Solving Partial Differential Equations by Self-Generated Stochasticity.

New physical principle for simulations of PDE has been introduced. It is based upon replacing the PDE to be solved by the system of ODE for which the PDE represents the corresponding Liouville equation. The proposed approach has a polynomial (rather than exponential) algorithmic complexity, and it is applicable to nonlinear parabolic, hyperbolic, and elliptic PDE.