Rational Krylov methods for functions of matrices with applications to fractional partial differential equations

Abstract

In this paper we propose a new choice of poles to define reliable rational Krylov methods. These methods are used for approximating function of positive definite matrices. In particular, the fractional power and the fractional resolvent are considered because of their importance in the numerical solution of fractional partial differential equations. The numerical experiments on some fractional partial differential equation models confirm that the proposed approach is promising.

Nonlocal diffusion of variable order on graphs

“…The second is to (i) extend the approach in [1] by working with a variable order integrator and the singular M-matrix L, by applying some of the techniques proposed in [6] and (ii) use computationally efficient techniques to solve (4.1) extending those proposed in [8].…”

“…For symmetric positive definite matrices there exist several efficient approaches, either based on various type of quadrature formulas [2,3,7,21,23], or on Krylov methods [1,25,26]. The case we want to deal with here needs an extra care because of the presence of the zero eigenvalue in L for which we refer to the strategies introduced in [6].…”

Exponentially convergent trapezoidal rules to approximate fractional powers of operators

“…Here L is a self-adjoint positive operator acting in an Hilbert space H in which the eigenfunctions of L form an orthonormal basis of H, so that L −α can be written through the spectral decomposition of L. In other words, for a given g ∈ H, we have (1) L −α g = +∞ j=1…”

“…Among the approaches recently introduced we quote here the methods based on the best uniform rational approximations of functions closely related to λ −α that have been studied in [6,7,8,9]. Another class of methods relies on quadrature rules arising from the Dunford-Taylor integral representation of λ −α [1,2,3,4,5,17,18]. Very recently, time stepping methods for a parabolic reformulation of fractional diffusion equations, proposed in [19], have been interpreted by Hofreither in [10] as rational approximations of λ −α .…”

Padé-type Approximations to the Resolvent of Fractional Powers of Operators

“…We experimentally show that using this new parameter sequence it is possible to improve the approximation attainable by taking τ k as in [1] for L −α and then using (1.1) to compute (I + hL α ) −1 . The latter approach has recently been used in [4]. In the applications, where one works with a discretization L N of L, if the largest eigenvalue λ N of L N (or an approximation of it) is known, then the theory developed for the unbounded case can be refined.…”