Summary
Partial differential equations (PDEs) are key tools in the mathematical modelling of physical processes in science and engineering. Traditional deterministic PDEbased models assume precise knowledge of all inputs (material properties, initial conditions, external forces, etc.). There exist an abundance of numerical methods that can be used to compute a solution to such models to any required accuracy. In practical applications, however, a complete characterisation of all the inputs to a PDE model may not be available. Examples include the modulus of elasticity of a stressed body (in linear elasticity models)and wave characteristics of inhomogeneous media (in wave propagation models). In these cases, simulations based on deterministic models are unable to estimate probabilities of undesirable events (e.g., the fracture of a stressed plate) and, hence, to perform a reliable risk assessment. The emergent area of uncertainty quantification (UQ) deals with mathematical modelling at a different level. It involves the use of probabilistic techniques in order to (i) determine and quantify uncertainties in the inputs to PDEbased models, and (ii) analyse how these uncertainties propagate to the outputs (either the solution to the PDE, or a quantity of interest derived from the solution). The models are then described by PDEs with random data, where both inputs and outputs take the form of random fields ... more
The project builds on the research in the precursor project
Analysis of Numerical Methods for Partial Differential Equations with Random Data
Research outputs to date

Efficient Adaptive Algorithms for Elliptic PDEs with Random Data, SIAM/ASA J. Uncertainty Quantification, 6(1), 243272, 2017

CBS constants & their role in error estimation for stochastic Galerkin finite element methods,
J. of Scientific Computing, 77, 10301054, 2018

Collocation methods for exploring perturbations
in linear stability analysis,
SIAM J. on Scientific Computing, 40, A2667A2693, 2018

Robust a posteriori error estimators for mixed approximation of nearly incompressible elasticity,
Int. J. Numerical Methods in Engineering, 2019

Goaloriented error estimation and adaptivity for elliptic PDEs with parametric or uncertain inputs,
Computer Methods in Applied Mechanics and Engineering, 345, 951982, 2019

Robust preconditioning for stochastic Galerkin formulations of parameterdependent linear elasticity equations,
SIAM J. on Scientific Computing, 41, A402A421, 2019

Adaptive BEM with optimal convergence rates for the Helmholtz equation,
Computer Methods in Applied Mechanics and Engineering, 346, 260287, 2019

Efficient adaptive multilevel stochastic Galerkin approximation using implicit a posteriori error estimation,
SIAM J. on Scientific Computing, 2019

Robust a posteriori error estimation for stochastic Galerkin formulations of parameterdependent
linear elasticity equations,
arXiv:1810.07440, 2018

Convergence of adaptive stochastic Galerkin FEM,
arXiv:1811.09462, 2018

A posteriori error estimation and adaptivity in stochastic Galerkin FEM for parametric
elliptic PDEs: beyond the affine case,
arXiv:1903.06520, 2019
Research software

SIFISS
solves diffusion problems with uncertain coefficients (square domain, needs
IFISS)

Stochastic TIFISS
solves diffusion problems with uncertain coefficients (arbitrary twodimensional domains)
Page last modified: 04 June 2019