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Symplectic model reduction of Hamiltonian systems using data-driven quadratic manifolds

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arXiv 2023年

作者： Sharma, Harsh Mu, Hongliang Buchfink, Patrick Geelen, Rudy Glas, Silke Kramer, Boris Department of Mechanical and Aerospace Engineering University of California San DiegoCA United States Department of Applied Mathematics University of Twente Enschede Netherlands Institute of Applied Analysis and Numerical Simulation University of Stuttgart Stuttgart Germany Oden Institute for Computational Engineering and Sciences University of Texas AustinTX United States

This work presents two novel approaches for the symplectic model reduction of high-dimensional Hamiltonian systems using data-driven quadratic manifolds. Classical symplectic model reduction approaches employ linear symplectic subspaces for representing the high-dimensional system states in a reduced-dimensional coordinate system. While these approximations respect the symplectic nature of Hamiltonian systems, linear basis approximations can suffer from slowly decaying Kolmogorov N-width, especially in wave-type problems, which then requires a large basis size. We propose two different model reduction methods based on recently developed quadratic manifolds, each presenting its own advantages and limitations. The addition of quadratic terms to the state approximation, which sits at the heart of the proposed methodologies, enables us to better represent intrinsic low-dimensionality in the problem at hand. Both approaches are effective for issuing predictions in settings well outside the range of their training data while providing more accurate solutions than the linear symplectic reduced-order models. © 2023, CC BY.

关键词： Hamiltonians

Adaptive meshfree approximation for linear elliptic partial differential equations with PDE-greedy kernel methods

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arXiv 2022年

作者： Wenzel, Tizian Winkle, Daniel Santin, Gabriele Haasdonk, Bernard Department of Mathematics Ludwig-Maximilians-Universität München Germany Institute for Stochastics and Applications University of Stuttgart Germany Department of Environmental Sciences Informatics and Statistics Ca’ Foscari University of Venice Italy Institute of Applied Analysis and Numerical Simulation University of Stuttgart Germany

We consider meshless approximation for solutions of boundary value problems (BVPs) of elliptic Partial Differential Equations (PDEs) via symmetric kernel collocation. We discuss the importance of the choice of the collocation points, in particular by using greedy kernel methods. We introduce a scale of PDE-greedy selection criteria that generalizes existing techniques, such as the PDE-P-greedy and the PDE-f-greedy rules for collocation point selection. For these greedy selection criteria we provide bounds on the approximation error in terms of the number of greedily selected points and analyze the corresponding convergence rates. This is achieved by a novel analysis of Kolmogorov widths of special sets of BVP point-evaluation functionals. Especially, we prove that target-data dependent algorithms that make use of the right hand side functions of the BVP exhibit faster convergence rates than the target-data independent PDE-P-greedy. The convergence rate of the PDE-f-greedy possesses a dimension independent rate, which makes it amenable to mitigate the curse of dimensionality. The advantages of these greedy algorithms are highlighted by numerical examples. Copyright © 2022, The Authors. All rights reserved.

关键词： Partial differential equations

A model reduction approach for inverse problems with operator valued data

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arXiv 2020年

作者： Dölz, Jürgen Egger, Herbert Schlottbom, Matthias Institute for Numerical Simulation University of Bonn Friedrich-Hirzebruch-Allee 7 Bonn53115 Germany Numerical Analysis and Scientific Computing Department of Mathematics TU Darmstadt Dolivostr. 15 Darmstadt64293 Germany Department of Applied Mathematics University of Twente P.O. Box 217 Enschede7500 AE Netherlands

We study the efficient numerical solution of linear inverse problems with operator valued data which arise, e.g., in seismic exploration, inverse scattering, or tomographic imaging. The high-dimensionality of the data space implies extremely high computational cost already for the evaluation of the forward operator which makes a numerical solution of the inverse problem, e.g., by iterative regularization methods, practically infeasible. To overcome this obstacle, we take advantage of the underlying tensor product structure of the problem and propose a strategy for constructing low-dimensional certified reduced order models of quasi-optimal rank for the forward operator which can be computed much more efficiently than the truncated singular value decomposition. A complete analysis of the proposed model reduction approach is given in a functional analytic setting and the efficient numerical construction of the reduced order models as well as of their application for the numerical solution of the inverse problem is discussed. In summary, the setup of a low-rank approximation can be achieved in an offline stage at essentially the same cost as a single evaluation of the forward operator, while the actual solution of the inverse problem in the online phase can be done with extremely high efficiency. The theoretical results are illustrated by application to a typical model problem in fluorescence optical tomography. Copyright © 2020, The Authors. All rights reserved.

