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Symplectic Model Reduction of Hamiltonian Systems on Nonlinear Manifolds

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arXiv 2021年

作者： Buchfink, Patrick Glas, Silke Haasdonk, Bernard Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany Department of Computer Science Cornell University IthacaNY14853 United States

Classical model reduction techniques project the governing equations onto linear subspaces of the high-dimensional state-space. For problems with slowly decaying Kolmogorov-nwidths such as certain transport-dominated problems, however, classical linear-subspace reduced-order models (ROMs) of low dimension might yield inaccurate results. Thus, the concept of classical linear-subspace ROMs has to be extended to more general concepts, like Model Order Reduction (MOR) on manifolds. Moreover, as we are dealing with Hamiltonian systems, it is crucial that the underlying symplectic structure is preserved in the reduced model, as otherwise it could become unphysical in the sense that the energy is not conserved or stability properties are lost. To the best of our knowledge, existing literature addresses either MOR on manifolds or symplectic model reduction for Hamiltonian systems, but not their combination. In this work, we bridge the two aforementioned approaches by providing a novel projection technique called symplectic manifold Galerkin (SMG), which projects the Hamiltonian system onto a nonlinear symplectic trial manifold such that the reduced model is again a Hamiltonian system. We derive analytical results such as stability, energy-preservation and a rigorous a-posteriori error bound. Moreover, we construct a weakly symplectic deep convolutional autoencoder as a computationally practical approach to approximate a nonlinear symplectic trial manifold. Finally, we numerically demonstrate the ability of the method to outperform (non-)structure-preserving linear-subspace ROMs and non-structure-preserving MOR on manifold *** Codes 65P10, 34C20, 37J25, 37M15, 37N30 Copyright © 2021, The Authors. All rights reserved.

关键词： Learning systems

A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws

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arXiv 2019年

In this article we consider one-dimensional random systems of hyperbolic conservation laws. We first establish existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws which involve random initial data and random flux functions. Based on these results we present an a posteriori error analysis for a numerical approximation of the random entropy admissible solution. For the stochastic discretization, we consider a non-intrusive approach, the Stochastic Collocation method. The spatio-temporal discretization relies on the Runge–Kutta Discontinuous Galerkin method. We derive the a posteriori estimator using continuous reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. We conclude with various numerical examples investigating the scaling properties of the residuals and illustrating the efficiency of the proposed adaptive *** Codes Primary 35L65, 35R60, Secondary 65M15, 65M60, 65M70 Copyright © 2019, The Authors. All rights reserved.

关键词： Stochastic systems

A century of research in sedimentation and thickening

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KONA Powder and Particle Journal 2002年第March期20卷 38-70页

作者： Concha, Fernando Bürger, Raimund Department of Metallurgical Engineering University of Concepción Casilla 53-C Correo 3 Concepción Chile Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 StuttgartD-70569 Germany

This paper provides a concise review of the contributions to research in sedimentation and thickening that were made during the 20th century, starting from the invention of the Dorr thickener in 1905. The different steps of progress that were made in understanding batch sedimentation and continuous thickening processes in mineral processing are reviewed. A major breakthrough was Kynch’s kinematic sedimentation theory published in 1952. The authors’ own contributions to sedimentation and thickening research are summarized, including the development of the appropriate mathematical framework for Kynch’s theory and its extensions to continuous sedimentation and to polydisperse suspensions of spheres. An even more general model framework is provided by the phenomenological theory of sedimentation, which permits the description and simulation of the transient behavior of flocculent suspensions. © 2002, Hosokawa Powder Technology Foundation. All rights reserved.

