Numerical analysis of adaptive UQ algorithms for PDEs with random inputs

Lead Research Organisation: University of Birmingham
Department Name: School of Mathematics

Abstract

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Description Partial differential equations (PDEs) are key tools in the mathematical modelling of physical processes in science and engineering. While traditional deterministic PDE-based models assume precise knowledge of all inputs (material properties, initial conditions, external forces, etc.), a complete characterisation of inputs may not be available in practical applications. In these cases, simulations must rely on probabilistic techniques in order to model the underlying uncertainties in the inputs (using random variables or random fields), analyse how the uncertainties propagate to the model outputs, estimate probabilities of undesirable events, and perform a reliable risk assessment. The models are then described by PDEs with random data, where both inputs and outputs take the form of random fields. Numerical solution of such PDE-based models is significantly more challenging than the solution of the deterministic analogues. The development of robust, accurate, and practical numerical methods for solving associated parameter-dependent PDEs has been the central focus of the project.

The key objectives of the project concerned the design and analysis of provably convergent adaptive algorithms for solving several classes of PDEs with random inputs. The work on the project followed four main strands: (1) a posteriori error analysis of approximations and design of adaptive algorithms; (2) convergence and rate optimality analysis of adaptive algorithms; (3) efficient linear algebra for solving the linear systems resulting from discretisations; (4) the implementation of algorithms and software development. Main objectives have been successfully achieved, resulting in 11 journal publications (with 2 more papers currently under review) and 1 software package.

(1) Rigorous a posteriori error analysis of computed solutions is at the heart of adaptive algorithms for PDE problems. We have introduced and analysed novel reliable and efficient a posteriori error estimators and practical error indicators for stochastic Galerkin approximations in various settings, including
- single-level and multilevel stochastic Galerkin finite element methods (SGFEMs) for general diffusion problems with uncertain permeability;
- error estimators that control approximation errors in the solution as well as the errors in quantities of interest (i.e., goal-oriented error estimators);
- parametric PDE problems with non-affine parameter dependence of coefficients;
- error estimation for parameter-dependent linear elasticity problems (the proposed error estimate has been shown to be robust in the incompressible limit).

Since the approximation spaces underlying SGFEM discretisations have a tensor-product structure, one of the main challenges in developing adaptive SGFEM algorithms is to identify enrichment strategies that reduce approximation errors most efficiently. We have developed a collection of problem-specific adaptive algorithms by exploiting the detailed information about computed solutions from the a posteriori error analysis in each of the above settings.

(2) The designed algorithms require theoretical guarantees that the associated adaptive strategies terminate. In addition, it is important to understand convergence behaviour of the generated approximations and address the question of optimality of adaptive strategies. In the setting of general diffusion problems with affine-parametric coefficients, we have proved that for any given error tolerance, the designed adaptive algorithms stop after a finite number of iterations. Moreover, we have proved that, under an appropriate saturation assumption, the proposed multilevel adaptive strategy yields optimal convergence rates with respect to the overall dimension of the underlying approximation spaces.

(3) An effective and bespoke iterative solver is a key ingredient of any SGFEM implementation. We have analysed and implemented a class of truncation preconditioners for SGFEMs. In the setting of general diffusion problems with affine-parametric representation of the diffusion coefficient, we have performed spectral analysis of the preconditioned matrices and established optimality of truncation preconditioners with respect to SGFEM discretisation parameters. Extensive numerical experiments for model diffusion problems with affine and non-affine parametric representations of the coefficient have shown that in terms of efficiency, truncation preconditioners compare favourably to other existing preconditioners for stochastic Galerkin matrices, such as the mean-based and the Kronecker product preconditioners.

(4) Software development and numerical experimentation have been an integral part of research on the project. We have implemented the developed algorithms in an open-source MATLAB toolbox Stochastic T-IFISS that is freely available to other researchers.
Exploitation Route The a posteriori error estimation framework and the methodology for analysis of adaptive algorithms that have been developed in this project in the context of stochastic Galerkin finite element methods can be extended to other numerical approximation schemes (e.g., stochastic collocation) for a wide range of PDEs with uncertain or parametric inputs. This will result in developing novel approximation schemes with guaranteed error control and will lead to provably convergent and optimal adaptive strategies for PDE problems stemming from the uncertainty quantification models of practical interest.

The open-source software package created within the project provides researchers with computational tools for experimentation and exploration, enabling them to apply our algorithms to other problems, develop new discretisations, and test alternative algorithms. The package is also valuable as a teaching and learning tool for students studying state-of-the-art computational techniques for uncertainty quantification.
Sectors Education

Energy

Environment
URL http://web.mat.bham.ac.uk/A.Bespalov/papers/index.html
 
Description Adaptive multilevel stochastic collocation methods for uncertainty quantification
Amount £56,461 (GBP)
Funding ID EP/W010925/1 
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 12/2021 
End 11/2022
 
Title Stochastic T-IFISS 
Description Stochastic T-IFISS is a Matlab package for solving steady-state diffusion problems with random coefficients using stochastic Galerkin finite element method. This is free software. It can be redistributed and/or modified under the terms of the GNU Lesser General Public License. 
Type Of Technology Software 
Year Produced 2022 
Open Source License? Yes  
Impact Stochastic T-IFISS extends the core version of T-IFISS (freely available from http://www.maths.manchester.ac.uk/~djs/ifiss/tifiss.html) to cover stochastic Galerkin approximations of diffusion problems with random coefficients, the associated a posteriori error estimation and adaptive algorithms, including goal-oriented adaptivity and multilevel adaptivity. The software has been used to generate numerical results in several papers. It provides researchers with computational tools for experimentation and exploration, enabling them to apply the algorithms to other problems, develop new discretisations, and test alternative algorithms. The package is also valuable as a teaching and learning tool for students studying state-of-the-art computational techniques for uncertainty quantification. 
URL https://github.com/albespalov/Stochastic_T-IFISS