Tuesday, September 5, 14:00 - 18:00
Instructor: Catherine Powell, University of Manchester
In many engineering applications, it is of interest to simulate a physical process that is well modelled by partial differential equations (PDEs). In this introductory lecture, we will discuss how to approach the numerical solution of such problems when there is uncertainty in the inputs, focusing on strategies that make use of standard finite element methods (FEMs) for the spatial discretisation.
We consider test problems consisting of PDEs with random coefficients and discuss simple stochastic FEMs based on sampling. We review computationally efficient numerical methods for generating realisations of random fields, focusing on the circulant embedding method and the Karhunen-Loeve expansion. We then discuss the computational costs associated with the basic Monte Carlo FEM and motivate the need for strategies that reduce the cost of the finite element solves required.
When we apply sampling methods such as Monte Carlo or stochastic collocation to PDE models with inputs that depend on uncertain parameters, we have to solve the PDE model for many (say R) choices of the parameters. If we apply finite element methods (FEMs) for the spatial discretisation, and assume there are N_h degrees of freedom to be computed then the work involved is R x O(N_h). In complex applications, this is infeasible.
Reduced basis methods (RBMs) are now a popular approach for speeding up simulations that require the repeated solution of a deterministic PDE for many choices of the input parameters. The basic idea is to compute a few FEM solutions corresponding to a low number of specific choices of parameters. These are then used as basis functions to construct low-dimensional approximations to all the other FEM solutions required.
We will discuss strategies for constructing reduced bases, focusing on computational efficiency and cost, and show how to combine them with smart sampling methods like stochastic collocation FEMs. We will also discuss recent innovations in combining reduced bases with stochastic Galerkin FEMs (so-called intrusive methods).