Andrew J. Wathen

University of Oxford, Oxford, England, United Kingdom

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Publications (102)115.94 Total impact

  • A. J. Wathen
    Acta Numerica 05/2015; 24:329-376. DOI:10.1017/S0962492915000021
  • Jennifer Pestana, Andrew J. Wathen
    SIAM Review 02/2015; 57(1):71-91. DOI:10.1137/130934921 · 4.79 Impact Factor
  • J. Pestana, A. J. Wathen
    SIAM Journal on Matrix Analysis and Applications 01/2015; 36(1):273-288. DOI:10.1137/140974213 · 1.81 Impact Factor
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    Andrew J Wathen, Sheng-Xin Zhu
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    ABSTRACT: This paper focuses on the spectral distribution of kernel matrices related to radial basis functions. The asymptotic behaviour of eigenvalues of kernel matrices related to radial basis functions with different smoothness are studied. These results are obtained by estimated the coefficients of an orthogonal expansion of the underlying kernel function. Beside many other results, we prove that there are exactly k+d−1 d−1 eigenvalues in the same order for analytic separable kernel functions like the Gaussian in R d . This gives theoretical support for how to choose the diagonal scaling matrix in the RBF-QR method (Fornberg et al, SIAM J. Sci. Comput. (33), 2011) which can stably compute Gaussian radial basis function interpolants.
    Numerical Algorithms 01/2015; DOI:10.1007/s11075-015-9970-0 · 1.01 Impact Factor
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    ABSTRACT: Saddle-point systems arise in many applications areas, in fact in any situation where extremum principle arises with constraints. The Stokes problem describing slow viscous flow of an incompressible fluid is an example coming form partial differential equations and in the area of Optimization such problems are ubiquitous. In this manuscript we show how new approaches for the solution of saddle-point systems arising in Optimization can be derived from the Bramble-Pasciak Conjugate Gradient approach widely used in PDEs and more recent generalizations thereof. In particular we derive a class of new solution methods based on the use of Preconditioned Conjugate Gradients in non-standard inner products and demonstrate how these can be understood through more standard machinery. We show connections to Constraint Preconditioning and give the results of numerical computations on a number of standard Optimization test examples.
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    ABSTRACT: For a prescribed porosity, the coupled magma/mantle flow equations can be formulated as a two field system of equations with velocity and pressure unknowns. Previous work has shown that while optimal preconditioners for the two field formulation can be constructed, the construction of preconditioners that are uniform with respect to model parameters is difficult. This limits the applicability of two field preconditioners in certain regimes of practical interest. We address this issue by reformulating the governing equations as a three field problem, which removes a term that was problematic in the two field formulation in favour of an additional equation for a pressure-like field. For the three-field problem, we develop and analyse new preconditioners and we show numerically that the new three-field preconditioners are optimal in terms of problem size and less sensitive to model parameters compared to the two-field preconditioner. This extends the applicability of optimal preconditioners for coupled mantle/magma dynamics into parameter regimes of important physical interest.
  • Shengxin Zhu, Andrew J. Wathen
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    ABSTRACT: It is known that interpolation with radial basis functions of the same shape can guarantee a nonsingular interpolation matrix, whereas little was known when one uses various shapes. In this paper, we prove that functions from a class of compactly supported radial basis functions are convex on a certain region; based on this local convexity and other local geometrical properties of the interpolation points, we construct a sufficient condition which guarantees diagonally dominant interpolation matrices for radial basis functions interpolation with different shapes. The proof is constructive and can be used to design algorithms directly. Numerical examples show that the scheme has a low accuracy but can be implemented efficiently. It can be used for inaccurate models where efficiency is more desirable. Large scale 3D implicit surface reconstruction problems are used to demonstrate the utility and reasonable results can be obtained efficiently.
    Journal of Scientific Computing 10/2014; DOI:10.1007/s10915-014-9919-9 · 1.70 Impact Factor
  • J. Pestana, A. J. Wathen
