Padé-type approximations to the resolvent of fractional powers of operators

TL;DR

The paper develops a reliable $(k-1,k)$ Padé-type rational approximation to the resolvent $(I+h\mathcal{L}^{\alpha})^{-1}$ for self-adjoint positive operators with $0<\alpha<1$, by constructing $\mathcal{S}_{k-1,k}(\mathcal{L})$ via a scale parameter $\tau$ that minimizes the worst-case error over $\lambda\in[c,\infty)$. It leverages a Gauss-Jacobi quadrature-based representation of $\mathcal{L}^{-\alpha}$ to obtain an explicit partial-fraction form with negative real poles, and derives a minimax-based rule for $\tau$, showing a sublinear $O(k^{-4\alpha})$ convergence in the unbounded case and a linear-like decay in the bounded/discretized case with $\tau_{k,N}$. The analysis yields sharp error estimates that guide the a priori choice of the number of poles (or inversions) and supports the construction of rational Krylov methods using the resulting poles. Numerical experiments on diagonal and Laplacian operators corroborate the theory and illustrate practical performance, including the applicability to rational Krylov subspaces for computing $ (I+h\mathcal{L}_{N}^{\alpha})^{-1}v$. The approach provides a principled framework for efficient, accurate resolvent computations in fractional diffusion contexts.

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.

Figures (5)

• Figure 1: Behavior of the function $g_k$ defined in (\ref{['gk']}) versus $\lambda$ (in logarithmic scale) for $\alpha=0.75, k=15, h=10^{-2},$$\tau_k$ as defined in (\ref{['tauk']}) and $c=1.$
• Figure 2: Error comparison using $\tau _{k}$ taken as in (\ref{['tauk']}) (solide line) and $\tilde{\tau _{k}}$ as defined in (\ref{['an0']}) (dashed line) for Example 1 with $N=100,$$p=7,$ and $h=10^{-2}$. The dotted line represents the values of the estimate given in Theorem \ref{['th1']}.
• Figure 3: Error comparison using $\tau =\tau _{k,N}$ as in (\ref{['tauk2']}) (solide line) and $\tilde{\tau}_{k,N}$ as defined in (\ref{['anN']}) (dashed line) for Example \ref{['e1']} with $N=100,$$p=3,$ and $h=10^{-2}.$ In the pictures we also plot the error estimate (\ref{['th2']}) (dotted line).
• Figure 4: Error comparison using $\tau _{k,N}$ taken as in ( \ref{['tauk2']}) (solide line) and $\tilde{\tau}_{k,N}$ as defined in (\ref{['anN']}) (dashed line) for Example \ref{['e2']} with $N=1000,$ and $h=10^{-2}.$
• Figure 5: Error comparison between SIKM, RKM with $\tilde{\tau}_{k}$ as in (\ref{['an0']}), RKM with $\tau _{k}$ as in (\ref{['tauk']}) for $\mathcal{L}_{N}$ defined in (\ref{['LN']}), with $N=3000,$ and $h=10^{-2}$.

Theorems & Definitions (20)

• Proposition 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4
• proof
• Remark 3.5
• Proposition 3.6
• proof