# Catherine E. Powell's Research Page

My research is mainly concerned with aspects of

**numerical analysis**related to the numerical solution of partial differential equations (PDEs). In particular, I am interested in the numerical solution of PDEs with uncertain/random inputs in engineering applications and efficient and accurate numerical algorithms for

**uncertainty quantification (UQ)**.

**Key Topics:**numerical analysis, UQ, finite elements, mixed finite elements, error estimation, a posteriori error estimation, adaptivity, stochastic finite elements, numerical linear algebra, saddlepoint systems, fast solvers, iterative solvers, preconditioning.

### Current & Recent Research Projects

- Multilevel Intrusive UQ Methods (EPSRC-funded).
- Models to Decisions (M2D) Network on Decision Making Under Uncertainty (UKRI-funded).
- Numerical Analysis of Adaptive UQ Algorithms for PDEs with Random Inputs (EPSRC-funded).

### Textbook

- An Introduction to Computational Stochastic PDEs (by G. Lord, C.E. Powell and T. Shardlow). Preface.
- Textbook designed to introduce recent graduates with a good grounding in applied mathematics and numerical analysis to stochastic differential equations.
- Assumes no prior knowledge of probability or statistics.

- MATLAB and python codes associated with the book are available HERE. Solutions to exercises are available for verified course instructions (by emailing one of the authors).

### Software

- IFISS3D: An add-on MATLAB toolbox that extends the IFISS (Incompressible Flow & Iterative Solver) finite element software package for (deterministic) PDEs to three dimensional spatial domains. This code is intended to be useful for teaching purposes and as a source of reproducible test matrices of arbitrarily large dimension arising from finite element discretisations.

- ML-SGFEM: A MATLAB toolbox for investigating adaptive multilevel stochastic Galerkin finite element approximation of parametric elliptic PDEs, driven by hierarchical error estimation. This code is associated with the paper: SIAM J. Comp. Sci., 41(3), A1681--A1705, (2019). A User Guide is HERE.

- S-IFISS: a simple extension of the IFISS software that implements (single level) stochastic Galerkin finite element discretisations of parametric elliptic PDEs. This code is also provided as supplementary material for the a posteriori error estimation strategy described in the paper: SIAM Journal Sci. Comp. 36(2), A339--A363 (2014).

- PIFISS: (Potential Incompressible Flow Software Library). This code illustrates the use of Krylov subspace solvers and preconditioning techniques based on algebraic multigrid for primal and mixed formulations of the steady-state diffusion problem. Test problems are included with discontinuous and anisotropic diffusion coefficients.

- Gaussian random field generators in the NAG Fortran Library. The routines G05ZMF, G05ZNF, G05ZPF G05ZQF, G05ZRF, G05ZSF and G05ZTF in Mark 24 of the NAG Fortran Library are based on the method of circulant embedding. These were developed during a collaboration between PhD student Phillip Taylor, Tony Shardlow and NAG (who partially funded the PhD project). The codes can be used to generate realisations of mean zero Gaussian random fields and stochastic processes with both standard and user-defined stationary covariance functions. Technical report in NAG news 119 , January 2014.

### Publications & Preprints

- Efficient Adaptive Stochastic Collocation Strategies for Advection-Diffusion Problems with Uncertain Inputs (with BM Kent, DJ Silvester and MJ Zimon). To appear in Journal of Scientific Computing (2023). arXiv preprint

- IFISS3D: A computational laboratory for investigating finite element approximation in three dimensions (with G Papanikos, DJ Silvester). To appear in ACM Transactions on Mathematical Software (2023).
arXiv preprint

- Alternating Energy Minimization Methods for Multi-term Matrix Equations (with K Lee, HC Elman, and D Lee).

BIT Numerical Mathematics, 1-30, (2022). arXiv preprint

- Parameter-Robust Stochastic Galerkin Mixed Approximation for Linear Poroelasticity with Uncertain Inputs (with Arbaz Khan).

SIAM J. Comp. Sci., 43(4), B855-B883, (2021) . arXiv preprint

- Robust A Posteriori Error Estimation for Stochastic Galerkin Formulations of Parameter-Dependent Linear Elasticity Equations (with A. Khan, D. Silvester, and A. Bespalov).
Math. Comp., 90(328), pp. 613-636, (2021). arXiv preprint.

- A Locally Adaptive Bayesian Cubature Method (with M. Fisher, C. Oates, A. Teckentrup).

Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), PMLR 108:1265-1275, (2020). arXiv preprint

- Efficient Adaptive Multilevel Stochastic Galerkin Approximation Using Implicit A Posteriori Error Estimation (with Adam Crowder, and A. Bespalov).

SIAM J. Comp. Sci., 41(3), A1681--A1705, (2019) . arXiv preprint.

- Surrogate Accelerated Bayesian Inversion for the Determination of the Thermal Diffusivity of a Material (with J. Rynn, S. Cotter, L. Wright).

Metrologia, 56 (1), 015018, (2019). (Open Access).

- Robust Preconditioning for Stochastic Galerkin Formulations of Parameter-dependent Linear Elasticity Equations (with Arbaz Khan and David Silvester). SIAM J. Comp. Sci., 41(1), A402-A421, (2019) . arXiv preprint.

- CBS Constants and Their Role in Error Estimation for Stochastic Galerkin Finite Element Methods (with Adam Crowder).

Journal of Scientific Computing, 77 (2), pp.1030–1054 (2018) . MIMS preprint.