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    ABSTRACT: Mastronardi and Van Dooren [SIAM J. Matrix Anal. Appl., 34 ( 2013), pp. 173-196] recently introduced the block antitriangular ("Batman") decomposition for symmetric indefinite matrices. Here we show the simplification of this factorization for saddle point matrices and demonstrate how it represents the common nullspace method. We show that rank-1 updates to the saddle point matrix can be easily incorporated into the factorization and give bounds on the eigenvalues of matrices important in saddle point theory. We show the relation of this factorization to constraint preconditioning and how it transforms but preserves the structure of block diagonal and block triangular preconditioners.
    SIAM Journal on Matrix Analysis and Applications 04/2014; 35(2):339-353. DOI:10.1137/130934933 · 1.81 Impact Factor
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    Jessica Bosch, David Kay, Martin Stoll, Andrew J. Wathen
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    ABSTRACT: The solution of Cahn-Hilliard variational inequalities is of interest in many applications. We discuss the use of them as a tool for binary image inpainting. This has been done before using double-well potentials but not for nonsmooth potentials as considered here. The existing bound constraints are incorporated via the Moreau-Yosida regularization technique. We develop effective preconditioners for the efficient solution of the Newton steps associated with the fast solution of the Moreau-Yosida regularized problem. Numerical results illustrate the efficiency of our approach. Moreover, precise eigenvalue intervals are given for the preconditioned system using a double-well potential. A comparison between the smooth and nonsmooth Cahn-Hilliard inpainting models shows that the latter achieves better results.
    SIAM Journal on Imaging Sciences 01/2014; 7(1). DOI:10.1137/130921842 · 2.87 Impact Factor
  • IMA Journal of Numerical Analysis 01/2014; 34(2). DOI:10.1093/imanum/drt025 · 1.25 Impact Factor
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    ABSTRACT: This article considers the iterative solution of a finite element discretisation of magma dynamics equations. In simplified form, the magma dynamics equations share some features of the Stokes equations. We therefore formulate, analyse and numerically test a Elman, Silvester and Wathen-type block preconditioner for magma dynamics. We prove analytically, and demonstrate numerically, optimality of the preconditioner. The presented analysis highlights the dependence of the preconditioner on parameters in the magma dynamics equations that can affect convergence of iterative linear solvers. The analysis is verified through a range of two- and three-dimensional numerical examples on unstructured grids, from simple illustrate problems through to large problems on subduction zone-like geometries. The computer code to reproduce all numerical examples is freely available as supporting material.
    SIAM Journal on Scientific Computing 11/2013; 36(4). DOI:10.1137/130946678 · 1.94 Impact Factor
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    ABSTRACT: Amongst recent contributions to preconditioning methods for saddle point sys-tems, standard iterative methods in nonstandard inner products have been use-fully employed. Krzy˙ zanowski (Numer. Linear Algebra Appl. 2011; 18:123–140) identified a two-parameter family of preconditioners in this context and Stoll and Wathen (SIAM J. Matrix Anal. Appl. 2008; 30:582–608) introduced combination preconditioning, where two preconditioners, self-adjoint with respect to different inner products, can lead to further preconditioners and associated bilinear forms or inner products. Preconditioners that render the preconditioned saddle point matrix nonsymmetric but self-adjoint with respect to a nonstandard inner prod-uct always allow a MINRES-type method (W-PMINRES) to be applied in the relevant inner product. If the preconditioned matrix is also positive definite with respect to the inner product a more efficient CG-like method (W-PCG) can be reliably used. We establish eigenvalue expressions for Krzy˙ zanowski precondition-ers and show that for a specific choice of parameters, although the Krzy˙ zanowski preconditioned saddle point matrix is self-adjoint with respect to an inner product, it is never positive definite. We provide explicit expressions for the combination of certain preconditioners and prove the rather counterintuitive result that the combination of two specific preconditioners for which only W-PMINRES can be reliably used leads to a preconditioner for which, for certain parameter choices, W-PCG is reliably applicable. That is, combining two indefinite preconditioners can lead to a positive definite preconditioner. This combination preconditioner out-performs either of the two preconditioners from which it is formed for a number of test problems.
    Numerical Linear Algebra with Applications 10/2013; 20(5). DOI:10.1002/nla.1843 · 1.42 Impact Factor
  • Jennifer Pestana, Andrew J. Wathen
