A Monte Carlo discretization method for nonlinear variational PDEs
A talk in the Bielefeld Stochastic Afternoon series by
Iulian Cimpean
| Abstract: | We introduce and test a numerical method for solving boundary
value problems for some classes of second order partial differential
equations in a bounded open set in $D$ in $\mathbb{R}^d$, including, for
example, those that arise as the Euler-Lagrange equation associated to
some energy functional of the type $ J(u)=\int_D F(x,u(x),\nabla
u(x))\;dx$ with prescribed boundary conditions.
Our approach is based on the orthogonal decomposition of $H^1(D)$,
probabilistic representations and the Monte Carlo method.
We test our method numerically on various PDEs, such as (non-)symmetric
second order linear elliptic PDEs, semilinear PDEs that admit multiple
solutions, quasilinear PDEs such as Poisson problems for the
$p(x)$-Laplace equation. Some of these examples will be presented in
this talk. Based on an ongoing work, jointly with A. Grecu (Bucharest,
Romania) and A. Zǎrnescu (Bilbao, Spain). Within the CRC this talk is associated to the project(s): A5, B1 |