Dr. Catherine Powell | The University of Manchester | United Kingdom
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.