Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem

We address the numerical discretization of the Allen-Cahn prob- lem with additive white noise in one-dimensional space. The discretization is conducted in two stages: (1) regularize the white noise and study the regularized problem, (2) approximate the regularized problem. We address (1) by introducing a piecewise constant random approximation of the white noise with respect to a space-time mesh. We analyze the regularized problem and study its relation to both the original problem and the deterministic Allen-Cahn problem. Step (2) is then performed leading to a practical Monte-Carlo method combined with a Finite Element-Implicit Euler scheme. The resulting numerical scheme is tested against theoretical benchmark results.Comment: 28 pages, 16 (4x4) figures, published in 2007; Interfaces and Free Boundaries 2007 vol. 9 (1

Formation of singularities for viscosity solutions of Hamilton-Jacobi equations in one space variable

Abstract: "In this work we study the generation and propagation of singularities (shock waves) of the solution of the Cauchy problem for Hamilton-Jacobi equations in one space variable, under no assumption on convexity or concavity of the hamiltonian. We study the problem in the class of viscosity solutions, which are the correct class of weak solutions. We obtain the exact global structure of the shock waves by studying the way the characteristics cross. We construct the viscosity solution by either selecting a single-valued branch of the multi-valued function given as a solution by the method of characteristics or constructing explicitly the proper rarefaction waves.

Formation of singularities for viscosity solutions of Hamilton-Jacobi equations in higher dimensions

Abstract: "In this work we study the generation of singularities (shock waves) of the solution of the Cauchy problem for Hamilton-Jacobi equations in several space variables, under no assumption on convexity or concavity of the hamiltonian. We study the problem in the class of viscosity solutions, which are the correct class of weak solutions. We first examine the way the characteristics cross by identifying the set of critical points of the characteristic manifold with the caustic set of the related lagrangian mapping. We construct the viscosity solution by selecting a single-valued branch of the multi-valued function given as a solution by the method of characteristics. We finally discuss how the shocks propagate and undergo catastrophe in the case of two space variables.