Kronecker product approximation preconditioners for convection–diffusion model problems

Xiang, Hua; Grigori, Laura

doi:10.1002/nla.666

Cited by 11 publications

(3 citation statements)

References 30 publications

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“…Remark. The derivation of the Laplace-like system from the CP decomposition can be extended from the Helmholtz equation to convection diffusion problems of the form [4,10,65]. For these problems we define the CP-decomposition…”

Section: Helmholtz Problemsmentioning

confidence: 99%

Fast global spectral methods for three-dimensional partial differential equations

Strössner,

Kressner

2021

Preprint

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Global spectral methods offer the potential to compute solutions of partial differential equations numerically to very high accuracy. In this work, we develop a novel global spectral method for linear partial differential equations on cubes by extending ideas of Chebop2 [Townsend and Olver, J. Comput. Phys., 299 (2015)] to the three-dimensional setting utilizing expansions in tensorized polynomial bases. Solving the discretized PDE involves a linear system that can be recast as a linear tensor equation. Under suitable additional assumptions, the structure of these equations admits for an efficient solution via the blocked recursive solver [Chen and Kressner, Numer. Algorithms, 84 (2020)]. In the general case, when these assumptions are not satisfied, this solver is used as a preconditioner to speed up computations.

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Section: Helmholtz Problemsmentioning

confidence: 99%

Fast global spectral methods for three-dimensional partial differential equations

Strössner,

Kressner

2021

Preprint

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“…The discretization of the partial differential equation

\{\begin{array}{ll} prefix- normalΔ u + c^{T} normal∇ u = f, & in normalΩ = {false[0, 1 false]}^{N}, \\ u = 0, & on \partial normalΩ \end{array}

leads to the following Sylvester tensor equation via the Tucker product 12,13

𝒳 \times_{1} A^{(1)} + 𝒳 \times_{2} A^{(2)} + \dots + 𝒳 \times_{N} A^{(N)} = 𝒟,

where the matrices

A^{(i)} \in ℝ^{I_{i} \times I_{i}} (i = 1, 2, \dots, N)

and the right‐hand side tensor

𝒟 \in ℝ^{I_{1} \times I_{2} \times \dots \times I_{N}}

are known, and

𝒳 \in ℝ^{I_{1} \times I_{2} \times \dots \times I_{N}}

is the unknown tensor to be determined.…”

Section: Introductionmentioning

confidence: 99%

Numerical subspace algorithms for solving the tensor equations involving Einstein product

Huang

2020

Numerical Linear Algebra App

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In this article, we propose some subspace methods such as the conjugate residual, generalized conjugate residual, biconjugate gradient, conjugate gradient squared and biconjugate gradient stabilized methods based on the tensor forms for solving the tensor equation involving the Einstein product. These proposed algorithms keep the tensor structure. The convergence analysis shows that the proposed methods converge to the solution of the tensor equation for any initial value. Some numerical results confirm the feasibility and applicability of the proposed algorithms in practice.

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“…The system (1.1) arises from many areas of computational sciences and engineerings, for example, in certain finite element and finite difference discretization of Navier-Stokes equations, Oseen equations, and mixed finite element discretization of second order convection-diffusion problems (cf. [3,8,13,14,16,18,19]). For the saddle point problem, there exist many algorithms, for example, the Krylov iteration methods with block diagonal, triangular block or constraint preconditioners (see [3] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Uzawa algorithms with variable relaxation for nonsymmetric generalized saddle point problems

Shen

Xiang

2015

Numerical Linear Algebra App

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In this paper, we extend the inexact Uzawa algorithm in [Q. Hu, J. Zou, SIAM J. Matrix Anal., 23(2001), pp. 317-338] to the nonsymmetric generalized saddle point problem. The techniques used here are similar to those in [Bramble et al, Math. Comput. 69(1999), pp. 667-689], where the convergence of Uzawa type algorithm for solving nonsymmetric case depends on the spectrum of the preconditioners involved. The main contributions of this paper focus on a new linear Uzawa type algorithm for nonsymmetric generalized saddle point problems and its convergence. This new algorithm can always converge without any prior estimate on the spectrum of two preconditioned subsystems involved, which may not be easy to achieve in applications. Numerical results of the algorithm on the Navier-Stokes problem are also presented.

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Kronecker product approximation preconditioners for convection–diffusion model problems

Cited by 11 publications

References 30 publications

Fast global spectral methods for three-dimensional partial differential equations

Fast global spectral methods for three-dimensional partial differential equations

Numerical subspace algorithms for solving the tensor equations involving Einstein product

Uzawa algorithms with variable relaxation for nonsymmetric generalized saddle point problems

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