Optimal Local Approximation Spaces for Parabolic Problems

Schleuß, Julia; Smetana, Kathrin

doi:10.1137/20m1384294

Cited by 10 publications

(5 citation statements)

References 57 publications

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“…Proof. The first paragraph closely follows the proof of Proposition 3.1 in [47]. Since w satisfies equation (A.1) integration by parts in time yields t * s w t (t), v H 1 0 (D) ϕ(t) dt+ t * s (κ(t)∇w(t), ∇v) L 2 (D) ϕ(t) dt = 0 for all v ∈ H 1 0 (D) and ϕ ∈ C ∞ 0 ((s, t * )).…”

Section: Discussionmentioning

confidence: 92%

“…Here, •, • H 1 0 (D) denotes the duality pairing between H 1 0 (D) and H −1 (D). Thanks to the Hahn-Banach theorem we have that u ∈ W 1,2,2 (I, H 1 0 (D), H −1 (D)) := v ∈ L 2 (I, H 1 0 (D)) | v t ∈ L 2 (I, H −1 (D)) and u ∈ C 0 ( Ī, L 2 (D)) (see, for instance, [47]). We highlight that the embedding W 1,2,2 (I, H 1 0 (D), H −1 (D)) → C 0 ( Ī, L 2 (D)) is not compact unless additional regularity is assumed.…”

Section: Discussionmentioning

confidence: 99%

“…Compactness of the transfer operator guarantees the existence of its SVD via the Hilbert-Schmidt theorem [42,Theorem 8.94]. Similar to [2,47,54] it can then be shown that the leading left singular vectors of T s→t span an optimal approximation space in the local solution space H t . Here, we use the concept of optimality in the sense of Kolmogorov [24]…”

Section: 1mentioning

confidence: 99%

“…The inequality can thus be used to assess whether the solution space at a point of time is amenable to approximation; facilitating the design of localizable multiscale and model order reduction methods. The combination of a Caccioppoli-type inequality with a suitable compactness theorem is usually used to show compactness of the transfer operator; see also [2,47,54,55].…”

Section: Appendix a Exemplary Model Problem: The Linear Heat Equationmentioning

confidence: 99%

“…Optimal spatially local approximation spaces have been introduced for elliptic [2,29,54] and parabolic [47] problems, and random sampling has been employed to efficiently approximate the optimal local spaces in the elliptic setting in [5,7]. Further spatially localizable multiscale methods for parabolic problems have been proposed in [8,32,38,39,40].…”

Section: Introductionmentioning

confidence: 99%

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Randomized quasi-optimal local approximation spaces in time

Schleuß¹,

Smetana²,

Maat³

2022

Preprint

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We target time-dependent partial differential equations (PDEs) with coefficients that are arbitrarily rough in both space and time. To tackle these problems, we construct reduced basis/ multiscale ansatz functions defined in space that can be combined with time stepping schemes within model order reduction or multiscale methods. To that end, we propose to perform several simulations of the PDE for few time steps in parallel starting at different, randomly drawn start points, prescribing random initial conditions; applying a singular value decomposition to a subset of the so obtained snapshots yields the reduced basis/ multiscale ansatz functions. This facilitates constructing the reduced basis/ multiscale ansatz functions in an embarrassingly parallel manner. In detail, we suggest using a data-dependent probability distribution based on the data functions of the PDE to select the start points. Each local in time simulation of the PDE with random initial conditions approximates a local approximation space in one time point that is optimal in the sense of Kolmogorov. The derivation of these optimal local approximation spaces which are spanned by the left singular vectors of a compact transfer operator that maps arbitrary initial conditions to the solution of the PDE in a later point of time, is one other main contribution of this paper. By solving the PDE locally in time with random initial conditions, we construct local ansatz spaces in time that converge provably at a quasi-optimal rate and allow for local error control. Numerical experiments demonstrate that the proposed method can outperform existing methods like the proper orthogonal decomposition even in a sequential setting.

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Section: Discussionmentioning

confidence: 92%

Section: Discussionmentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: Appendix a Exemplary Model Problem: The Linear Heat Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Randomized quasi-optimal local approximation spaces in time

Schleuß¹,

Smetana²,

Maat³

2022

Preprint

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Fully discrete heterogeneous multiscale method for parabolic problems with multiple spatial and temporal scales

Eckhardt

Verfürth

2023

Bit Numer Math

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The aim of this work is the numerical homogenization of a parabolic problem with several time and spatial scales using the heterogeneous multiscale method. We replace the actual cell problem with an alternate one, using Dirichlet boundary and initial values instead of periodic boundary and time conditions. Further, we give a detailed a priori error analysis of the fully discretized, i.e., in space and time for both the macroscopic and the cell problem, method. Numerical experiments illustrate the theoretical convergence rates.

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