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Absolute-value based preconditioner for complex-shifted Laplacian systems

Xuelei Lin, Congcong Li, Sean Hon

TL;DR

The paper addresses efficiently solving complex-shifted Laplacian systems by reformulating them as a $2\times 2$ block real system and applying an absolute-value based preconditioner. For constant coefficients on uniform grids, the preconditioner is rapidly invertible via the discrete sine transform, while for general coefficients an averaged coefficient yields a practical preconditioner with spectral bounds that are independent of mesh size, leading to matrix-size independent convergence of MINRES. Theoretical results establish that the ideal preconditioner restricts eigenvalues to $\{-1,1\}$, giving exact two-step convergence, and the practical preconditioner has provable spectral bounds ensuring robust performance under specified shift conditions. Numerical experiments demonstrate strong performance on both constant and variable coefficient problems, showing improved efficiency and robustness compared to established preconditioners such as multigrid with Vanka smoothing and GMRES-based methods, and indicating near $O(M\log M)$ complexity on uniform grids. The work provides a solid foundation for fast, robust solvers for Helmholtz-like equations and points toward domain-decomposition extensions for irregular domains.

Abstract

The complex-shifted Laplacian systems arising in a wide range of applications. In this work, we propose an absolute-value based preconditioner for solving the complex-shifted Laplacian system. In our approach, the complex-shifted Laplacian system is equivalently rewritten as a $2\times 2$ block real linear system. With the Toeplitz structure of uniform-grid discretization of the constant-coefficient Laplacian operator, the absolute value of the block real matrix is fast invertible by means of fast sine transforms. For more general coefficient function, we then average the coefficient function and take the absolute value of the averaged matrix as our preconditioner. With assumptions on the complex shift, we theoretically prove that the eigenvalues of the preconditioned matrix in absolute value are upper and lower bounded by constants independent of matrix size, indicating a matrix-size independent linear convergence rate of MINRES solver. Interestingly, numerical results show that the proposed preconditioner is still efficient even if the assumptions on the complex shift are not met. The fast invertibility of the proposed preconditioner and the robust convergence rate of the preconditioned MINRES solver lead to a linearithmic (nearly optimal) complexity of the proposed solver. The proposed preconditioner is compared with several state-of-the-art preconditioners via several numerical examples to demonstrate the efficiency of the proposed preconditioner.

Absolute-value based preconditioner for complex-shifted Laplacian systems

TL;DR

The paper addresses efficiently solving complex-shifted Laplacian systems by reformulating them as a block real system and applying an absolute-value based preconditioner. For constant coefficients on uniform grids, the preconditioner is rapidly invertible via the discrete sine transform, while for general coefficients an averaged coefficient yields a practical preconditioner with spectral bounds that are independent of mesh size, leading to matrix-size independent convergence of MINRES. Theoretical results establish that the ideal preconditioner restricts eigenvalues to , giving exact two-step convergence, and the practical preconditioner has provable spectral bounds ensuring robust performance under specified shift conditions. Numerical experiments demonstrate strong performance on both constant and variable coefficient problems, showing improved efficiency and robustness compared to established preconditioners such as multigrid with Vanka smoothing and GMRES-based methods, and indicating near complexity on uniform grids. The work provides a solid foundation for fast, robust solvers for Helmholtz-like equations and points toward domain-decomposition extensions for irregular domains.

Abstract

The complex-shifted Laplacian systems arising in a wide range of applications. In this work, we propose an absolute-value based preconditioner for solving the complex-shifted Laplacian system. In our approach, the complex-shifted Laplacian system is equivalently rewritten as a block real linear system. With the Toeplitz structure of uniform-grid discretization of the constant-coefficient Laplacian operator, the absolute value of the block real matrix is fast invertible by means of fast sine transforms. For more general coefficient function, we then average the coefficient function and take the absolute value of the averaged matrix as our preconditioner. With assumptions on the complex shift, we theoretically prove that the eigenvalues of the preconditioned matrix in absolute value are upper and lower bounded by constants independent of matrix size, indicating a matrix-size independent linear convergence rate of MINRES solver. Interestingly, numerical results show that the proposed preconditioner is still efficient even if the assumptions on the complex shift are not met. The fast invertibility of the proposed preconditioner and the robust convergence rate of the preconditioned MINRES solver lead to a linearithmic (nearly optimal) complexity of the proposed solver. The proposed preconditioner is compared with several state-of-the-art preconditioners via several numerical examples to demonstrate the efficiency of the proposed preconditioner.
Paper Structure (6 sections, 8 theorems, 45 equations)

This paper contains 6 sections, 8 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Theoretical results
  3. Spectral analysis of the ideal preconditioner $\mathcal{\tilde{A}}^{-1}\mathcal{A}$
  4. Spectral analysis of $\mathcal{P}^{-1}\mathcal{A}$
  5. Implementations
  6. Numerical experiments

Key Result

lemma thmcounterlemma

elman2014finite Let ${\bf P}\in\mathbb{R}^{N\times N}$ and ${\bf A}\in\mathbb{R}^{N\times N}$ be a symmetric positive definite matrix and a symmetric nonsingular matrix, respectively. Suppose $\Sigma({\bf P}^{-1}{\bf A})\in[-a_1,-a_2]\cup[a_3,a_4]$ with $a_4\geq a_3>0$, $a_1\geq a_2>0$ and $a_1-a_2= where ${\bf r}_{k}={\bf P}^{-1}{\bf y}-{\bf P}^{-1}{\bf A}{\bf x}_k$ denotes the residual vector at

Theorems & Definitions (12)

  • lemma thmcounterlemma
  • theorem 1
  • proof
  • theorem 2
  • proof
  • lemma thmcounterlemma
  • proof
  • proposition thmcounterproposition
  • lemma thmcounterlemma
  • theorem 3
  • ...and 2 more