule to recover the hyperbolic limit. Non-overlapping Schwarz DDMs with Robin transmission conditions (TCs) applied to advectiondiffusion equations were analyzed in [1, 10] and a stabilized finite-element method for singularly perturbed problems was discussed in [9], see also [3, 4] and references therein for heterogeneous couplings. However, the behavior of these DDMs in the limit of vanishing diffusion has not been addressed. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.

关键词： Advection

TRACE ESTIMATES FOR HARMONIC FUNCTIONS ALONG CIRCULAR ARCS WITH APPLICATIONS TO DOMAIN DECOMPOSITION ON OVERLAPPING DISKS

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arXiv 2024年

作者： Corso, Thiago Carvalho Hassan, Muhammad Jha, Abhinav Stamm, Benjamin Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany Institute of Mathematics EPF Lausanne Switzerland Department of Mathematics Indian Institute of Technology Gandhinagar Palaj Gujarat Gandhinagar382355 India

In this paper we derive several (and in many cases sharp) estimates for the L2-trace norm of harmonic functions along circular arcs. More precisely, we obtain geometry-dependent estimates on the norm, spectral radius, and numerical range of the Dirchlet-to-Dirichlet (DtD) operator sending data on the boundary of the disk to the restriction of its harmonic extension along circular arcs inside the disk. The estimates we derive here have applications in the convergence analysis of the Schwarz domain decomposition method for overlapping disks in two dimensions. In particular, they allow us to establish a rigorous convergence proof for the discrete parallel Schwarz method applied to the Conductor-like Screening Model (COSMO) from theoretical chemistry in the two-disk case, and to derive error estimates with respect to the discretization parameter, the number of Schwarz iterations, and the geometry of the domain. Our analysis addresses challenges beyond classical domain decomposition theory, especially the weak enforcement of boundary *** Codes 35J05, 30C40, 65N55 © 2024, CC BY-NC-SA.

关键词： Harmonic functions

Towards hybrid two-phase modelling using linear domain decomposition

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arXiv 2021年

作者： Seus, David Radu, Florin A. Rohde, Christian Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany Department of Mathematics University of Bergen P.O. Box 7800 BergenN-5020 Norway

The viscous flow of two immiscible fluids in a porous medium on the Darcy scale is governed by a system of nonlinear parabolic equations. If infinite mobility of one phase can be assumed (e.g. in soil layers in contact with the atmosphere) the system can be substituted by the scalar Richards model. Thus, the domain of the porous medium may be partitioned into disjoint subdomains with either the full two-phase or the simplified Richards model dynamics. Extending the one-model approach from [1, 2] we suggest coupling conditions for this hybrid model approach. Based on an Euler implicit discretisation, a linear iterative (-type) domain decomposition scheme is proposed, and proven to be convergent. The theoretical findings are verified by a comparative numerical study that in particular confirms the efficiency of the hybrid ansatz as compared to full two-phase model computations. © 2021, CC BY-NC-ND.

关键词： Two phase flow

ON REDUCED BASIS METHODS FOR EIGENVALUE PROBLEMS, AND ON ITS COUPLING WITH PERTURBATION THEORY

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arXiv 2024年

作者： Garrigue, Louis Stamm, Benjamin Laboratoire "analyse géométrie modélisation" CY Cergy Paris Université Cergy-Pontoise95302 France Institute of Applied Analysis and Numerical Simulation University of Stuttgart Stuttgart70569 Germany

We study eigenvalue problems and their approximation obtained by subspace projection, as in the reduced basis method. We provide bounds on the error between the exact eigenmodes and the approximated ones. Self-adjoint operators and degenerate cases are treated. We apply the bounds to the setting where the reduced space contains the vectors extracted from perturbation theory. © 2024, CC BY.

关键词： Vector spaces

COMBINING HYPERBOLIC QUADRATURE METHOD OF MOMENTS AND DISCRETE-VELOCITY-DIRECTION MODELS FOR SOLVING BGK-TYPE EQUATIONS

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arXiv 2024年

作者： Li, Tianshu Chen, Yihong Huang, Qian Weiyang College Tsinghua University Beijing100084 China Department of Mathematical Sciences Tsinghua University Beijing100084 China Institute of Applied Analysis and Numerical Simulation University of Stuttgart Stuttgart70569 Germany

This paper introduces the discrete-velocity-direction model (DVDM) in conjunction with the hyperbolic quadrature method of moments (HyQMOM) to develop a multidimensional spatial-temporal approximation of the BGK equation, termed DVD-HyQMOM. Serving as a multidimensional extension of HyQMOM, DVD-HyQMOM model achieves higher accuracy than other DVDM submodels, especially with an increased number of abscissas. The efficiency and effectiveness of this model are demonstrated through various numerical tests. Copyright © 2024, The Authors. All rights reserved.

关键词： Hyperbolic functions

A Quantitative analysis of a Virtual Programming Lab

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A Quantitative Analysis of a Virtual Programming Lab

IEEE International Symposium on Multimedia

作者： Jan Vanvinkenroye Christoph Grüninger Claus-Justus Heine Thomas Richter Technische Informations-und Kommunikationsdienste University of Stuttgart Stuttgart Germany Department of Hydromechanics and Modelling of Hydrosystems University of Stuttgart Stuttgart Germany Institute for Applied Analysis and Numerical Simulation University of Stuttgart Stuttgart Germany

We implemented a survey with one learning group using the web-based tools and a control group working with a traditional setup based on editor and compiler. In a recent publication, we described the design and implementation of a web-based programming lab (ViPLab) targeted at undergraduate Engineering and Mathematics courses. This work provides a quantitative analysis of the user feedback, experience and learning success. The survey shows that web-based installations are as efficient as classical tools, while Windows users prefer the web-based chain over the editor/compiler installation on Linux. This justifies the use of web-based installations in programming beginner courses, if the learning target focuses on programming and not a particular tool chain.

关键词： Educational institutions Programming profession Computers Statistical analysis

ON A VARIATIONAL PROBLEM RELATED TO THE CWIKEL–LIEB–ROZENBLUM AND LIEB–THIRRING INEQUALITIES

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arXiv 2024年

作者： Corso, Thiago Carvalho Ried, Tobias Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany Max-Planck-Institut für Mathematik in den Naturwissenschaften Inselstrasse 22 Leipzig04103 Germany

We explicitly solve a variational problem related to upper bounds on the optimal constants in the Cwikel–Lieb–Rozenblum (CLR) and Lieb–Thirring (LT) inequalities, which has recently been derived in [HKRV23] and [FHJN21]. We achieve this through a variational characterization of the L1 norm of the Fourier transform of a function and duality, from which we obtain a reformulation in terms of a variant of the Hadamard three lines lemma. By studying Hardy-like spaces of holomorphic functions in a strip in the complex plane, we are able to provide an analytic formula for the minimizers, and use it to get the best possible upper bounds for the optimal constants in the CLR and LT inequalities achievable by the method of [HKRV23] and [FHJN21].MSC Codes Primary 35P15, Secondary 81Q10, 30D05 Copyright © 2024, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions