Random Series and Discrete Path Integral methods: The Levy-Ciesielski implementation

Abstract

We perform a thorough analysis of the relationship between discrete and series representation path integral methods, which are the main numerical techniques used in connection with the Feynman-Kac formula. First, a new interpretation of the so-called standard discrete path integral methods is derived by direct discretization of the Feynman-Kac formula. Second, we consider a particular random series technique based upon the Levy-Ciesielski representation of the Brownian bridge and analyze its main implementations, namely the primitive, the partial averaging, and the reweighted versions. It is shown that the n=2^k-1 subsequence of each of these methods can also be interpreted as a discrete path integral method with appropriate short-time approximations. We therefore establish a direct connection between the discrete and the random series approaches. In the end, we give sharp estimates on the rates of convergence of the partial averaging and the reweighted Levy-Ciesielski random series approach for sufficiently smooth potentials. The asymptotic rates of convergence are found to be O(1/n^2), in agreement with the rates of convergence of the best standard discrete path integral techniques.

20 pages, 4 figures; the two equations before Eq. 14 are corrected; other typos are removed

References in corpus (3)

• Optimal series representations for numerical path integral simulations
• Free energy of the Fröhlich polaron in two and three dimensions
• Quantum Monte Carlo Method for Attractive Coulomb Potentials

Cited by in corpus (7)

• Numerical implementation of some reweighted path integral methods
• Energy estimators for random series path-integral methods
• Upon the existence of short-time approximations of any polynomial order for the computation of density matrices by path integral methods
• Accelerating ab initio path integral molecular dynamics with multilevel sampling of potential surface
• The fast sampling algorithm for Lie-Trotter products
• Convergence Characteristics of the Cumulant Expansion for Fourier Path Integrals
• On the efficient Monte Carlo implementation of path integrals