Numerical Methods for Partial Differential Equations with Random Data

Traditional methods of mathematical modeling depend on the assumption that components of models such as diffusion coefficients or boundary conditions are known. In practice, however, such quantities may not be known with certainty and instead they may be represented as random functions, that is, a random variable for each point in the physical domain. A standard approach for handling this situation, Monte Carlo simulation, is known to be slow. We discuss an alternative methodology, known generally as the stochastic finite element method, through which uncertainty in models is handled by introducing an m-dimensional space associated with the stochastic aspect of the model. For d-dimensional spatial models, this leads to a problem posed in a (d+m)-dimensional tensor product space that can be handled in a style analogous to standard treatment of deterministic spatial models. We present an overview of how this methodology is defined and how it can be used to obtain statistical properties of solutions, discuss discretization methods designed to compute numerical approximations to solutions, and outline solution algorithms and costs for solving the algebraic systems that arise using this methodology.

Speaker Bio: Howard Elman is a Professor in the Computer Science Department at the University of Maryland, College Park. He received his doctorate in Computer Science from Yale University in 1982. He has had visiting positions at Stanford University, the University of Manchester Institute of Science and Technology, and the University of Oxford. He has been selected as a SIAM Fellow and has served on the editorial boards of SIAM Journal on Scientific Computing, where he was editor-in-chief from 1998-2004, Mathematics of Computation, and Numerical Linear Algebra and Applications. His research concerns numerical solution algorithms for partial differential equations, computational fluid dynamics and sparse matrix problems.

Presentation Slides: PDF

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