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    ABSTRACT: We consider the solution of left preconditioned linear systems P−1Cx=P−1cP−1Cx=P−1c, where P,C∈Cn×nP,C∈Cn×n are non-Hermitian, c∈Cnc∈Cn, and CC, PP, and P−1CP−1C are diagonalisable with spectra symmetric about the real line. We prove that, when PP and CC are self-adjoint with respect to the same Hermitian sesquilinear form, the convergence of a minimum residual method in a particular nonstandard inner product applied to the preconditioned linear system is bounded by a term that depends only on the spectrum of P−1CP−1C. The inner product is related to the spectral decomposition of PP. When PP is self-adjoint with respect to a nearby Hermitian sesquilinear form to CC, the convergence of a minimum residual method in this nonstandard inner product applied to the preconditioned linear system is bounded by a term involving the eigenvalues of P−1CP−1C and a constant factor. The size of this factor is related to the nearness of the Hermitian sesquilinear forms. Numerical experiments indicate that for certain matrices eigenvalue-dependent convergence is observed both for the nonstandard method and for standard GMRES.
    Journal of Computational and Applied Mathematics 09/2013; 249:57–68. DOI:10.1016/j.cam.2013.02.020 · 1.08 Impact Factor
  • John W. Pearson, Andrew J. Wathen
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    ABSTRACT: In this manuscript, we describe effective solvers for the optimal control of stabilized convection-diffusion control problems. We employ the local projection stabilization, which results in the same matrix system whether the discretize-then-optimize or optimize-then-discretize approach for this problem is used. We then derive two effective preconditioners for this problem, the first to be used with MINRES and the second to be used with the Bramble-Pasciak Conjugate Gradient method. The key components of both preconditioners are an accurate mass matrix approximation, a good approximation of the Schur complement, and an appropriate multigrid process to enact this latter approximation. We present numerical results to illustrate that these preconditioners result in convergence in a small number of iterations, which is robust with respect to the step-size h and the regularization parameter β for a range of problems.
    Electronic transactions on numerical analysis ETNA 01/2013; 40. · 0.89 Impact Factor
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    John W. Pearson, Martin Stoll, Andrew J. Wathen
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    ABSTRACT: The fast iterative solution of optimal control problems, and in particular PDE-constrained optimization problems, has become an active area of research in applied mathematics and numerical analysis. In this paper, we consider the solution of a class of time-dependent PDE-constrained optimization problems, specifically the distributed control of the heat equation. We develop a strategy to approximate the (1, 1)-block and Schur complement of the saddle point system that results from solving this problem, and therefore derive a block diagonal preconditioner to be used within the MINRES algorithm. We present numerical results to demonstrate that this approach yields a robust solver with respect to step-size and regularization parameter. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)
    PAMM 12/2012; 12(1). DOI:10.1002/pamm.201210002
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    John W Pearson, Martin Stoll, Andrew J Wathen
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    ABSTRACT: In this article, we motivate, derive and test effective preconditioners to be used with the Minres algorithm for solving a number of saddle point systems, which arise in PDE constrained optimization problems. We consider the distributed control problem involving the heat equation with two different functionals, and the Neumann boundary control problem involving Poisson's equation and the heat equation. Crucial to the effectiveness of our preconditioners in each case is an effective approximation of the Schur complement of the matrix system. In each case, we state the problem being solved, propose the preconditioning approach, prove relevant eigenvalue bounds, and provide numerical results which demonstrate that our solvers are effective for a wide range of regularization parameter values, as well as mesh sizes and time-steps.
    SIAM Journal on Matrix Analysis and Applications 10/2012; 33(4). DOI:10.1137/110847949 · 1.81 Impact Factor
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    John W. Pearson, Andrew J. Wathen
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    ABSTRACT: Saddle point systems arise widely in optimization problems with constraints. The utility of Schur complement approximation is now broadly appreciated in the context of solving such saddle point systems by iteration. In this short manuscript, we present a new Schur complement approximation for PDE-constrained optimization, an important class of these problems. Block diagonal and block triangular preconditioners have previously been designed to be used to solve such problems along with MINRES and non-standard Conjugate Gradients, respectively; with appropriate approximation blocks, these can be optimal in the sense that the time required for solution scales linearly with the problem size, however small the mesh size we use. In this paper, we extend this work to designing such preconditioners for which this optimality property holds independently of both the mesh size and the Tikhonov regularization parameter β that is used. This also leads to an effective symmetric indefinite preconditioner that exhibits mesh and β independence. We motivate the choice of these preconditioners based on observations about approximating the Schur complement obtained from the matrix system, derive eigenvalue bounds that verify the effectiveness of the approximation and present numerical results that show that these new preconditioners work well in practice. Copyright © 2011 John Wiley & Sons, Ltd.
