Stability of sparse space-time finite element discretizations of linear parabolic evolution equations

Andreev, Roman

doi:10.1093/imanum/drs014

Cited by 72 publications

(120 citation statements)

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“…By means of a reduction to a boundary integral equation [20], stable space-time compressive algorithms were constructed in [18]. Well-posed space-time variational formulations and stable a priori Petrov-Galerkin discretizations were derived and implemented in [36,40,4,5], space-time compressive variants in [2,3,6]. Quasi-optimality in spacetime norms and space-time adaptivity were investigated in [49].…”

Section: 4mentioning

confidence: 99%

“…We will content ourselves with the full tensor product subspace X L as in (16), but we mention that space-time compressive discretizations X L and Y L could be used for (19) and for (20), see [3]. Let Φ be a basis for X L , and Ψ a basis for Y L .…”

Section: 4mentioning

confidence: 99%

“…Minimal residual variational formulation. For details and proofs for this section we refer to [3,2]. Contrary to the aforegoing section we now assume that dim Y L ≥ dim X L , both finite but possibly unequal.…”

Section: 4mentioning

confidence: 99%

“…L is the norm of the H-orthogonal projector from V to V L ⊂ V , see [3,Lemma 6.2] and the discussion therein, as well as [10] for a quantitative analysis for finite element subspaces.…”

Section: Parabolic Space-time Preconditionersmentioning

confidence: 99%

“…See [2,3,4,5,6] for further numerical experiments. We briefly comment on the symmetric one (14) in Section 4.2.6.…”

Section: Resolution Of Parabolic Evolution Equationsmentioning

confidence: 99%

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Wavelet-In-Time Multigrid-In-Space Preconditioning of Parabolic Evolution Equations

Andreev¹

2016

SIAM J. Sci. Comput.

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Abstract. Two space-time variational formulations of linear parabolic evolution equations are considered, one is symmetric and elliptic on the trial space while the other is not. In each case, a space-time Petrov-Galerkin discretization using suitable tensor product trial and test functions leads to a large linear system of equations. The well-posedness of this system with respect to parabolic norms induces a canonical preconditioner for the algebraic equations that arise after a choice of basis. For the iterative resolution of this algebraic system with parallelization in the temporal direction we propose a sparse algebraic wavelet-in-time transformation on possibly nonuniform temporal meshes. This transformation approximately block-diagonalizes the preconditioner, and the individual spatial blocks can then be inverted for instance by standard spatial multigrid methods in parallel. The performance of the preconditioner is documented in a series of numerical experiments.

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Section: 4mentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

“…L is the norm of the H-orthogonal projector from V to V L ⊂ V , see [3,Lemma 6.2] and the discussion therein, as well as [10] for a quantitative analysis for finite element subspaces.…”

Section: Parabolic Space-time Preconditionersmentioning

confidence: 99%

“…See [2,3,4,5,6] for further numerical experiments. We briefly comment on the symmetric one (14) in Section 4.2.6.…”

Section: Resolution Of Parabolic Evolution Equationsmentioning

confidence: 99%

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Wavelet-In-Time Multigrid-In-Space Preconditioning of Parabolic Evolution Equations

Andreev¹

2016

SIAM J. Sci. Comput.

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Multilevel preconditioning and low‐rank tensor iteration for space–time simultaneous discretizations of parabolic PDEs

Andreev

Tobler

2014

Numerical Linear Algebra App

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SummaryThis paper addresses the solution of parabolic evolution equations simultaneously in space and time as may be of interest in, for example, optimal control problems constrained by such equations. As a model problem, we consider the heat equation posed on the unit cube in Euclidean space of moderately high dimension. An a priori stable minimal residual Petrov–Galerkin variational formulation of the heat equation in space–time results in a generalized least squares problem. This formulation admits a unique, quasi‐optimal solution in the natural space–time Hilbert space and serves as a basis for the development of space–time compressive solution algorithms. The solution of the heat equation is obtained by applying the conjugate gradient method to the normal equations of the generalized least squares problem. Starting from stable subspace splittings in space and in time, multilevel space–time preconditioners for the normal equations are derived. In order to reduce the complexity of the full space–time problem, all computations are performed in a compressed or sparse format called the hierarchical Tucker format, supposing that the input data are available in this format. In order to maintain sparsity, compression of the iterates within the hierarchical Tucker format is performed in each conjugate gradient iteration. Its application to vectors in the hierarchical Tucker format is detailed. Finally, numerical results in up to five spatial dimensions based on the recently developed htucker toolbox for MATLAB are presented. Copyright © 2014 John Wiley & Sons, Ltd.

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Efficient space‐time adaptivity for parabolic evolution equations using wavelets in time and finite elements in space

Venetië

Westerdiep

2022

Numerical Linear Algebra App

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Considering the space-time adaptive method for parabolic evolution equations we introduced in Stevenson et al., this work discusses an implementation of the method in which every step is of linear complexity. Exploiting the tensor-product structure of the space-time cylinder, the method allows for a family of trial spaces given as spans of wavelets-in-time tensorized with finite element spaces-in-space. On spaces whose bases are indexed by double-trees, we derive an algorithm that applies the resulting bilinear forms in linear complexity.We provide extensive numerical experiments to demonstrate the linear runtime of the resulting adaptive loop.

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Stability of sparse space-time finite element discretizations of linear parabolic evolution equations

Cited by 72 publications

References 0 publications

Wavelet-In-Time Multigrid-In-Space Preconditioning of Parabolic Evolution Equations

Wavelet-In-Time Multigrid-In-Space Preconditioning of Parabolic Evolution Equations

Multilevel preconditioning and low‐rank tensor iteration for space–time simultaneous discretizations of parabolic PDEs

Efficient space‐time adaptivity for parabolic evolution equations using wavelets in time and finite elements in space

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