Anomalous Shock Displacement Probabilities for a Perturbed Scalar Conservation Law

TL;DR

This paper investigates the probabilities of rare shock profile displacements in a stochastic conservation law using large deviations theory, providing theoretical bounds, numerical calculations, and an efficient importance sampling method.

Contribution

It introduces a detailed analysis of the rate function scaling for shock displacements and develops a robust importance sampling Monte Carlo strategy based on large deviations.

Findings

Rate function scales quadratically for small displacements.

Rate function scales linearly for large displacements.

Importance sampling significantly improves probability estimation efficiency.

Abstract

We consider an one-dimensional conservation law with random space-time forcing and calculate using large deviations the exponentially small probabilities of anomalous shock profile displacements. Under suitable hypotheses on the spatial support and structure of random forces, we analyze the scaling behavior of the rate function, which is the exponential decay rate of the displacement probabilities. For small displacements we show that the rate function is bounded above and below by the square of the displacement divided by time. For large displacements the corresponding bounds for the rate function are proportional to the displacement. We calculate numerically the rate function under different conditions and show that the theoretical analysis of scaling behavior is confirmed. We also apply a large-deviation-based importance sampling Monte Carlo strategy to estimate the displacement probabilities. We use a biased distribution centered on the forcing that gives the most probable transition path for the anomalous shock profile, which is the minimizer of the rate function. The numerical simulations indicate that this strategy is much more effective and robust than basic Monte Carlo.