A Primer on Stochastic Galerkin Methods

In the last five years, the scientific computing community has taken great interest in the so-called stochastic Galerkin schemes (SGS). These schemes are typically used to address the following type of problem: Suppose that I have a deterministic differential equation model for which I can find an approximate solution using standard numerical discretization techniques. Now suppose that, in an attempt to improve the model, I replace some deterministic model inputs with stochastic quantities that better represent my understanding of the problem. In other words, there may be some intrinsic uncertainty in the problem that I would like to account for in the model. One example of this may be an equation parameter that admits some measurement error, which we can model with a stochastic representation. Another example may be a randomly perturbed initial or boundary condition. Or perhaps the domain of the problem may have some “roughness” on the boundaries that we can represent with some stochastic quantities. If the inputs to the model are now random, then the solution to the model will also be random. In effect, we have introduced a new stochastic dimension to the problem, or more accurately, we have accounted for the known uncertainties in the model using a probabilistic framework. So how can I adapt the standard numerical techniques for solving the original deterministic problem to the new problem with randomness? Or, how can I alter the numerical techniques to propagate the uncertainty in the inputs and quantify the uncertainty in the solution? The stochastic Galerkin schemes offer one approach to these questions. Before we move on, you may be asking why these schemes are called stochastic Galerkin. In a deterministic finite element framework, a standard Galerkin method projects the solution to the given differential equation onto a finite-dimensional basis. Then we can compute this “discretized” solution on our finite computers. The stochastic Galerkin schemes perform a similar projection in the random dimensions. In other words, they discretize the random dimensions to allow computation.