arXiv subjects

Xue-Lei Lin

Publications and source records attributed to Xue-Lei Lin.

4 recordsLinked to original sources

arXiv2024

An efficient preconditioner for evolutionary partial differential equations with $θ$-method in time discretization

In this study, the $θ$-method is used for discretizing a class of evolutionary partial differential equations. Then, we transform the resultant all-at-once linear system and introduce a novel one-sided preconditioner, which can be fast implemented in a parallel-in-time way. By introducing an auxiliary two-sided preconditioned system, we provide theoretical insights into the relationship between the residuals of the generalized minimal residual (GMRES) method when applied to both one-sided and two-sided preconditioned systems. Moreover, we show that the condition number of the two-sided preconditioned matrix is uniformly bounded by a constant that is independent of the matrix size, which in turn implies that the convergence behavior of the GMRES method for the one-sided preconditioned system is guaranteed. Numerical experiments confirm the efficiency and robustness of the proposed preconditioning approach.

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arXiv2024

A $τ$-preconditioner for space fractional diffusion equation with non-separable variable coefficients

In this paper, we study a $τ$-matrix approximation based preconditioner for the linear systems arising from discretization of unsteady state Riesz space fractional diffusion equation with non-separable variable coefficients. The structure of coefficient matrices of the linear systems is identity plus summation of diagonal-times-multilevel-Toeplitz matrices. In our preconditioning technique, the diagonal matrices are approximated by scalar identity matrices and the Toeplitz matrices are approximated by τ-matrices (a type of matrices diagonalizable by discrete sine transforms). The proposed preconditioner is fast invertible through the fast sine transform (FST) algorithm. Theoretically, we show that the GMRES solver for the preconditioned systems has an optimal convergence rate (a convergence rate independent of discretization stepsizes). To the best of our knowledge, this is the first preconditioning method with the optimal convergence rate for the variable-coefficients space fractional diffusion equation. Numerical results are reported to demonstrate the efficiency of the proposed method.

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arXiv2023

A parallel-in-time preconditioner for Crank-Nicolson discretization of a parabolic optimal control problem

In this paper, a fast solver is studied for saddle point system arising from a second-order Crank-Nicolson discretization of an initial-valued parabolic PDE constrained optimal control problem, which is indefinite and ill-conditioned. Different from the saddle point system arising from the first-order Euler discretization, the saddle point system arising from Crank-Nicolson discretization has a dense and non-symmetric Schur complement, which brings challenges to fast solver designing. To remedy this, a novel symmetrization technique is applied to the saddle point system so that the new Schur complement is symmetric definite and the well-known matching-Schur-complement (MSC) preconditioner is applicable to the new Schur complement. Nevertheless, the new Schur complement is still a dense matrix and the inversion of the corresponding MSC preconditioner is not parallel-in-time (PinT) and thus time consuming. For this concern, a modified MSC preconditioner for the new Schur complement system. Our new preconditioner can be implemented in a fast and PinT way via a temporal diagonalization technique. Theoretically, the eigenvalues of the preconditioned matrix by our new preconditioner are proven to be lower and upper bounded by positive constants independent of matrix size and the regularization parameter. With such spectrum, the preconditioned conjugate gradient (PCG) solver for the Schur complement system is proven to have a convergence rate independent of matrix size and regularization parameter. To the best of my knowledge, it is the first time to have an iterative solver with problem-independent convergence rate for the saddle point system arising from Crank-Nicolson discretization of the optimal control problem. Numerical results are reported to show that the performance of the proposed preconditioner.

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arXiv2021

Spectral analysis for preconditioning of multi-dimensional Riesz fractional diffusion equations

In this paper, we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations. The finite difference method is employed to approximate the multi-dimensional Riesz fractional derivatives, which will generate symmetric positive definite ill-conditioned multi-level Toeplitz matrices. The preconditioned conjugate gradient method with a preconditioner based on the sine transform is employed to solve the resulting linear system. Theoretically, we prove that the spectra of the preconditioned matrices are uniformly bounded in the open interval (1/2,3/2) and thus the preconditioned conjugate gradient method converges linearly. The proposed method can be extended to multi-level Toeplitz matrices generated by functions with zeros of fractional order. Our theoretical results fill in a vacancy in the literature. Numerical examples are presented to demonstrate our new theoretical results in the literature and show the convergence performance of the proposed preconditioner that is better than other existing preconditioners.

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