[Paper Review] Error Bounds for the Krylov Subspace Methods for Computations of Matrix Exponentials

Hao Wang, Qiang Ye|arXiv (Cornell University)|Mar 23, 2016

Matrix Theory and Algorithms22 references3 citations

TL;DR

This paper presents new a priori and a posteriori error bounds for Krylov subspace methods in computing matrix exponentials $e^{-\tau A}v$, using the eigenvalues of the Hermitian and skew-Hermitian parts of $A$ to define a rectangular field of values enclosure. The bounds explain superlinear convergence and initial stagnation, significantly improving on classical bounds by Saad and Hochbruch-Lubich.

ABSTRACT

In this paper, we present new a posteriori and a priori error bounds for the Krylov subspace methods for computing $e^{-τA}v$ for a given $τ>0$ and $v \in C^n$, where $A$ is a large sparse non-Hermitian matrix. The {\em a priori} error bounds relate the convergence to $λ_{\min}\left(\frac{A+A^*}{2} ight)$, $λ_{\max}\left(\frac{A+A^*}{2} ight)$ (the smallest and the largest eigenvalue of the Hermitian part of $A$) and $|λ_{\max}\left(\frac{A-A^*}{2} ight)|$ (the largest eigenvalue in absolute value of the skew-Hermitian part of $A$), which define a rectangular region enclosing the field of values of $A$. In particular, our bounds explain an observed superlinear convergence behavior where the error may first stagnate for certain iterations before it starts to converge. The special case that $A$ is skew-Hermitian is also considered. Numerical examples are given to demonstrate the theoretical bounds.

Motivation & Objective

• To develop sharper a priori and a posteriori error bounds for Krylov subspace methods in computing $e^{-\tau A}v$ for large sparse non-Hermitian matrices.
• To explain the observed superlinear convergence behavior, including initial stagnation, through spectral properties of the matrix's Hermitian and skew-Hermitian parts.
• To improve upon existing bounds—particularly those by Saad and Hochbruch-Lubich—by using a rectangular enclosure of the field of values instead of circular or norm-based estimates.
• To provide computable and sharp a posteriori error estimates based on decay properties of banded matrix functions.

Proposed method

• Derive a posteriori error bounds using the decay properties of functions of banded matrices, enabling sharp and computable error estimation.
• Construct Faber polynomial approximations of the exponential function over a rectangular region enclosing the field of values of $A$, defined by $a = \lambda_{\min}\left(\frac{A+A^*}{2}\right)$, $b = \lambda_{\max}\left(\frac{A+A^*}{2}\right)$, and $c = \left|\lambda_{\max}\left(\frac{A-A^*}{2}\right)\right|$.
• Utilize Jacobi elliptic functions to create a conformal mapping for the rectangular region, which is then simplified to yield tractable error bounds.
• Establish simplified a priori bounds for non-Hermitian positive definite and skew-Hermitian matrices, linking convergence speed to the size and shape of the bounding rectangle.
• Validate the bounds numerically using examples with varying $\tau$, matrix types (non-Hermitian positive definite, skew-Hermitian), and Krylov iterations.
• Compare the new bounds with classical bounds (Saad, Hochbruch-Lubich) to demonstrate significant improvements in tightness and accuracy.

Experimental results

Research questions

• RQ1 How can a priori error bounds for Krylov subspace methods in matrix exponential computation be improved by using the field of values' rectangular enclosure?
• RQ2 Why does the error in Krylov approximations sometimes exhibit initial stagnation before converging, and can this be explained by spectral properties of $A$?
• RQ3 Can a posteriori error bounds be constructed that are both sharp and computable using banded matrix decay properties?
• RQ4 How do the new bounds compare quantitatively with existing bounds by Saad and Hochbruch-Lubich in terms of tightness and practical utility?
• RQ5 To what extent do the bounds explain convergence behavior for skew-Hermitian matrices, where solutions conserve norm?

Key findings

• The a priori error bounds are derived in terms of the extreme eigenvalues of the Hermitian and skew-Hermitian parts of $A$, providing a geometric interpretation via the field of values rectangle.
• The bounds explain the observed superlinear convergence and initial stagnation: convergence typically begins at $k \approx 2\tau\rho$, where $\rho$ is the spectral gap of the Hermitian part.
• For $\tau = 50$, the a priori bound remains significantly tighter than Saad’s norm-based bound, which becomes overly pessimistic due to the large $\|\tau A\|$.
• The a posteriori error estimate is sharp across all test cases, consistently tracking the true error in both non-Hermitian positive definite and skew-Hermitian examples.
• The new a priori bound improves substantially on Hochbruch and Lubich’s circular-enclosure bound, especially for large $\tau$, due to the tighter rectangular enclosure of the field of values.
• In the skew-Hermitian case ($A = iH$), the bounds match the observed convergence curve closely, with stagnation followed by rapid decay, and the predicted onset of convergence at $k \approx 2\tau\rho$ aligning with numerical results.