Rank Structured Approximation Method for Quasi-Periodic Elliptic Problems

Khoromskij, Boris N.; Repin, Sergey

doi:10.1515/cmam-2017-0014

Cited by 13 publications

(8 citation statements)

References 26 publications

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“…The theoretical analysis of quisi-periodic and stochastic/parametric problems can be found in [3,29,18,10,7] and in references therein. The rank structured tensor methods for quasi-periodic geometric homogenization methods and for the elliptic equations with highly oscillating coefficients were considered in [27,20]. Data sparse and tensor methods for stochastic/parametric elliptic problems have been considered in [26,28,25,33,8,9].…”

Section: Introductionmentioning

confidence: 99%

Tensor-based techniques for fast discretization and solution of 3D elliptic equations with random coefficients

Khoromskaia¹,

Khoromskij²

2020

Preprint

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In this paper, we propose and analyze the numerical algorithms for fast solution of periodic elliptic problems in random media in R d , d = 2, 3. We consider the stochastic realizations using checkerboard configuration of the equation coefficients built on a large L × L × L lattice, where L is the size of representative volume elements. The Kronecker tensor product scheme is introduced for fast generation of the stiffness matrix for FDM discretization on a tensor grid. We describe tensor techniques for the construction of the low Kronecker rank spectrally equivalent preconditioner in periodic setting to be used in the framework of PCG iteration. In our construction the diagonal matrix of the discrete Laplacian inverse represented in the Fourier basis is reshaped into a 3D tensor, which is then approximated by a low-rank canonical tensor, calculated by the multigrid Tucker-to-canonical tensor transform. The FDM discretization scheme on a tensor grid is described in detail, and the computational characteristics in terms of L for the 3D Matlab implementation of the PCG iteration are illustrated. The present work continues the developments in [22], where the numerical primer to study the asymptotic convergence rate vs. L for the homogenized matrix for 2D elliptic PDEs with random coefficients was investigated numerically. The presented elliptic problem solver can be applied for calculation of long sequences of stochastic realizations in numerical analysis of 3D stochastic homogenization problems for ergodic processes, for solving 3D quasi-periodic geometric homogenization problems, as well as in the numerical simulation of dynamical many body interaction processes and multi-particle electrostatics.

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Section: Introductionmentioning

confidence: 99%

Tensor-based techniques for fast discretization and solution of 3D elliptic equations with random coefficients

Khoromskaia¹,

Khoromskij²

2020

Preprint

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“…For the above constructions, which apply to any dimension d, we are able to prove the following storage complexity and Kronecker rank estimates for the stiffness matrix A. In the following lemma we consider the case of rather general coefficient such that the number of cells in (17), K, is large enough. In this case the stiffness matrix is stored in sparse data format by applying the fast algorithm by using a sum of K rank-1 Kronecker terms, where each triple of three-diagonal matrices in every rank-1 terms is stored in sparse format.…”

Section: Fast Matrix Assembling For the Stochastic Part In The Stiffn...mentioning

confidence: 99%

Fast solution of three‐dimensional elliptic equations with randomly generated jumping coefficients by using tensor‐structured preconditioners

Khoromskij

Khoromskaia

2022

Numerical Linear Algebra App

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In this paper, we propose and analyze the numerical algorithms for fast solution of periodic elliptic problems in random media in ℝd$$ {\mathbb{R}}^d $$, d=2,3$$ d=2,3 $$. Both the two‐dimensional (2D) and three‐dimensional (3D) elliptic problems are considered for the jumping equation coefficients built as a checkerboard type configuration of bumps randomly distributed on a large Lprefix×L$$ L\times L $$, or Lprefix×Lprefix×L$$ L\times L\times L $$ lattice, respectively. The finite element method discretization procedure on a 3D nprefix×nprefix×n$$ n\times n\times n $$ uniform tensor grid is described in detail, and the Kronecker tensor product approach is proposed for fast generation of the stiffness matrix. We introduce tensor techniques for the construction of the low Kronecker rank spectrally equivalent preconditioner in a periodic setting to be used in the framework of the preconditioned conjugate gradient iteration. The discrete 3D periodic Laplacian pseudo‐inverse is first diagonalized in the Fourier basis, and then the diagonal matrix is reshaped into a fully populated third‐order tensor of size nprefix×nprefix×n$$ n\times n\times n $$. The latter is approximated by a low‐rank canonical tensor by using the multigrid Tucker‐to‐canonical tensor transform. As an example, we apply the presented solver in numerical analysis of stochastic homogenization method where the 3D elliptic equation should be solved many hundred times, and where for every random sampling of the equation coefficient one has to construct the new stiffness matrix and the right‐hand side. The computational characteristics of the presented solver in terms of a lattice parameter L$$ L $$ and the grid‐size, nd$$ {n}^d $$, in both 2D and 3D cases are illustrated in numerical tests. Our solver can be used in various applications where the elliptic problem should be solved for a number of different coefficients for example, in many‐particle dynamics, protein docking problems or stochastic modeling.

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“…In general, the applications of low-rank approximations are very broad, e.g. for stochastic problems in higher dimensions [40,36,41,42], acceleration of solutions to PDEs [43,44], or model order reduction [45], but its application to FFT-based homogenisation is new. However, an alternative low-rank representation has been studied recently in [46].…”

Section: Introductionmentioning

confidence: 99%

FFT-based homogenisation accelerated by low-rank tensor approximations

Vondřejc

Liu

Ladecký

et al. 2020

Computer Methods in Applied Mechanics and Engineering

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Fast Fourier transform (FFT) based methods have turned out to be an effective computational approach for numerical homogenisation. In particular, Fourier-Galerkin methods are computational methods for partial differential equations that are discretised with trigonometric polynomials. Their computational effectiveness benefits from efficient FFT based algorithms as well as a favourable condition number. Here these kind of methods are accelerated by low-rank tensor approximation techniques for a solution field using canonical polyadic, Tucker, and tensor train formats. This reduced order model also allows to efficiently compute suboptimal global basis functions without solving the full problem. It significantly reduces computational and memory requirements for problems with a material coefficient field that admits a moderate rank approximation. The advantages of this approach against those using full material tensors are demonstrated using numerical examples for the model homogenisation problem that consists of a scalar linear elliptic variational problem defined in two and three dimensional settings with continuous and discontinuous heterogeneous material coefficients. This approach opens up the potential of an efficient reduced order modelling of large scale engineering problems with heterogeneous material.

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Rank Structured Approximation Method for Quasi-Periodic Elliptic Problems

Cited by 13 publications

References 26 publications

Tensor-based techniques for fast discretization and solution of 3D elliptic equations with random coefficients

Tensor-based techniques for fast discretization and solution of 3D elliptic equations with random coefficients

Fast solution of three‐dimensional elliptic equations with randomly generated jumping coefficients by using tensor‐structured preconditioners

FFT-based homogenisation accelerated by low-rank tensor approximations

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