Tensor product method for fast solution of optimal control problems with fractional multidimensional Laplacian in constraints

Heidel, Gennadij; Khoromskaia, Venera; Khoromskij, Boris N.; Schulz, Volker

doi:10.1016/j.jcp.2020.109865

Cited by 16 publications

(40 citation statements)

References 32 publications

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“…by diagonalizing the discrete anisotropic Laplacian Δ B , see (34), in the Fourier basis. Our construction generalizes the scheme in 44 applied to the case of classical Laplacian; (B) Making use of the direct low K-rank approximation to the reciprocal matrix-valued function…”

Section: Preconditioner In the Low-rank Kronecker Formmentioning

confidence: 99%

“…The typical choice R = 6 was sufficient to provide the uniform convergence rate in various model and discretization parameters. We refer to paper, 44 where results for different preconditioner ranks are presented for the case of 3D fractional Laplacian in constraints. In general, the effective range of R depends on the variations in the coefficients of the individual operators A 𝓁 .…”

Section: Preconditioner Via Diagonalization Of Anisotropic Laplacian (Case A)mentioning

confidence: 99%

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Tensor method for optimal control problems constrained by fractional three‐dimensional elliptic operator with variable coefficients

Schmitt

Khoromskij

Khoromskaia

et al. 2021

Numerical Linear Algebra App

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We introduce the tensor numerical method for solving optimal control problems that are constrained by fractional two-(2D) and three-dimensional (3D) elliptic operators with variable coefficients. We solve the governing equation for the control function which includes a sum of the fractional operator and its inverse, both discretized over large 3D n × n × n spacial grids. Using the diagonalization of the arising matrix-valued functions in the eigenbasis of the one-dimensional Sturm-Liouville operators, we construct the rank-structured tensor approximation with controllable precision for the discretized fractional elliptic operators and the respective preconditioner. The right-hand side in the constraining equation (the optimal design function) is supposed to be represented in a form of a low-rank canonical tensor. Then the equation for the control function is solved in a tensor structured format by using preconditioned CG iteration with the adaptive rank truncation procedure that also ensures the accuracy of calculations, given an 𝜀-threshold. This method reduces the numerical cost for solving the control problem to O(n log n) (plus the quadratic term O(n 2 ) with a small weight), which outperforms traditional approaches with O(n 3 log n) complexity in the 3D case. The storage for the representation of all 3D nonlocal operators and functions involved is also estimated by O(n log n).This essentially outperforms the traditional methods operating with fully populated n 3 × n 3 matrices and vectors in R n 3 . Numerical tests for 2D/3D control problems indicate the almost linear complexity scaling of the rank truncated preconditioned conjugate gradient iteration in the univariate grid size n.

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Section: Preconditioner In the Low-rank Kronecker Formmentioning

confidence: 99%

Section: Preconditioner Via Diagonalization Of Anisotropic Laplacian (Case A)mentioning

confidence: 99%

Tensor method for optimal control problems constrained by fractional three‐dimensional elliptic operator with variable coefficients

Schmitt

Khoromskij

Khoromskaia

et al. 2021

Numerical Linear Algebra App

Self Cite

View full text Add to dashboard Cite

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“…Here, low‐rank variants of iterative schemes are derived, where the iterates are replaced by low‐rank tensors, and the potency of these schemes depends on the derivation of effective preconditioners. Additionally, recent research has been undertaken on tensor‐based schemes for PDE‐constrained optimization problems [409,410], as well as FDE‐constrained optimization problems [411,412]. High‐performance computing: Some problems are so large and complex that they necessitate HPC and heterogeneous architectures. This brings new challenges for iterative methods: memory is usually limited and communication, particularly between nodes and processors, is slow relative to the speed of floating point operations.…”

Section: Preconditioners With “Nonstandard” Goalsmentioning

confidence: 99%

Section: Preconditioners With “Nonstandard” Goalsmentioning

confidence: 99%

Preconditioners for Krylov subspace methods: An overview

Pearson

Pestana

2020

GAMM-Mitteilungen

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When simulating a mechanism from science or engineering, or an industrial process, one is frequently required to construct a mathematical model, and then resolve this model numerically. If accurate numerical solutions are necessary or desirable, this can involve solving large-scale systems of equations. One major class of solution methods is that of preconditioned iterative methods, involving preconditioners which are computationally cheap to apply while also capturing information contained in the linear system. In this article, we give a short survey of the field of preconditioning. We introduce a range of preconditioners for partial differential equations, followed by optimization problems, before discussing preconditioners constructed with less standard objectives in mind.

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“…Other decomposition methods include canonical polyadic (CP), Tucker and hierarchical Tucker decompositions; see [19,20] for more details. Up to now, tensor techniques are applied by more and more researchers to scientific computing [21][22][23][24][25][26].…”

mentioning

confidence: 99%

Fast preconditioned iterative methods for fractional Sturm–Liouville equations

Zhang

Liang

2020

Numerical Methods Partial

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In this paper, we have considered fast solutions of the linear system arising from the fractional Sturm-Liouville problem, whose coefficient matrix contains the product of Toeplitz-like matrices. Based on suitable circulant approximations of the related coefficient matrix, we establish a matching preconditioner of matrix-free form. In theory, the spectrum of the preconditioned matrix is shown to cluster around [1/2, 1), which suggests the fast convergence speed of the proposed preconditioner within Krylov subspace acceleration. In addition, to reduce the computation time and storage, we consider an all-at-once discretized system and explore its low-rank tensor structure and alternating iterative tensor algorithms. Numerical experiments are given to show the effectiveness of the proposed solution techniques compared with some existing methods.

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Tensor product method for fast solution of optimal control problems with fractional multidimensional Laplacian in constraints

Cited by 16 publications

References 32 publications

Tensor method for optimal control problems constrained by fractional three‐dimensional elliptic operator with variable coefficients

Tensor method for optimal control problems constrained by fractional three‐dimensional elliptic operator with variable coefficients

Preconditioners for Krylov subspace methods: An overview

Fast preconditioned iterative methods for fractional Sturm–Liouville equations

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