Numerical analysis of adaptive UQ algorithms for PDEs with random inputs |
This is an EPSRC funded research project running from April 2017 to June 2021.
Research Team: Alex Bespalov Arbaz Khan Catherine Powell David Silvester Feng Xu
Summary
Partial differential equations (PDEs) are key tools in the mathematical modelling of physical processes in science and engineering. Traditional deterministic PDE-based 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 PDE-based 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
Selected research outputs
Efficient Adaptive Algorithms for Elliptic PDEs with Random Data, SIAM/ASA J. Uncertainty Quantification, 6(1), 243-272, 2017
CBS constants & their role in error estimation for stochastic Galerkin finite element methods, J. of Scientific Computing, 77, 1030-1054, 2018
Collocation methods for exploring perturbations in linear stability analysis, SIAM J. on Scientific Computing, 40, A2667-A2693, 2018
Goal-oriented error estimation and adaptivity for elliptic PDEs with parametric or uncertain inputs, Computer Methods in Applied Mechanics and Engineering, 345, 951-982, 2019
Robust preconditioning for stochastic Galerkin formulations of parameter-dependent linear elasticity equations, SIAM J. on Scientific Computing, 41, A402-A421, 2019
Efficient adaptive multilevel stochastic Galerkin approximation using implicit a posteriori error estimation, SIAM J. on Scientific Computing, 41, A1681-A1705, 2019
Convergence of adaptive stochastic Galerkin FEM, SIAM J. on Numerical Analysis, 57, 2359-2382, 2019
A posteriori error estimation and adaptivity in stochastic Galerkin FEM for parametric elliptic PDEs: beyond the affine case, Computers and Mathematics with Applications, 2020
A fully adaptive multilevel collocation strategy for solving elliptic PDEs with random data, J. of Computational Physics, 2020
T-IFISS: a toolbox for adaptive FEM computation, Computers and Mathematics with Applications, 2020
Latest preprints
Two-level a posteriori error estimation for adaptive multilevel stochastic Galerkin FEM, arXiv:2006.02255, 2020
Truncation preconditioners for stochastic Galerkin finite element discretizations, arXiv:2006.06428, 2020
Alternating energy minimization methods for multi-term matrix equations, arXiv:2006.08531, 2020
PhD theses
Adaptive algorithms for partial differential equations with parametric uncertainty, Leonardo Rocchi, 2019
Adaptive & multilevel stochastic Galerkin finite element methods, Adam Crowder, 2020
Research software
Stochastic T-IFISS solves diffusion problems with uncertain coefficients (arbitrary two-dimensional domains)
Link to Manchester UQ group.
Page last modified: 14 July 2020