A dynamical polynomial chaos approach for long-time evolution of SPDEs
For researchers in computational stochastic PDEs, this method enables long-time simulations that were previously infeasible due to dimensionality growth.
The paper introduces a Dynamical generalized Polynomial Chaos (DgPC) method for long-time simulation of SPDEs with white noise, addressing the curse of dimensionality in standard PC methods. The method achieves accurate long-time simulations, including invariant measures, and compares favorably with Monte Carlo for stochastic Burgers and Navier-Stokes equations.
We propose a Dynamical generalized Polynomial Chaos (DgPC) method to solve time-dependent stochastic partial differential equations (SPDEs) with white noise forcing. The long-time simulation of SPDE solutions by Polynomial Chaos (PC) methods is notoriously difficult as the dimension of the stochastic variables increases linearly with time. Exploiting the markovian property of white noise, DgPC [1] implements a restart procedure that allows us to expand solutions at future times in terms of orthogonal polynomials of the measure describing the solution at a given time and the future white noise. The dimension of the representation is kept minimal by application of a Karhunen--Loeve (KL) expansion. Using frequent restarts and low degree polynomials on sparse multi-index sets, the method allows us to perform long time simulations, including the calculation of invariant measures for systems which possess one. We apply the method to the numerical simulation of stochastic Burgers and Navier--Stokes equations with white noise forcing. Our method also allows us to incorporate time-independent random coefficients such as a random viscosity. We propose several numerical simulations and show that the algorithm compares favorably with standard Monte Carlo methods.