Accuracy and convergence of the backward Monte-Carlo method

TL;DR

This paper analyzes the backward Monte-Carlo method for parabolic partial differential equations, showing it reduces statistical error when the quantity of interest lies in a small region of phase space and avoids the finite-bin error intrinsic to forward Monte-Carlo methods. Grounded in the Feynman-Kac representation, the backward approach samples $f(x,T)$ via trajectories initialized at $X^{\leftarrow}(T)=x$ and propagated backward to sample the initial condition, providing efficient targeting of relevant regions. Validation on a Lorentz diffusion model demonstrates comparable global accuracy to forward MC but superior tail accuracy, with backward MC avoiding empty-bin issues and maintaining consistent error scaling with $N$. The authors further develop a higher-order backward update that improves time-step convergence to $\mathcal{O}(\Delta t^{3/2})$, while showing that a similar scheme does not improve forward MC, underscoring the method’s practical advantages for localized problems and gradient computations.

Abstract

The recently introduced backward Monte-Carlo method [Johan Carlsson, arXiv:math.NA/0010118] is validated, benchmarked, and compared to the conventional, forward Monte-Carlo method by analyzing the error in the Monte-Carlo solutions to a simple model equation. In particular, it is shown how the backward method reduces the statistical error in the common case where the solution is of interest in only a small part of phase space. The forward method requires binning of particles, and linear interpolation between the bins introduces an additional error. Reducing this error by decreasing the bin size increases the statistical error. The backward method is not afflicted by this conflict. Finally, it is shown how the poor time convergence can be improved for the backward method by a minor modification of the Monte-Carlo equation of motion that governs the stochastic particle trajectories. This scheme does not work for the conventional, forward method.

Figures (6)

• Figure 1: The conventional, forward, weighted Monte-Carlo method. The trajectories as given by the Monte-Carlo equation of motion (\ref{['eq:forward_motion']}).
• Figure 2: The backward Monte-Carlo method. The trajectories as given by the backward Monte-Carlo equation of motion (\ref{['eq:backward_motion']}).
• Figure 3: Black line is analytic solution, red line is backward, and blue line is forward Monte-Carlo solution. Global solutions are to the left and tail solutions to the right.
• Figure 4: Scaling of the error in the forward (blue) and backward (red) solutions. The error as a function of the number of particles ($N$) is plotted in the left frame, and as a function of the number of bins ($N_{\mathrm{bin}}$) to the right.
• Figure 5: The error as a function of the time step ($\Delta{}t$) with the forward (blue) and backward (red) methods.