CSE 2007: Numerical Methods for Stochastic PDEsJune 12, 2007
One of the themes of this year's SIAM Conference on Computational Science and Engineering (see related articles) was numerical methods for stochastic partial differential equations (SPDEs). Sessions devoted to this theme included the plenary talk of Hermann Matthies, a four-part mini-symposium, and several other minisymposia and contributed talks and posters. Collectively, these presentations highlighted the important applications of SPDEs and the latest numerical methods for determining approximate solutions of problems involving SPDEs.
Simply put, an SPDE is a PDE for which some inputs, e.g., coefficients, right-hand sides, boundary descriptions, are random---that is, their values are known only in a statistical sense. Such equations are used to model problems in all areas of science and engineering. Uncertainty arises in the inputs for two reasons.
First, in many situations, it is impossible or very costly to determine an input precisely. For example, the amount of future rainfall over a specified land area is impossible to predict; it is also impossible, cost-wise, to determine precise properties of subsurface media, e.g., porosity or permeability distribution, through measurements. Instead, the quantities in both cases are modeled as stochastic inputs to many groundwater flow models.
A second source of uncertainty is unresolved scales: Small (temporal and spatial) effects are not precisely accounted for, either because computer resources are limited or because they are not of immediate interest. In many situations, the effects of small-scale phenomena on large-scale behaviors are modeled through random inputs. An example is surface roughness---because accurate resolution would necessitate very fine grids near the boundary, it is often modeled instead through random inputs.
Solutions of SPDEs are used to determine (from given statistical information about system inputs) statistical information about the state of a system or about functionals of that state. As such, SPDEs are an important tool for increasing our understanding of basic (physical, chemical, biological, economic, social, environmental, etc.) processes, and they can play a central role in uncertainty quantification and risk assessment. It is not surprising, then, that the development, analysis, implementation, and application of computational methods for the solution of SPDEs have become a very active area of research and thus provided a very timely theme for the SIAM CSE conference.
The randomness that can occur in inputs can be seen simplistically as falling into two categories. First, we can have (a finite number of) random parameters appearing in the inputs; these can be thought of as "knobs" that can be turned to determine the inputs. Randomness comes into play because in many situations we have only probabilistic information about how the knobs should be set.
Second, we can have random field inputs: At each point in the spatial domain and at each instant in time, the value of the field is independently sampled from a given probability distribution. If the values are uncorrelated (in space–time), we have white noise; if they are correlated, we have what is often referred to as "colored noise." In a computation, random fields must be discretized. Truncated Karhunen–Loeve expansions can be used to express colored random fields in terms of a finite number of parameters. For white noise fields, we usually have to be content with random samples of the values on a grid.
Most of the SPDE-related presentations at the conference focused on stochastic finite element methods (SFEMs). Loosely defined, SFEMs (this and other acronyms are mine and may differ from those used by others), are methods in which spatial discretization is effected via finite element methods. (Of course, other methods for this purpose, e.g., finite volume or spectral methods, could be substituted.)
Roughly speaking, SFEMs themselves can be divided into two classes. The first consists of descendants of Monte Carlo-based methods, the most obvious means for solving SPDEs. Here, one simply samples the values of the random inputs and then, for each such sample, solves (approximately, of course) the PDE to obtain realizations of the solution. It is not the individual realizations that are of interest, but rather ensemble averages, e.g., statistical information, of the solution or of functionals of the solution. The values of the random parameters could be obtained by random sampling, leading to the Monte Carlo–finite element method for SPDEs. Other sampling techniques developed for high-dimensional integration, such as quasi-Monte Carlo sequences, Latin hypercube sampling, orthogonal arrays, or lattice rules, could be substituted. Another approach to sampling is through the quadrature points used to approximate the stochastic integrals that appear in the functionals of the solution and in other quantities of interest. Sparse-grid sampling techniques based on Smolyak quadrature rules, one very promising member of this class, were discussed extensively at the conference.
In the second class of SFEMs are stochastic Galerkin methods (SGMs), in which the dependence both on physical (space–time) variables and on probabilistic variables is treated with a Galerkin method. Thus, the approximate solution (and sometimes the input fields) are expressed as a sum of terms involving products of basis functions depending on the spatial coordinates and on the random parameters; the coefficients in the sum are determined by applying a Galerkin method, i.e., through a residual "orthogonalization" method, with respect to both the spatial and the probabilistic domains. Polynomial chaos methods make up one popular class of SGMs.
All in all, the SIAM conference has increased awareness within the overall CSE community of the importance of stochastic partial differential equations and of the wonderful opportunities for further research on numerical techniques for treating them.
Max Gunzburger is the Francis Eppes Distinguished Professor of Mathematics in the School of Computational Science and the Department of Mathematics at Florida State University.