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An error control framework for computing the exponential of matrices arising from the finite element discretization

Fuminori Tatsuoka, Yuto Miyatake, Tomohiro Sogabe

Abstract

Several methods for computing the action of the matrix exponential $\mathrm{e}^{\boldsymbol{A}} \boldsymbol{b}$ are expressed by substituting $\boldsymbol{A}$ into a rational approximation of the scalar exponential function. The error of such methods can be estimated using the numerical range of $\boldsymbol{A}$, which enables the computation of $\mathrm{e}^{\boldsymbol{A}}\boldsymbol{b}$ with a prescribed accuracy. However, when the input matrix has the structure $\boldsymbol{A} = τ\boldsymbol{M}^{-1} \boldsymbol{K}$, this approach is challenging because computing the bounding box of numerical range is difficult and the numerical range may be too large to construct rational approximations on it. In this paper, focusing on the case where $\boldsymbol{M}$ is a well-conditioned symmetric positive definite matrix, we propose considering the numerical range of a similarity transformed matrix of $\boldsymbol{A}$. The numerical range of transformed matrix is not only numerically computable but can also be theoretically bounded depending on properties of $\boldsymbol{K}$. Numerical experiments confirm that the computations can be performed within the prescribed error tolerance.

An error control framework for computing the exponential of matrices arising from the finite element discretization

Abstract

Several methods for computing the action of the matrix exponential are expressed by substituting into a rational approximation of the scalar exponential function. The error of such methods can be estimated using the numerical range of , which enables the computation of with a prescribed accuracy. However, when the input matrix has the structure , this approach is challenging because computing the bounding box of numerical range is difficult and the numerical range may be too large to construct rational approximations on it. In this paper, focusing on the case where is a well-conditioned symmetric positive definite matrix, we propose considering the numerical range of a similarity transformed matrix of . The numerical range of transformed matrix is not only numerically computable but can also be theoretically bounded depending on properties of . Numerical experiments confirm that the computations can be performed within the prescribed error tolerance.
Paper Structure (9 sections, 3 theorems, 14 equations, 6 figures, 4 tables, 2 algorithms)

This paper contains 9 sections, 3 theorems, 14 equations, 6 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Error control framework
  3. Numerical examples
  4. Test problems
  5. Computational methods of the matrix exponential
  6. Results for checking mathematical validity
  7. Results for
  8. Results for the large matrices
  9. Conclusion

Key Result

Theorem 1

For $\bm{A} = \tau \bm{M}^{-1}\bm{K}$, where $\tau > 0$, $\bm{M} \in \mathbb{R}^{n\times n}$ is an SPD matrix, and $\bm{K} \in \mathbb{R}^{n\times n}$, define $\hat{\bm{A}} = \bm{M}^{1/2}\bm{A} \bm{M}^{-1/2}$. Then, for any rational function $r(z)$ that is bounded on $\mathcal{W}(\hat{\bm{A}})$, it where $\kappa(\bm{M}) = \|\bm{M}\|_2\|\bm{M}^{-1}\|_2$.

Figures (6)

  • Figure 1: Comparison of $\mathcal{W}(\bm{A})$ and $\mathcal{W}(\hat{\bm{A}})$. The matrix $\bm{A}$ is a test matrix described in Section 3.1, obtained by discretizing the two dimensional advection–diffusion equation on a square domain with a P2 finite element method. The set $\mathcal{W}(\hat{\bm{A}})$ lies in the left half plane.
  • Figure 2: The square domain and the star-shaped domain we used.
  • Figure 3: Eigenvalues and numerical ranges $\mathcal{W}(\bm{A})$, $\mathcal{W}(\hat{\bm{A}})$ for the test matrices in Tab. \ref{['tab:test_matrices']}. Black dots represent eigenvalues, while blue and orange regions denote $\mathcal{W}(\bm{A})$ and $\mathcal{W}(\hat{\bm{A}})$, respectively. For these matrices, $\mathcal{W}(\hat{\bm{A}})$ is contained within $\mathcal{W}(\bm{A})$.
  • Figure 4: Errors of each method for $\tau = \bar{h}$. Each point represents the result for one test matrix. Although overestimations occur, the error is smaller than the tolerance for all matrices and methods.
  • Figure 5: Approximation errors for $\tau = 10\bar{h}$. Each point represents the result for one test matrix.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Remark 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof