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 PDE models with uncertain/random inputs, and efficient algorithms for uncertainty quantification (UQ).
Topics: numerical analysis, uncertainty quantification, 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 Research Projects
An Introduction to Computational Stochastic PDEs (by G. Lord, C.E. Powell and T. Shardlow). Preface.
This is a textbook designed to introduce recent graduates with a good grounding in applied mathematics and numerical analysis to stochastic differential equations. It assumes no prior knowledge of probability or statistics.
Note that although electronic versions of the book seems to be available from various vendors, the mathematics only displays correctly in pdf format!
MATLAB files associated with the book and solutions (for verified course instructors only) are available HERE .
Journal Papers & Preprints
Most of the following pieces of work can be obtained from journals or as preprints. Copies of older papers which have no active links can be obtained from me by sending me an e-mail and asking nicely.
- Parameter-Robust Stochastic Galerkin Mixed Approximation for Linear Poroelasticity with Uncertain Inputs (with Arbaz Khan). To appear in SISC (2021).
- Alternating Energy Minimization Methods for Multi-term Matrix Equations (with K Lee, HC Elman, and D Lee). Submitted (2020). arXiv preprint
- A Locally Adaptive Bayesian Cubature Method (with M. Fisher, C. Oates, A. Teckentrup), Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics (AISTATS), PMLR 108:1265-1275, (2020). 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.
- 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, the 8th European Conference on Numerical Mathematics and Advanced Applications. 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 .
- 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