关键词： Sedimentation

analysis of Structured Deep Kernel Networks

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arXiv 2021年

作者： Wenzel, Tizian Santin, Gabriele Haasdonk, Bernard Institute for Applied Analysis and Numerical Simulation University of Stuttgart Germany Deparment of Mathematics Ludwig-Maximilians-Universität München Germany Ca’ Foscari University of Venice Venice Italy

In this paper, we leverage a recent deep kernel representer theorem to connect kernel based learning and (deep) neural networks in order to understand their interplay. In particular, we show that the use of special types of kernels yields models reminiscent of neural networks that are founded in the same theoretical framework of classical kernel methods, while benefiting from the computational advantages of deep neural networks. Especially the introduced Structured Deep Kernel Networks (SDKNs) can be viewed as neural networks (NNs) with optimizable activation functions obeying a representer theorem. This link allows us to analyze also NNs within the framework of kernel networks. We prove analytic properties of the SDKNs which show their universal approximation properties in three different asymptotic regimes of unbounded number of centers, width and depth. Especially in the case of unbounded depth, more accurate constructions can be achieved using fewer layers compared to corresponding constructions for ReLU neural networks. This is made possible by leveraging properties of kernel approximation. Copyright © 2021, The Authors. All rights reserved.

关键词： Neural network models

Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds

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arXiv 2023年

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

For projection-based linear-subspace model order reduction (MOR), it is well known that the Kolmogorov n-width describes the best-possible error for a reduced order model (ROM) of size n. In this paper, we provide approximation bounds for ROMs on polynomially mapped manifolds. In particular, we show that the approximation bounds depend on the polynomial degree p of the mapping function as well as on the linear Kolmogorov n-width for the underlying problem. This results in a Kolmogorov (n, p)-width, which describes a lower bound for the best-possible error for a ROM on polynomially mapped manifolds of polynomial degree p and reduced size *** Codes 41A45, 41A46, 41A65, 65L70, 65M15, 65N15 Copyright © 2023, The Authors. All rights reserved.

关键词： Polynomial approximation

A p-multigrid method enhanced with an ILUT smoother and its comparison to h-multigrid methods within Isogeometric analysis

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arXiv 2019年

作者： Tielen, R. Möller, M. Göddeke, D. Vuik, C. Delft Institute of Applied Mathematics Delft University of Technology Van Mourik Broekmanweg 6 Delft2628XE Netherlands Institute for Applied Analysis and Numerical Simulation University of Stuttgart Allmandring 5b Stuttgart70569 Germany

Over the years, Isogeometric analysis has shown to be a successful alternative to the Finite Element Method (FEM). However, solving the resulting linear systems of equations efficiently remains a challenging task. In this paper, we consider a p-multigrid method, in which coarsening is applied in the approximation order p instead of the mesh width h. Since the use of classical smoothers (e.g. Gauss-Seidel) results in a p-multigrid method with deteriorating performance for higher values of p, the use of an ILUT smoother is investigated. numerical results and a spectral analysis indicate that the resulting p-multigrid method exhibits convergence rates independent of h and p. In particular, we compare both coarsening strategies (e.g. coarsening in h or p) adopting both smoothers for a variety of two and threedimensional benchmarks. Copyright © 2019, The Authors. All rights reserved.

关键词： Coarsening

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

DISSIPATIVENESS OF THE HYPERBOLIC QUADRATURE METHOD OF MOMENTS FOR KINETIC EQUATIONS

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arXiv 2024年

作者： Zhang, Ruixi Chen, Yihong Huang, Qian Yong, Wen-An Department of Mathematical Sciences Tsinghua University Beijing100084 China Institute of Applied Analysis and Numerical Simulation University of Stuttgart Stuttgart70569 Germany Beijing Institute of Mathematical Sciences and Applications Beijing101408 China

This paper presents a dissipativeness analysis of a quadrature method of moments (called HyQMOM) for the one-dimensional BGK equation. The method has exhibited its good performance in numerous applications. However, its mathematical foundation has not been clarified. Here we present an analytical proof of the strict hyperbolicity of the HyQMOM-induced moment closure systems by introducing a polynomial-based closure technique. As a byproduct, a class of numerical schemes for the HyQMOM system is shown to be realizability preserving under CFL-type conditions. We also show that the system preserves the dissipative properties of the kinetic equation by verifying a certain structural stability condition. The proof uses a newly introduced affine invariance and the homogeneity of the HyQMOM and heavily relies on the theory of orthogonal polynomials associated with realizable moments, in particular, the moments of the standard normal distribution. © 2024, CC BY.

关键词： Stability

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