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On the Inversion of Polynomials of Discrete Laplace Matrices

Sabia Asghar, Qiyao Peng, Fred Vermolen, Cornelis Vuik

TL;DR

Problem: efficient inversion of matrix polynomials in discretized Laplacians; Approach: spectral (eigenvector-eigenvalue) expansion of a symmetric positive definite $A$ yields closed-form expressions for $A^{-1}$ and $P(A)^{-1}$, including time-stepping extensions. Contributions: a general inverses-principle for matrix polynomials, exact 1D/2D/3D case studies via tensor-product eigenstructure, and fast implicit time-integration formulations. Impact: enables fast, exact spectral solutions for large linear systems from PDE discretizations and provides a foundation for extensions to higher dimensions and more complex time schemes. The work connects discrete and continuous spectral perspectives and offers practical closed-form tools for a broad class of Laplacian-based problems.

Abstract

The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on eigenvector and eigenvalue expansions. The method is consistent with previous expressions of the inverse discretized Laplacian in one spatial dimension \citep{Vermolen_2022}. Several examples are given.

On the Inversion of Polynomials of Discrete Laplace Matrices

TL;DR

Problem: efficient inversion of matrix polynomials in discretized Laplacians; Approach: spectral (eigenvector-eigenvalue) expansion of a symmetric positive definite yields closed-form expressions for and , including time-stepping extensions. Contributions: a general inverses-principle for matrix polynomials, exact 1D/2D/3D case studies via tensor-product eigenstructure, and fast implicit time-integration formulations. Impact: enables fast, exact spectral solutions for large linear systems from PDE discretizations and provides a foundation for extensions to higher dimensions and more complex time schemes. The work connects discrete and continuous spectral perspectives and offers practical closed-form tools for a broad class of Laplacian-based problems.

Abstract

The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on eigenvector and eigenvalue expansions. The method is consistent with previous expressions of the inverse discretized Laplacian in one spatial dimension \citep{Vermolen_2022}. Several examples are given.

Paper Structure

This paper contains 9 sections, 3 theorems, 94 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The eigenvector expansion technique
  3. The Principle for Inverses of Matrix Polynomials
  4. Case Studies
  5. One dimensional Laplacian matrix
  6. Two dimensional Laplace on a rectangle
  7. Higher dimensional Laplace matrix on a (hyper) cube
  8. Time Dependent Problems
  9. Discussion and Conclusions

Key Result

Theorem 1

Let $A$ be an $n \times n$ matrix over $\mathbb{R}$ with eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n$ (counting algebraic multiplicities). For any polynomial the matrix $P(A)$ has eigenvalues $P(\lambda_i)$ for $i = 1,\ldots,n$.

Figures (5)

  • Figure 1: Solutions to the equation $-u" + \alpha u = 1$ for different $\alpha$-values $\alpha = 0, 1, 10, 100, 1000$.
  • Figure 2: Solutions to the Poisson equation $-\Delta u = 1$ with $u|_{\partial \Omega} = 0$. The top figure represents the solution obtained by the current procedure, the bottom figure represents the solution by classical solution methods.
  • Figure 3: Solutions to the fourth-order equation $\Delta^2 u - \Delta u + u = 1$ with $u|_{\partial \Omega} = 0$ and $\frac{\partial u}{\partial n}|_{\partial \Omega}=0$. The top figure represents the solution obtained by the current procedure, the bottom figure represents the solution by classical solution methods.
  • Figure 4: Snapshot at consecutive times of the heat equation $u_t - u_{xx}=0$ with $u|_{\partial \Omega} = 0$ and $u = 1$ at $t = 0$ using the backward Euler time-integration method for different $\tau$-values $\tau = 1, 10, 100, 1000, 10000$.
  • Figure 5: Computation times (wall clock times) using tic-toc in Matlab for a second and fourth order PDE.

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Corollary 1: Extension to Diagonalizable Matrices
  • Theorem 2
  • proof