Solving PDEs with random data by stochastic collocation

• Main Author: Gordon, Andrew
• Other Authors: Powell, Catherine; Higham, Nicholas
• Published: University of Manchester 2013
• Subjects: 515
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.568594

In many science and engineering problems there is uncertainty in the input data. The ability to suitably model and handle this uncertainty is crucial for obtaining meaningful information about solutions. In this thesis, we consider the numerical approximation of statistics of solutions to partial differential equations (PDEs) with uncertain inputs. We focus on PDEs with random coefficients and random domains.We consider a general set of numerical methods known collectively as stochastic finite element methods. We can distinguish stochastic Galerkin methods and stochastic sampling methods. For the latter, samples of the random inputs are generated and the deterministic PDE is solved for each one. Averages of the quantities of interest are calculated using solutions obtained from the samples. We focus in particular on a specific type of sampling method, the stochastic collocation method.The main computational cost associated with solving PDEs with random data using stochastic finite element methods is the solution of the resulting linear system(s) obtained from the fully discrete problem. The main aim in this thesis is to identify efficient and robust techniques for solving the sequence of linear systems obtained from stochastic collocation methods and to reduce the computational costs by recycling as much information as possible. New iterative solution strategies are presented for stochastic collocation discretisations of PDEs with random coefficients and for stochastic collocation discretisations of PDEs on random domains. Substantial savings on computing costs have been demonstrated.