Exponentially convergent trapezoidal rules to approximate fractional powers of operators

TL;DR

This work shows how to improve the existing error estimates for the scalar case and also extend the analysis to operators, and reviews some theoretical aspects in order to refine the choice of parameters that allow a faster convergence.

Abstract

In this paper we are interested in the approximation of fractional powers of self-adjoint positive operators. Starting from the integral representation of the operators, we apply the trapezoidal rule combined with a single-exponential and a double-exponential transform of the integrand function. For the first approach our aim is only to review some theoretical aspects in order to refine the choice of the parameters that allow a faster convergence. As for the double exponential transform, in this work we show how to improve the existing error estimates for the scalar case and also extend the analysis to operators. We report some numerical experiments to show the reliability of the estimates obtained.

Figures (4)

• Figure 1: Error for the trapezoidal rule applied with the single-exponential transform with $d= \pi /4$ and $d=\pi /2$, and error estimate given by (\ref{['stima']}).
• Figure 2: Error for the trapezoidal rule applied with the double-exponential transform (error DE), estimates (\ref{['ere']}) and (\ref{['ere2']}) vs the number of inversions, for the computation of $\lambda ^{-\alpha }$ with $\lambda =10^{12}$ and $\tau =100$.
• Figure 3: Plot of the function $\varphi (\lambda,\tau ^{\ast })$ for $n=40$ and $\alpha =1/2$. The asterisk represents the approximation of the local maximum given by (\ref{['m2']}), that is, the point $\left( \lambda ^{\ast },\varphi (\lambda ^{\ast },\tau ^{\ast })\right)$. The diamond represents the approximation of $\varphi (\lambda ^{\ast },\tau ^{\ast })$ stated in (\ref{['errm2']}). Finally the circle is the approximation of $\varphi (1,\tau ^{\ast })$ given in (\ref{['t']}).
• Figure 4: Error for the trapezoidal rule applied with the double-exponential transform (error DE), with the single-exponential transform (error SE) and error estimate given by (\ref{['fest']}).

Theorems & Definitions (5)

• Definition 1
• Theorem 1
• Theorem 2
• proof
• Proposition 1