    Numerical Linear Algebra with Applications 10/2012; 19(5). DOI:10.1002/nla.814 · 1.42 Impact Factor
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    Faisal A. Fairag, Andrew J. Wathen
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    ABSTRACT: Iterative methods of Krylov-subspace type can be very effective solvers for matrix systems resulting from partial differential equations if appropriate preconditioning is employed. We describe and test block preconditioners based on a Schur complement approximation which uses a multigrid method for finite element approximations of the linearized incompressible Navier-Stokes equations in streamfunction and vorticity formulation. By using a Picard iteration, we use this technology to solve fully nonlinear Navier-Stokes problems. The solvers which result scale very well with problem parameters. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011
    Numerical Methods for Partial Differential Equations 05/2012; 28(3):888 - 898. DOI:10.1002/num.20661 · 1.06 Impact Factor
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    ABSTRACT: Solving efficiently the incompressible Navier–Stokes equations is a major challenge, especially in the three-dimensional case. The approach investigated by Elman et al. (Finite Elements and Fast Iterative Solvers. Oxford University Press: Oxford, 2005) consists in applying a preconditioned GMRES method to the linearized problem at each iteration of a nonlinear scheme. The preconditioner is built as an approximation of an ideal block-preconditioner that guarantees convergence in 2 or 3 iterations. In this paper, we investigate the numerical behavior for the three-dimensional lid-driven cavity problem with wedge elements; the ultimate motivation of this analysis is indeed the development of a preconditioned Krylov solver for stratified oceanic flows which can be efficiently tackled using such meshes. Numerical results for steady-state solutions of both the Stokes and the Navier–Stokes problems are presented. Theoretical bounds on the spectrum and the rate of convergence appear to be in agreement with the numerical experiments. Sensitivity analysis on different aspects of the structure of the preconditioner and the block decomposition strategies are also discussed. Copyright © 2011 John Wiley & Sons, Ltd.
    International Journal for Numerical Methods in Fluids 01/2012; 68(3):269 - 286. DOI:10.1002/fld.2505 · 1.33 Impact Factor
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    Tyrone Rees, Andrew J. Wathen
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    ABSTRACT: Solving problems regarding the optimal control of partial differential equations (PDEs)—also known as PDE-constrained optimization—is a frontier area of numerical analysis. Of particular interest is the problem of flow control, where one would like to effect some desired flow by exerting, for example, an external force. The bottleneck in many current algorithms is the solution of the optimality system—a system of equations in saddle point form that is usually very large and ill conditioned. In this paper we describe two preconditioners—a block diagonal preconditioner for the minimal residual method and a block lower-triangular preconditioner for a nonstandard conjugate gradient method—which can be effective when applied to such problems where the PDEs are the Stokes equations. We consider only distributed control here, although we believe other problems could be treated in the same way. We give numerical results, and we compare these with those obtained by solving the equivalent forward problem using similar techniques.
    SIAM Journal on Scientific Computing 01/2011; 33:2903-2926. DOI:10.1137/100798491 · 1.94 Impact Factor

Publication Stats

3k Citations
115.94 Total Impact Points

Institutions

  • 1970–2014
    • University of Oxford
      • Mathematical Institute
      Oxford, England, United Kingdom
  • 2001
    • Uppsala University
      • Department of Information Technology
      Uppsala, Uppsala, Sweden
  • 2000
    • Franklin and Marshall College
      Lancaster, California, United States
  • 1993–1999
    • The University of Manchester
      • School of Mathematics
      Manchester, England, United Kingdom
  • 1987–1995
    • University of Bristol
      • School of Mathematics
      Bristol, England, United Kingdom
  • 1991
    • Stanford University
      • Department of Computer Science
      Palo Alto, CA, United States
  • 1986
    • University of Reading
      • Department of Mathematics and Statistics
      Reading, England, United Kingdom