Multiplicative cascades applied to PDEs (two numerical examples)
Numerical approximations to the Fourier transformed solution of partial differential equations are obtained via Monte Carlo simulation of certain random multiplicative cascades. Two particular equations are considered: linear diffusion equation and viscous Burgers equation. The algorithms proposed exploit the structure of the branching random walks in which the multiplicative cascades are defined. The results show initial numerical approximations with errors less than 5% in the leading Fourier coefficients of the solution. This approximation is then improved substantially using a Picard iteration scheme on the integral equation associated with the representation of the respective PDE in Fourier space.
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