Branching diffusion representation of semilinear PDEs and Monte Carlo approximation

Abstract

We provide a probabilistic representations of the solution of some semilinear hyperbolic and high-order PDEs based on branching diffusions. These representations pave the way for a Monte-Carlo approximation of the solution, thus bypassing the curse of dimensionality. We illustrate the numerical implications in the context of some popular PDEs in physics such as nonlinear Klein-Gordon equation, a simplified scalar version of the Yang-Mills equation, a fourth-order nonlinear beam equation and the Gross-Pitaevskii PDE as an example of nonlinear Schrödinger equations.

Nesting Monte Carlo for high-dimensional non-linear PDEs

“…In order to have a converging method we will see that we will have to take u < 1 in ρ expression (38) excluding the exponential distribution. This a weaker constraint than in [18] where, using branching for some polynomial non-linearities, converging results were only obtained for u < 0.5.…”

“…The goal of this section is to study the underlying algorithm when the resolution of equation (9) is achieved by nesting Monte Carlo. Starting from the ideas used in [18] we propose a nesting algorithm calculating all u p n by Monte Carlo. We have to show the bias associated to the algorithm goes to zero and that the global variance induced is controlled.…”

“…where the last inequality is obtained by Jensen. Plugging (19) in (18) and using the relation E(|x − E(x)| 2 ) ≤ E(x 2 ) we get :…”

“…• The second is based on branching methods and is effective for non linearities polynomial in the solution u and its gradient Du. Convergence results are given in [18] and numerical results show that the PDEs can be solved in dimension 100. However the authors showed that the variance of the method explodes rapidly when the maturity grows or when the non linearity becomes important: numerical tests confirm that it is in fact the case.…”

“…As for the branching method, the methodology proposed here is based on the Feynman-Kac representation of the PDEs coupled with the randomization of the time step proposed in [18]. This approach is combined with nesting Monte Carlo with a given depth.…”