Numerical analysis of adaptive UQ algorithms for PDEs with random inputs
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Research Team:  Alex Bespalov Catherine Powell David Silvester Feng Xu
This is an EPSRC funded research project.

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.
Numerical methods based on a parametric reformulation of such PDE problems emerged in the engineering literature in the 1990s as more efficient and rapidly convergent alternatives to Monte-Carlo sampling in cases where the dimension of the stochastic space is moderate (of the order of 10 random parameters). Recent research into these methods suggests that their advantageous approximation properties can best be achieved by using an adaptive refinement strategy, when spatial and stochastic components of the approximate solution are judiciously chosen in the course of numerical computation. The design of optimal adaptive algorithms remains an open question however. The proposed research programme aims at the design, theoretical analysis and efficient implementation of the state-of-the-art adaptive algorithms applicable to a range of PDE problems with random inputs.

The project builds on our work done on the precursor project Analysis of Numerical Methods for Partial Differential Equations with Random Data

Research outputs
  • Efficient adaptive algorithms for parametric PDEs with spatial singularities, Birmingham preprint, 2017
  • CBS constants and their role in error estimation for stochastic Galerkin finite element methods, MIMS preprint, 2017

Research software
Our S-IFISS software for solving stochastic diffusion problems is freely available. To install the latest release, 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 UQ group.

Page last modified: 10 August 2017