Branching diffusion representation for nonlinear Cauchyproblems and Monte Carlo approximation
For researchers solving high-dimensional nonlinear PDEs, this work provides a novel Monte Carlo method that bypasses the curse of dimensionality, though it is limited to specific PDE classes.
The authors develop probabilistic representations for semilinear hyperbolic and high-order PDEs using branching diffusions, enabling Monte Carlo approximation that avoids the curse of dimensionality. Numerical tests on Klein-Gordon, Yang-Mills, beam, and Gross-Pitaevskii equations demonstrate the method's applicability.
We provide a probabilistic representations of the solution of some semilinear hyperbolicand high-order PDEs based on branching diffusions. These representations pave theway for a Monte-Carlo approximation of the solution, thus bypassing the curse ofdimensionality. We illustrate the numerical implications in the context of some popularPDEs in physics such as nonlinear Klein-Gordon equation, a simplied scalar versionof the Yang-Mills equation, a fourth-order nonlinear beam equation and the Gross-Pitaevskii PDEas an example of nonlinear Schrodinger equations.