Analysis of Numerical Methods for PDEs with Random Data |
Research Team: Alex Bespalov Catherine Powell David Silvester This was an EPSRC funded research project.Summary
Recently, sophisticated numerical methods for approximating solutions to PDEs with correlated random data have been proposed. To date, this work has been restricted to scalar, elliptic PDEs and the question of efficient linear algebra for the resulting linear systems of equations has been largely overlooked. So-called stochastic Galerkin methods, in particular, have attractive approximation properties but have been somewhat ignored due to a lack of robust solvers. More simplistic schemes which require less user know-how but ultimately more computing time to implement, have been popularised. The aim of this project is to investigate approximation schemes for quantifying uncertainty in more complex engineering problems modelled by systems of PDEs with two output variables (e.g. groundwater flow in a random porous medium). We will extend and test the efficiency of approximation schemes introduced for scalar PDEs with random data, paying significant attention to the development of efficient linear algebra techniques for solving the resulting linear systems of equations.
The project built on previous work done on a DAAD/British Council project entitled Uncertainty quantification in computer simulations of groundwater flow problems with emphasis on contaminant transport in collaboration with Oliver Ernst and Elisabeth Ullmann.
Research outputs
A priori error analysis of stochastic Galerkin mixed approximations of elliptic PDEs with random data,
SIAM J. on Numerical Analysis, 50: 2039-2063, 2012A framework for the development of implicit solvers for incompressible flow problems
Discrete and Continuous Dynamical Systems --- Series S, 5: 1195-1221, 2012Preconditioning steady-state Navier-Stokes equations with random data,
SIAM J. on Scientific Computing, 34: A2482-A2506, 2012.Energy norm a posteriori error estimation for parametric operator equations,
SIAM J. on Scientific Computing, 36: A339-A363, 2014.
Research software
Our S-IFISS software for solving stochastic diffusion problems is freely available. To install release 1.02, simply move the stoch_diffusion directory to make it a subdirectory of the current release of IFISS and then type "install_sifiss" (in Matlab) or "octave_sifiss" (in Octave).
Link to Manchester NA group.
Page last modified: 27 July 2015