Error estimates on ergodic properties of discretized Feynman–Kac semigroups

Abstract

We consider the numerical analysis of the time discretization of Feynman-Kac semigroups associated with diffusion processes. These semigroups naturally appear in several fields, such as large deviation theory, Diffusion Monte Carlo or non-linear filtering. We present error estimates à la Talay-Tubaro on their invariant measures when the underlying continuous stochastic differential equation is discretized; as well as on the leading eigenvalue of the generator of the dynamics, which corresponds to the rate of creation of probability. This provides criteria to construct efficient integration schemes of Feynman-Kac dynamics, as well as a mathematical justification of numerical results already observed in the Diffusion Monte Carlo community. Our analysis is illustrated by numerical simulations.

Adaptive Sampling of Large Deviations

“…The left plot in this figure shows the evolution in time of the estimated SCGF from the eigenvalue iteration (49), while the right plot shows the final estimated eigenfunction h k for different time steps ∆t, compared with the reference Fourier solution. We see that the results for the eigenvalue and the eigenvector significantly depart from the reference solutions for ∆t = 0.05, but converge to them as ∆t is decreased, in accordance with Assumption 2 and the theoretical results of [36]. For the same time step used for the Ornstein-Uhlenbeck process, namely, ∆t = 5 × 10 −3 , we see no notable difference between the estimated eigenfunction and the reference values, leading to a precise estimation of the SCGF.…”

“…This assumption holds for many systems, in particular when X is bounded [36,40] or when b and g are gradient fields with appropriate growth conditions; see [41,Sec. 2.5].…”

“…Many discretizations also exist for the Feynman-Kac semi-group P k T . Here, we use the natural scheme where the diffusion is discretized as above and the integral of A T is discretized as a Riemann sum with the left-point rule for the integral involving f and the mid-point rule for the Stratonovich integral involving g [36]. The action of P k T is thus replaced by…”

“…This applies, for example, when X is compact. In this case, precise estimates for the errors in (31) are obtained for g = 0 in [36] and can be extended to g = 0.…”

“…The calculation of error bars for the estimated quantities is also simplified compared to other techniques, as the algorithm is based on simple time averages and stochastic approximations [33][34][35]. Finally, the errors incurred by discretizing continuous-time processes and functionals can be analysed in a precise way, in principle, via Feynman-Kac semi-groups [36].…”