Optimal Solvers for Linear Systems with Fractional Powers of Sparse SPD Matrices

Harizanov, Stanislav; Lazarov, Raytcho; Marinov, Pencho; Margenov, Svetozar; Vutov, Yavor

doi:10.48550/arxiv.1612.04846

Cited by 7 publications

(18 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

View full text Add to dashboard Cite

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.

show abstract

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

View full text Add to dashboard Cite

show abstract

“…However, as the pay-off for higher modeling accuracy in applications, fractional operators in PDEs also imply nonlocality to the given equation, which after discretization leads to dense problem structures resulting in quadratic complexity in the number of degrees of freedom in R d , so that they are hard to handle especially when large grids are considered in many dimensions. As a result, common numerical solution approaches lead to severe problems, such that a number of special techniques has been advocated [26,45,16,27,4].…”

Section: Introductionmentioning

confidence: 99%

“…For an overview about several characterizations of the fractional elliptic operators and the respective algebra see [18,28,26,40,41]. A number of application fields motivating the use of fractional power of elliptic operators, for example in biophysics, mechanics, nonlocal electrostatics and image processing have been discussed in the literature [3,2,15,45,16,27,30]. In such applications control problems arise naturally.…”

Section: Introductionmentioning

confidence: 99%

Tensor Method for Optimal Control Problems Constrained by Fractional 3D Elliptic Operator with Variable Coefficients

Schmitt,

Khoromskij,

Khoromskaia

et al. 2020

Preprint

View full text Add to dashboard Cite

We introduce the tensor numerical method for solving optimal control problems that are constrained by fractional 2D and 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 1D 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 is superior to the approaches based on the traditional linear algebra tools that yield at least 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 PCG iteration in the univariate grid size n.

show abstract

“…The method is shown to generate non-polynomial approximations of A s . The contour integral method of [18] and the extended Krylov method of [23] are then related to rational function approximations of A s , while [21] consider the best uniform rational aproximations of the trasformed function A → A β−s . In general, the approximation properties of these methods depend on the condition number of A and thus computations of extremal eigenvalues are often part of the algorithm.…”

Section: Introductionmentioning

confidence: 99%

Multigrid Methods for Discrete Fractional Sobolev Spaces

Bærland¹,

Kuchta²,

Mardal³

2019

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Coupled multiphysics problems often give rise to interface conditions naturally formulated in fractional Sobolev spaces. Here, both positive-and negative fractionality are common. When designing efficient solvers for discretizations of such problems it would then be useful to have a preconditioner for the fractional Laplacian. In this work, we develop an additive multigrid preconditioner for the fractional Laplacian with positive fractionality, and show a uniform bound on the condition number. For the case of negative fractionality, we re-use the preconditioner developed for the positive fractionality and left-right multiply a regular Laplacian with a preconditioner with positive fractionality to obtain the desired negative fractionality. Implementational issues are outlined in details as the differences between the discrete operators and their corresponding matrices must be addressed when realizing these algorithms in code. We finish with some numerical experiments verifying the theoretical findings.

show abstract

Optimal Solvers for Linear Systems with Fractional Powers of Sparse SPD Matrices

Cited by 7 publications

References 26 publications

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

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

Tensor Method for Optimal Control Problems Constrained by Fractional 3D Elliptic Operator with Variable Coefficients

Multigrid Methods for Discrete Fractional Sobolev Spaces

Contact Info

Product

Resources

About