- Robust A Posteriori Error Estimators for Mixed Approximation of Nearly
Incompressible Elasticity (with Arbaz Khan and David Silvester).

Int. J. Numer. Methods Engrg. To appear (2019). Related technical reports : Report 1 . Report 2 (lowest-order case) .

- Efficient Reduced Basis methods for Saddle Point Problems with Applications in Groundwater Flow (with Craig Newsum).

SIAM/ASA Journal on Uncertainty Quantification, 5(1), pp 1248-1278 (2017) . MIMS preprint .

- An Efficient Reduced Basis Solver for Stochastic Galerkin Matrix Equations (with Valeria Simoncini and David Silvester).

SIAM J. Comp. Sci. 39(1), pp A141-A163 (2017). MIMS preprint .

- Solving log-transformed random diffusion problems by stochastic Galerkin mixed finite element methods (with Elisabeth Ullmann).

SIAM/ASA J. Uncertainty Quantification 3-1 (2015), pp. 509--534 . MIMS preprint .

- Preface to "An Introduction to Computational Stochastic PDEs".

Extract from: An Introduction to Computational Stochastic PDEs, T. Shardlow, C.E. Powell, G.J. Lord, Cambridge University Press, 2014.

- A preconditioner for fictitious domain formulations of elliptic PDEs on uncertain parameterized domains (with Andrew Gordon).

SIAM/ASA J. Uncertainty Quantification 2-1 (2014), pp.622--646. MIMS preprint .

- Energy norm a posteriori error estimation for parametric operator equations (with Alex Bespalov and David Silvester).

SIAM Journal Sci. Comp. 36(2), A339--A363 (2014) . MIMS preprint .

- Preconditioning steady-state Navier-Stokes equations with random data (with David Silvester).

SIAM Journal Sci. Comp. 34(5). (2012) . MIMS preprint .

- A framework for the development of implicit solvers for incompressible flow problems (with Alex Bespalov and David Silvester).

Discrete and Continuous Dynamical Systems - Series S. Vol 5 (6), pp. 1195--1221. (2012). (OPEN ACCESS)

- A Priori Error Analysis of Stochastic Galerkin Mixed Approximations of Elliptic PDEs with Random Data (with Alex Bespalov and David Silvester).

SIAM Journal on Numerical Analysis. 50(4), 2039--2063, (2012). MIMS preprint .

- Solving Stochastic Collocation Systems with Algebraic Multigrid (with Andrew Gordon).

IMA Journal of Numerical Analysis. Volume 32 (3). pp 1051-1070. (2011) . MIMS preprint .

- Preconditioning Stochastic Galerkin Saddle Point Systems (with Elisabeth Ullmann).

SIAM J. Matrix. Anal., 31, pp. 2813--2840 (2010) . MIMS preprint .

- Solving Stochastic Collocation Systems with Algebraic Multigrid (with Andrew Gordon).

Numerical Mathematics and Advanced Applications. Proceedings of ENUMATH 2009. Springer.

- H(div) Preconditioning for a Mixed Finite Element Formulation of The Stochastic Diffusion Problem.

(With Darran Furnival and Howard Elman). Mathematics of Computation. 79, pp, 733--760 (2010). (electronic 2009). MIMS preprint .

- Efficient Solvers for a Linear Stochastic Galerkin Mixed Formulation of Diffusion Problems with Random Data.

(With O.G. Ernst, E. Ullmann, D.J. Silvester). SIAM Journal Sci. Comp., 31,2, pp. 1424-1447, (2009) . MIMS preprint .

- Block-diagonal preconditioning for spectral stochastic finite element systems. (With Howard Elman).

IMA Journal of Numerical Analysis. 29 (2), pp. 350--375. , April 2009. (Submitted May**2007,**electronic access 2008).

- PIFISS. Potential (Incompressible) Flow & Iterative Solution Software Guide. MIMS Preprint 2007.

- Robust Preconditioning for Second-Order Elliptic PDEs with Random Field Coefficients. MIMS Preprint 2006.

- Parameter-free H(div) preconditioning for mixed finite element formulation of diffusion problems.

IMA J. Numer. Anal.,25(4), pp.783-796, 2005. MIMS preprint.

- Improving the Forward Solver for the Complete Electrode Model in EIT using Algebraic Multigrid

(with M. Soleimani and N. Polydorides). IEEE Transactions on Medical Imaging, 24(5), pp.577-584, 2005. MIMS preprint .

- Black-Box Preconditioning for Self-Adjoint Elliptic PDEs
*(with David Silvester)*

Lecture Notes in Computational Science and Engineering, 35, Springer, 2003 (CISC 2002). MIMS preprint .

- Optimal Preconditioning for Raviart-Thomas Mixed Formulation of Second-Order Elliptic Problems
*(with David Silvester)*

SIMAX, Vol. 25, No.3, pp.718-738, 2003. MIMS preprint .

### Theses

- Optimal Preconditioning for Mixed Finite Element Formulations of Second-Order Elliptic Problems

*Ph.D thesis, UMIST, 2003.*(Available at MIMS E-print server or request by e-mail.)

- An analysis of a mixed finite element method for the biharmonic equation.

MSc. Dissertation. UMIST, 2000.

### Past Research Projects

- Analysis of Numerical Methods for Partial Differential Equations with Random Data (EPSRC-funded).

- UQ in computer simulations of groundwater flow problems with emphasis on contaminant transport (British Council & DAAD funded).