Prediction of Shock Formation From Boundary Measurements

We propose a method for the Bayesian prediction of shocks in scalar partial differential equations (PDEs) representing conservation equations from noisy observations of the boundary conditions. By considering the implicit transformation from boundary conditions to shocks, we construct an arrival process interpretation of shocks as well as an associated arrival rate function. We then introduce a Monte Carlo method to approximate the arrival rate of shocks based on the probability of a sufficiently large range of values in an epsilon ball conditioned on noisy boundary measurements. We illustrate the method with simulations of Burgers' equation with initial conditions set by Brownian motion. Despite the non-smooth boundary, our proposed method constructs a sparse and readily interpretable probabilistic structure of shock arrival and propagation.