关键词： Inverse problems

MODEL REDUCTION ON MANIFOLDS: A DIFFERENTIAL GEOMETRIC FRAMEWORK

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arXiv 2023年

作者： Buchfink, Patrick Glas, Silke Haasdonk, Bernard Unger, Benjamin Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany Department of Applied Mathematics University of Twente P.O. Box 217 Enschede7500 AE Netherlands University of Stuttgart Uni-versitätsstr. 32 Stuttgart70569 Germany

Using nonlinear projections and preserving structure in model order reduction (MOR) are currently active research fields. In this paper, we provide a novel differential geometric framework for model reduction on smooth manifolds, which emphasizes the geometric nature of the objects involved. The crucial ingredient is the construction of an embedding for the low-dimensional submanifold and a compatible reduction map, for which we discuss several options. Our general framework allows capturing and generalizing several existing MOR techniques, such as structure preservation for Lagrangian- or Hamiltonian dynamics, and using nonlinear projections that are, for instance, relevant in transport-dominated problems. The joint abstraction can be used to derive shared theoretical properties for different methods, such as an exact reproduction result. To connect our framework to existing work in the field, we demonstrate that various techniques for data-driven construction of nonlinear projections can be included in our *** Codes 34A26, 34C20, 37C05, 37N30, 65P10 Copyright © 2023, The Authors. All rights reserved.

关键词： Hamiltonians

Space-time stochastic Galerkin boundary elements for acoustic scattering problems

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arXiv 2024年

作者： Gimperlein, Heiko Meyer, Fabian Özdemir, Ceyhun University of Innsbruck Engineering Mathematics Technikerstraße 13 Innsbruck6020 Austria Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany Institute for Mechanics and Graz Center of Computational Engineering Graz University of Technology Graz8010 Austria

Acoustic emission or scattering problems naturally involve uncertainties about the sound sources or boundary conditions. This article initiates the study of time domain boundary elements for such stochastic boundary problems for the acoustic wave equation. We present a space-time stochastic Galerkin boundary element method which is applied to sound-hard, sound-soft and absorbing scatterers. Uncertainties in both the sources and the boundary conditions are considered using a polynomial chaos expansion. The numerical experiments illustrate the performance and convergence of the proposed method in model problems and present an application to a problem from traffic noise. Copyright © 2024, The Authors. All rights reserved.

关键词： Boundary conditions

A phase-field approach to model evaporation from porous media: Modeling and upscaling

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arXiv 2021年

作者： Ghosh, Tufan Bringedal, Carina Rohde, Christian Helmig, Rainer Institute for Modelling Hydraulic and Environmental Systems University of Stuttgart Pfaffenwaldring 61 Stuttgart70569 Germany Department of Computer Science Electrical Engineering and Mathematical Sciences Western Norway University of Applied Sciences Inndalsveien 28 Bergen5063 Norway Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

We develop a phase-field model for evaporation from a porous medium by explicitly considering a vapor component together with the liquid and gas phases in the system. The phase-field model consists of the conservation of mass (for phases and vapor component), momentum, and energy. In addition, the evolution of the phase field is described by the Allen-Cahn equation. In the limit of vanishing interface width, matched asymptotic expansions reveal that the phase-field model reduces to the sharp-interface model with all the relevant transmission conditions on the moving interface. An energy estimate is derived, which suggests that for the diffusion-dominated regime, energy always decreases with time. However, this is not trivial in the case of other regimes. Through numerical examples, we analyze the efficiency of the developed phase-field formulation in modeling the evaporation process. We observe that our formulation is able to capture shrinking liquid droplet, in other words evaporation. Further, the phase-field model is upscaled to the Darcy scale using periodic homogenization for the diffusion-dominated regime. The effective parameters at the Darcy scale are connected to the pore scale through corresponding cell *** Codes 35C20, 35Q35, 35R35, 76N99, 76R50, 76S05, 76T10 Copyright © 2021, The Authors. All rights reserved.

