Monte Carlo Methods for Partial Differential Equations Monte Carlo Methods and Linear Algebra PDEs 1 Courant, Friedrichs, and Lewy: Their pivotal 1928 paper has probabilistic interpretations and MC algorithms for linear elliptic and parabolic problems 2 Fermi Ulam von Neumann: Atomic bomb calculations were done Derived a MCM for solving special linear systems related to discrete elliptic
Deep Learning-Based Algorithms for High-Dimensional PDEs and Control . . . Related Work in High-dimensional Case •Linear parabolic PDEs: Monte Carlo methods based on theFeynman-Kac formula •Semilinear parabolic PDEs: 1 branching diffusionapproach (Henry-Labord`ere 2012, Henry-Labord `ere et al 2014) 2 multilevel Picard approximation(E and Jentzen et al 2015) •Hamilton-Jacobi PDEs: usingHopf formulaand fast convex nonconvex
Monte Carlo Methods and Partial Differential Equations: Algorithms and . . . Monte Carlo Methods and Partial Differential Equations: Algorithms and Implications for High-Performance Computing probabilistic interpretations and MC algorithms for linear elliptic and parabolic problems 2 Fermi Ulam von Neumann: Atomic bomb calculations were done 4 Forsythe and Leibler: Derived a MCM for solving special linear
High-Dimensional Partial Differential Equations - Deep PDE Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach; Actor-critic method for high dimensional static Hamilton–Jacobi–Bellman partial differential equations based on neural networks; A derivative-free method for solving elliptic partial differential equations with deep neural networks
Using Simple SDEs (Stochastic Differential Equations) to Solve . . . - NIST linear elliptic and parabolic problems 2 Fermi Ulam von Neumann: Atomic bomb calculations were done using Monte Carlo methods for neutron transport, their success inspired much post-War work especially in nuclear reactor design 3 Kac and Donsker: Used large deviation calculations to estimate eigenvalues of a linear Schrödinger equation
Monte Carlo Methods for Partial Differential Equations: A . . . - FSUSciComp Monte Carlo Methods for Partial Differential Equations: A Personal Journey probabilistic interpretations and MC algorithms for linear elliptic and parabolic problems 2 Fermi Ulam von Neumann: Atomic bomb calculations were done Monte Carlo methods for solving Poisson and linearized I
A MULTILEVEL MONTE CARLO ENSEMBLE SCHEME FOR - University of South Carolina 9 solved more e ciently than a sequence of linear systems In this paper, we pursue in the same 10 direction and develop a new multilevel Monte Carlo ensemble method for solving random parabolic 11 partial di erential equations Comparing with the approach in [26], this method possesses a second-12 order accuracy in time and further reduces the
Solving High-dimensional PDEs Using Deep Learning •Linear parabolic PDEs: Monte Carlo methods based on the Feynman-Kac formula •Semilinear parabolic PDEs: 1 branching diffusionapproach (Henry-Labord`ere 2012, Henry-Labord`ere et al 2014) 2 multilevel Picard approximation(E et al 2016) •Hamilton-Jacobi PDEs: usingHopf formulaand fast convex nonconvex optimization methods (Darbon Osher
Monte Carlo Methods - School of Mathematics So far we have not discussed the convergence of Monte Carlo algorithms It is clear that the \usual" notions of convergence are insu cient when analysing Monte Carlo methods No matter how large sample we take, we can always be extremely \unlucky", draw a \unrepresentative" sample and get a bad estimate for the true solution of our problem
Monte Carlo Methods for Partial Differential Equations Monte Carlo Methods and Linear Algebra probabilistic interpretations and MC algorithms for linear elliptic and parabolic problems 2 Fermi Ulam von Neumann: Atomic bomb calculations were done 4 Forsythe and Leibler: Derived a MCM for solving special linear systems related to discrete elliptic PDE problems