Article2017

SDE Based Regression for Linear Random PDEs

Felix Anker · Christian Bayer · Martin Eigel · Marcel Ladkau · Johannes Neumann · John G. M. Schoenmakers

7 citations0/0 files storedSIAM Journal on Scientific Computing

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Probabilistic and Robust Engineering Design Stochastic processes and financial applications Model Reduction and Neural Networks

Abstract

A simulation based method for the numerical solution of PDEs with random coefficients is presented. By the Feynman--Kac formula, the solution can be represented as conditional expectation of a functional of a corresponding stochastic differential equation (SDE) driven by independent noise. A time discretization of the SDE for a set of points in the domain and a subsequent Monte Carlo regression lead to an approximation of the global solution of the random PDE. We provide an initial error and complexity analysis of the proposed method along with numerical examples illustrating its behavior.

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References

1991Stochastic Finite Elements: A Spectral Approach 2001Valuing American Options by Simulation: A Simple Least-Squares Approach 2007A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data 2008A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data 2004Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations 2001Regression methods for pricing complex American-style options 2004Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations 2008An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data

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