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, 2012
-
A framework for the development of implicit solvers for incompressible flow problems
Discrete and Continuous Dynamical Systems --- Series S,
5: 1195-1221, 2012
-
Preconditioning 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).
Page last modified: 27 July 2015
