Rational Krylov for Stieltjes matrix functions: convergence and pole selection

• Published in: BIT Vol. 61; no. 1; pp. 237 - 273
• Main Authors: Massei, Stefano, Robol, Leonardo
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.03.2021; Springer Nature B.V
• Subjects: Approximation; Computational Mathematics and Numerical Analysis; Convergence; Error analysis; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrix methods; Numeric Computing; Subspaces; Tensors; Zolotarev problem; 65F60; 30E20; Pole selection; Rational Krylov; Function of matrices; 65E05; Kronecker sum; Stieltjes functions
• ISSN: 0006-3835; 1572-9125
• DOI: 10.1007/s10543-020-00826-z

Evaluating the action of a matrix function on a vector, that is  x = f ( M ) v , is an ubiquitous task in applications. When  M  is large, one usually relies on Krylov projection methods. In this paper, we provide effective choices for the poles of the rational Krylov method for approximating  x  when  f ( z ) is either Cauchy–Stieltjes or Laplace–Stieltjes (or, which is equivalent, completely monotonic) and  M  is a positive definite matrix. Relying on the same tools used to analyze the generic situation, we then focus on the case  M = I ⊗ A - B T ⊗ I , and  v obtained vectorizing a low-rank matrix; this finds application, for instance, in solving fractional diffusion equation on two-dimensional tensor grids. We see how to leverage tensorized Krylov subspaces to exploit the Kronecker structure and we introduce an error analysis for the numerical approximation of  x . Pole selection strategies with explicit convergence bounds are given also in this case.