关键词： Porous materials

Certified Nonlinear Parameter Optimization with Reduced Basis Surrogate Models

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PAMM 2013年第1期13卷

作者： Markus Dihlmann Bernard Haasdonk Institute for Applied Analysis and Numerical Simulation Pfaffenwaldring 57 70569 Stuttgart Germany

PDE-constrained parameter optimization problems suffer from the high dimensionality of the corresponding discretizations, which results in long optimization runtimes. One possible approach to solve such large scale optimization problems more rapidly is to replace the PDE constraint by a low-dimensional model constraint obtained via model reduction. We present a general technique for certification of such surrogate optimization results by a-posteriori error estimation based on Reduced Basis (RB) models. We allow arbitrary PDEs and optimization functionals, in particular cover nonlinear optimization problems. Experiments on a stationary heat-conduction problem demonstrate the applicability of the error bound. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

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Bayesian inversion for electromyography using low-rank tensor formats

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arXiv 2020年

作者： Rörich, Anna Werthmann, Tim A. Göddeke, Dominik Grasedyck, Lars Institute of Applied Analysis and Numerical Simulation University of Stuttgart Allmandring 5b Stuttgart70569 Germany Institut für Geometrie und Praktische Mathematik Rwth Aachen University Templergraben 55 Aachen52056 Germany Stuttgart Center for Simulation Science University of Stuttgart Pfaffenwaldring 5a Stuttgart70569 Germany

The reconstruction of the structure of biological tissue using electromyographic data is a non-invasive imaging method with diverse medical applications. Mathematically, this process is an inverse problem. Furthermore, electromyographic data are highly sensitive to changes in the electrical conductivity that describes the structure of the tissue. Modeling the inevitable measurement error as a stochastic quantity leads to a Bayesian approach. Solving the discretized Bayesian inverse problem means drawing samples from the posterior distribution of parameters, e.g., the conductivity, given measurement data. Using, e.g., a Metropolis-Hastings algorithm for this purpose involves solving the forward problem for different parameter combinations which requires a high computational effort. Low-rank tensor formats can reduce this effort by providing a data-sparse representation of all occurring linear systems of equations simultaneously and allow for their efficient solution. The application of Bayes' theorem proves the well-posedness of the Bayesian inverse problem. The derivation and proof of a low-rank representation of the forward problem allow for the precomputation of all solutions of this problem under certain assumptions, resulting in an efficient and theory-based sampling algorithm. numerical experiments support the theoretical results, but also indicate that a high number of samples is needed to obtain reliable estimates for the parameters. The Metropolis-Hastings sampling algorithm, using the precomputed forward solution in a tensor format, draws this high number of samples and therefore enables solving problems which are infeasible using classical methods. Copyright © 2020, The Authors. All rights reserved.

关键词： Tensors

ASYMPTOTIC EXPANSION FOR CONVECTION-DOMINATED TRANSPORT IN A THIN GRAPH-LIKE JUNCTION

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arXiv 2022年

作者： Mel'Nyk, Taras Rohde, Christian Department of Mathematical Physics Faculty of Mathematics and Mechanics Taras Shevchenko National University of Kyiv Volodymyrska str. 64 Kyiv01601 Ukraine Institute of Applied Analysis and Numerical Simulation Faculty of Mathematics and Physics Suttgart University Pfaffenwaldring 57 Suttgart70569 Germany

We consider for a small parameter Ε > 0 a parabolic convection-diffusion problem with Péclet number of order O(Ε−1) in a three-dimensional graph-like junction consisting of thin curvilinear cylinders with radii of order O(Ε) connected through a domain (node) of diameter O(Ε). Inhomogeneous Neumann type boundary conditions, that involve convective and diffusive fluxes, are prescribed both on the lateral surfaces of the thin cylinders and the boundary of the node. The asymptotic behaviour of the solution is studied as Ε → 0, i.e., when the diffusion coefficients are eliminated and the thin junction is shrunk into a three-part graph connected in a single vertex. A rigorous procedure for the construction of the complete asymptotic expansion of the solution is developed and the corresponding energetic and uniform pointwise estimates are proven. Depending on the directions of the limit convective fluxes, the corresponding limit problems (Ε = 0) are derived in the form of first-order hyperbolic differential equations on the one-dimensional branches with novel gluing conditions at the vertex. These generalize the classical Kirchhoff transmission conditions and might require the solution of a three-dimensional cell-like problem associated with the vertex to account for the local geometric inhomogeneity of the node and the physical processes in the node. The asymptotic ansatz consists of three parts, namely, the regular part, node-layer part, and boundary-layer one. Their coefficients are classical solutions to mixed-dimensional limit problems. The existence and other properties of those solutions are *** Codes 35K20, 35R02, 35B40, 35B25, 35B45, 35K57, 35Q49 © 2022, CC BY.

关键词： Boundary layers