Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces

Ehrlacher, Virginie; Lombardi, Damiano; Mula, Olga; Vialard, François-Xavier

doi:10.1051/m2an/2020013

Cited by 40 publications

(30 citation statements)

References 63 publications

(87 reference statements)

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“…In addition to the algebraic limitations associated with the solution of a rank-deficient system, the fact that the matrix S might be singular or ill conditioned prevents the reduced basis from evolving on the manifold of the orthosymplectic matrices. As shown in [26,Proposition 4.3], if U (t τ −1 ) ∈ U τ then U (t) ∈ R 2N ×2nτ solution of (7b) in T τ satisfies U (t) ∈ U τ for all t ∈ T τ , owing to the fact that F(U, Z; η h ) belongs to the space H U in (9).…”

Section: Reduced Dynamics Under Rank-deficiencymentioning

confidence: 94%

Rank-adaptive structure-preserving model order reduction of Hamiltonian systems

2022

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This work proposes an adaptive structure-preserving model order reduction method for finite-dimensional parametrized Hamiltonian systems modeling non-dissipative phenomena. To overcome the slowly decaying Kolmogorov width typical of transport problems, the full model is approximated on local reduced spaces that are adapted in time using dynamical low-rank approximation techniques. The reduced dynamics is prescribed by approximating the symplectic projection of the Hamiltonian vector field in the tangent space to the local reduced space. This ensures that the canonical symplectic structure of the Hamiltonian dynamics is preserved during the reduction. In addition, accurate approximations with low-rank reduced solutions are obtained by allowing the dimension of the reduced space to change during the time evolution. Whenever the quality of the reduced solution, assessed via an error indicator, is not satisfactory, the reduced basis is augmented in the parameter direction that is worst approximated by the current basis. Extensive numerical tests involving wave interactions, nonlinear transport problems, and the Vlasov equation demonstrate the superior stability properties and considerable runtime speedups of the proposed method as compared to global and traditional reduced basis approaches.

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Section: Reduced Dynamics Under Rank-deficiencymentioning

confidence: 94%

Rank-adaptive structure-preserving model order reduction of Hamiltonian systems

2022

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“…We remark that such preprocessing stage might be performed once during the offline stage, at the beginning of the online stage for any new µ ∈ P, or at each time step in combination with a suitable time-marching scheme. A fourth class of methods considers directly nonlinear approximations in combination with specialized methods to compute the solution during the online stage: to provide concrete references, we refer to the approaches based on convolutional autoencoders, [24,30,32,34], and to the approach in [20] based on optimal transport and nonlinear interpolation. As explained below (cf.…”

Section: Methods Based On Nonlinear Approximations: Expressivity and ...mentioning

confidence: 99%

Registration-based model reduction of parameterized two-dimensional conservation laws

Ferrero,

Taddei,

Zhang

2021

Preprint

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We propose a nonlinear registration-based model reduction procedure for rapid and reliable solution of parameterized two-dimensional steady conservation laws. This class of problems is challenging for model reduction techniques due to the presence of nonlinear terms in the equations and also due to the presence of parameter-dependent discontinuities that cannot be adequately represented through linear approximation spaces. Our approach builds on a general (i.e., independent of the underlying equation) registration procedure for the computation of a mapping Φ that tracks moving features of the solution field and on an hyper-reduced least-squares Petrov-Galerkin reduced-order model for the rapid and reliable computation of the solution coefficients. The contributions of this work are twofold. First, we investigate the application of registration-based methods to two-dimensional hyperbolic systems. Second, we propose a multi-fidelity approach to reduce the offline costs associated with the construction of the parameterized mapping and the reduced-order model. We discuss the application to an inviscid supersonic flow past a parameterized bump, to illustrate the many features of our method and to demonstrate its effectiveness.

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“…Let us point out that for all µ ∈ P, f µ is a C 1 function on [0, 1]. In this case, it is known [13] that there exists a constant c > 0 such that d n (M) ≤ cn −2 for all n ∈ N * .…”

Section: Second Test Casementioning

confidence: 99%

Influence of sampling on the convergence rates of greedy algorithms for parameter-dependent random variables

Blel¹,

Ehrlacher²,

Lelièvre³

2021

Preprint

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The main focus of this article is to provide a mathematical study of the algorithm proposed in [6] where the authors proposed a variance reduction technique for the computation of parameter-dependent expectations using a reduced basis paradigm. We study the effect of Monte-Carlo sampling on the theoretical properties of greedy algorithms. In particular, using concentration inequalities for the empirical measure in Wasserstein distance proved in [14], we provide sufficient conditions on the number of samples used for the computation of empirical variances at each iteration of the greedy procedure to guarantee that the resulting method algorithm is a weak greedy algorithm with high probability. These theoretical results are not fully practical and we therefore propose a heuristic procedure to choose the number of Monte-Carlo samples at each iteration, inspired from this theoretical study, which provides satisfactory results on several numerical test cases.

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Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces

Cited by 40 publications

References 63 publications

Rank-adaptive structure-preserving model order reduction of Hamiltonian systems

Rank-adaptive structure-preserving model order reduction of Hamiltonian systems

Registration-based model reduction of parameterized two-dimensional conservation laws

Influence of sampling on the convergence rates of greedy algorithms for parameter-dependent random variables

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