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

Abstract

We study a reliable pole selection for the rational approximation of the resolvent of fractional powers of operators in both the finite and infinite dimensional setting. The analysis exploits the representation in terms of hypergeometric functions of the error of the Padé approximation of the fractional power. We provide quantitatively accurate error estimates that can be used fruitfully for practical computations. We present some numerical examples to corroborate the theoretical results. The behavior of the rational Krylov methods based on this theory is also presented.

A Gaussian method for the operator square root

“…Assuming that F (L) ⊂ Σ β,a , a > 0, we define τ (depending on n) such that the rational approximation is still reliable for L − 1 2 . Similar approaches, based on quadrature rules arising from the Dunford-Taylor integral representation of λ −α , have been considered for instance in [1,3,2,4,7,8]. Other methods that rely on the best uniform rational approximations of functions closely related to λ −α have been treated in [10,11,13,14].…”

Exponentially convergent trapezoidal rules to approximate fractional powers of operators

“…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 .…”

“…In this work, starting from the integral representation (2) L −α = 2 sin(απ) π +∞ 0 t 2α−1 (I + t 2 L) −1 dt, α ∈ (0, 1),…”