Reweighted techniques: definition and asymptotic convergence

TL;DR

This paper introduces reweighted techniques for the Feynman-Kac formula's random series methods, achieving improved asymptotic convergence without modifying the potential, especially for Sobolev-regular potentials.

Contribution

It defines and characterizes reweighted methods, proves their convergence properties, and demonstrates superior asymptotic convergence rates for Levy-Ciesielski and Wiener-Fourier series representations.

Findings

Reweighted techniques achieve o(1/n^2) convergence for certain potentials.

Potential for reaching O(1/n^3) convergence with second order Sobolev derivatives.

Reweighted methods preserve computational scaling and outperform partial averaging in convergence constants.

Abstract

I define and characterize the reweighted methods, which are techniques used in conjunction with the random series implementation of the Feynman-Kac formula. I prove several convergence results valid for all series representations and then I specialize the results for the Levy-Ciesielski and Wiener-Fourier series. As opposed to the partial averaging method on which they are based, the reweighted techniques do not involve any modification of the physical potential. Rather, the underlying idea is to develop some specialized constructions of the Brownian bridge that enters the Feynman-Kac formula, so as to simulate the partial averaging effect. For the Levy-Ciesielski series representation, I develop a reweighted technique which has o(1/n^2) convergence for potentials having first order Sobolev derivatives. It is suggested that the asymptotic convergence may reach O(1/n^3) for potentials having second order Sobolev derivatives. The method preserves the favorable log(n) scaling for the time necessary to compute a path at a given discretization point. For the Wiener-Fourier series representation, the particular reweighted method designed in the present article is shown to have O(1/n^3) convergence if the potential has second order Sobolev derivatives. The convergence constant has superior dependence with the inverse temperature as compared to the partial averaging method for the same series. Because the expression of the convergence constant does not actually involve the second order derivatives of the potential, it is conjectured that the O(1/n^3) convergence extends to the potentials having first order Sobolev derivatives only.