Rational Krylov for Stieltjes matrix functions: convergence and pole selection

Abstract

Evaluating the action of a matrix function on a vector, that is $$x=f({\mathcal {M}})v$$ x = f ( M ) v , is an ubiquitous task in applications. When $${\mathcal {M}}$$ 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 $${\mathcal {M}}$$ M is a positive definite matrix. Relying on the same tools used to analyze the generic situation, we then focus on the case $${\mathcal {M}}=I \otimes A - B^T \otimes I$$ 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.

Rational Krylov methods for fractional diffusion problems on graphs

“…A pole selection strategy for the evaluation of f (A)b was recently proposed in [24] for the case of a Hermitian positive definite matrix A and a Cauchy-Stieltjes or Laplace-Stieltjes function f . For a matrix A with spectrum contained in the positive interval [a, b], the choice of poles described in [24] gives after k iterations an error…”

“…However, the poles that satisfy the error bound for iteration k + 1 are not obtained by adding a new pole to the ones of iteration k, so in order to effectively use this pole selection strategy one would need to decide a priori the number of iterations to be performed. In order to overcome this drawback, in [24,Section 3.5] the authors use the method of equidistributed sequences (EDS) to construct an infinite sequence of poles with the same asymptotic rate of convergence, that can be more easily used in practice. For the details on the construction of this pole sequence, we refer to the discussion in [24,Section 3.5].…”

“…In order to overcome this drawback, in [24,Section 3.5] the authors use the method of equidistributed sequences (EDS) to construct an infinite sequence of poles with the same asymptotic rate of convergence, that can be more easily used in practice. For the details on the construction of this pole sequence, we refer to the discussion in [24,Section 3.5].…”

“…In this section we observe that the function f (z) = e −tz α is a Laplace-Stieltjes function, and hence we can use the pole sequence proposed in [24] for the fractional diffusion problem (3.2). Even though there are no guarantees on the effectiveness of this pole sequence for general matrices, in our numerical experiments we observed that it provides a good convergence rate even when A is the (singular and nonsymmetric) Laplacian of a directed graph.…”

“…Moreover, since f is not analytic, it is preferable to approximate f (L T )u(0) using rational Krylov methods instead of polynomial methods. In our experiments, in addition to the polynomial Krylov method, we investigate the performance of the Shift-and-Invert Krylov method with two different shifts, and of a rational Krylov method with asymptotically optimal poles presented in [24] for Laplace-Stieltjes functions.…”