Fast tensor product solvers for optimization problems with fractional differential equations as constraints

Dolgov, Sergey; Pearson, John W.; Savostyanov, Dmitry; Stoll, Martin

doi:10.1016/j.amc.2015.09.042

Cited by 37 publications

(28 citation statements)

References 74 publications

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“…This system contains the necessary first-order optimality conditions for the minimizers solving the discretized optimal control problem ( 12), (13).…”

Section: Discrete Optimality Conditionsmentioning

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

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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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“…This system contains the necessary first-order optimality conditions for the minimizers solving the discretized optimal control problem ( 12), (13).…”

Section: Discrete Optimality Conditionsmentioning

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

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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%

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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“…Recently the numerical analysis of optimal control problems constrained by spacefractional elliptic operators was intensively discusses in the literature [2,21,12,18,29,3,5,10], see also papers on the discretization and analysis of fractional PDEs [13,17] and [28,27,4].…”

Section: Introductionmentioning

confidence: 99%

Tensor numerical method for optimal control problems constrained by an elliptic operator with general rank-structured coefficients

Khoromskij,

Schmitt,

Schulz

2021

Preprint

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We introduce tensor numerical techniques for solving optimal control problems constrained by elliptic operators in R d , d = 2, 3, with variable coefficients, which can be represented in a low rank separable form. We construct a preconditioned iterative method with an adaptive rank truncation for solving the equation for the control function, governed by a sum of the elliptic operator and its inverse M = A + A −1 , both discretized over large n ⊗d , d = 2, 3, spatial grids. Two basic solution schemes are proposed and analyzed. In the first approach, one solves iteratively the initial linear system of equations with the matrix M such that the matrix vector multiplication with the elliptic operator inverse, y = A −1 u, is performed as an embedded iteration by using a rank-structured solver for the equation of the form Ay = u. The second numerical scheme avoids the embedded iteration by reducing the initial equation to an equivalent one with the polynomial system matrix of the form A 2 + I. For both schemes, a low Kronecker rank spectrally equivalent preconditioner is constructed by using the corresponding matrix valued function of the anisotropic Laplacian diagonalized in the Fourier basis. Numerical tests for control problems in 2D setting confirm the linearquadratic complexity scaling of the proposed method in the univariate grid size n. Further, we numerically demonstrate that for our low rank solution method, a cascadic multigrid approach reduces the number of PCG iterations considerably, however the total CPU time remains merely the same as for the unigrid iteration.

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Fast tensor product solvers for optimization problems with fractional differential equations as constraints

Cited by 37 publications

References 74 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

Tensor numerical method for optimal control problems constrained by an elliptic operator with general rank-structured coefficients

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