Tensorized low-rank circulant preconditioners for multilevel Toeplitz linear systems from high-dimensional fractional Riesz equations

The solution of d-dimensional fractional partial differential equations (FPDEs) is challenging in numerical computation. This paper proposes a low-rank preconditioned conjugate gradient (PCG) method for discretizations with a low-rank right-hand side stemming from the d-dimensional fractional Riesz...

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Bibliographic Details
Published inComputers & mathematics with applications (1987) Vol. 110; pp. 64 - 76
Main Authors Zhang, Lei, Zhang, Guo-Feng, Liang, Zhao-Zheng
Format Journal Article
LanguageEnglish
Published Oxford Elsevier Ltd 15.03.2022
Elsevier BV
Subjects
Circulant preconditioning
Fast Fourier transformations
Fourier transforms
Fractional Riesz equation
Iterative methods
Linear systems
Mathematical analysis
Matrices (mathematics)
Multilevel Toeplitz matrix
Numerical analysis
Partial differential equations
PCG method
Tensor train format
Tensors
Tensor train format
PCG method
Circulant preconditioning
Fractional Riesz equation
Multilevel Toeplitz matrix
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Summary:The solution of d-dimensional fractional partial differential equations (FPDEs) is challenging in numerical computation. This paper proposes a low-rank preconditioned conjugate gradient (PCG) method for discretizations with a low-rank right-hand side stemming from the d-dimensional fractional Riesz equation. At each iteration step, a combination of low-rank in Tensor Train (TT) format technique and Fast Fourier Transform (FFT) is used to accelerate multilevel Toeplitz matrix-vector multiplications. Furthermore, a low-rank circulant preconditioner is constructed based on the spectrum of the coefficient matrix. The reported numerical results demonstrate the competitiveness of the new method compared with a state-of-the-art matrix-free approach based on FFT.
ISSN:0898-1221
1873-7668
DOI:10.1016/j.camwa.2022.01.003