### Topics:

numerical analysis, finite elements, error estimation, mixed finite elements, stochastic finite elements, numerical linear algebra, saddlepoint systems, fast solvers, preconditioning, multigrid, algebraic multigrid### Applications:

groundwater flow modelling, fluid flow, image processing

This is a new graduate 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.

More information can be found at the CUP book web page, the Amazon UK book web page or the Amazon US web page .

An electronic version of the book in secure pdf format is also available from ebooks.com. Note that although other electronic formats are available, the mathematics only displays correctly in pdf format.

MATLAB files associated with the book and solutions (for verified course instructors only) are available HERE .

- S-IFISS: a simple extension of the Incompressible Flow & Iterative Solver Software (IFISS) for stochastic Galerkin finite element discretisations of elliptic PDEs with random diffusion coefficients. This code is also provided as supplementary material for the a posteriori error estimation strategy recently proposed in the paper: 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) .

- 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 new 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 the PhD student Phillip Taylor 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 NAGnews 119 , January 2014.

- Numerical Methods for generating Gaussian Random Fields. Lecture notes from the Porous Media-Processes and Mathematics (PMPM) annual meeting 2014 (held at the ICMS, Edinburgh). Part I and Part II .

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

- Analysis of Numerical Methods for Partial Differential Equations with Random Data. (3 year EPSRC funded). See research outputs .

- Uncertainty quantification in computer simulations of groundwater flow problems with emphasis on contaminant transport