Publications (77)50.76 Total impact

Technical Report: A BramblePasciaklike method with applications in optimization
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ABSTRACT: Saddlepoint 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 saddlepoint systems arising in Optimization can be derived from the BramblePasciak 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 nonstandard 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. 
Article: Convexity and Solvability for Compactly Supported Radial Basis Functions with Different Shapes
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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; · 1.71 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The solution of CahnHilliard 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 doublewell potentials but not for nonsmooth potentials as considered here. The existing bound constraints are incorporated via the MoreauYosida regularization technique. We develop effective preconditioners for the efficient solution of the Newton steps associated with the fast solution of the MoreauYosida regularized problem. Numerical results illustrate the efficiency of our approach. Moreover, precise eigenvalue intervals are given for the preconditioned system using a doublewell potential. A comparison between the smooth and nonsmooth CahnHilliard inpainting models shows that the latter achieves better results.SIAM Journal on Imaging Sciences 01/2014; 7(1). · 2.97 Impact Factor  IMA Journal of Numerical Analysis 01/2014; 34(2). · 1.33 Impact Factor
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ABSTRACT: Amongst recent contributions to preconditioning methods for saddle point systems, standard iterative methods in nonstandard inner products have been usefully employed. Krzy˙ zanowski (Numer. Linear Algebra Appl. 2011; 18:123–140) identified a twoparameter 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, selfadjoint 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 selfadjoint with respect to a nonstandard inner product always allow a MINREStype method (WPMINRES) 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 CGlike method (WPCG) can be reliably used. We establish eigenvalue expressions for Krzy˙ zanowski preconditioners and show that for a specific choice of parameters, although the Krzy˙ zanowski preconditioned saddle point matrix is selfadjoint 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 WPMINRES can be reliably used leads to a preconditioner for which, for certain parameter choices, WPCG is reliably applicable. That is, combining two indefinite preconditioners can lead to a positive definite preconditioner. This combination preconditioner outperforms 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). · 1.20 Impact Factor  [Show abstract] [Hide abstract]
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 nonHermitian, 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 selfadjoint 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 selfadjoint 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 eigenvaluedependent convergence is observed both for the nonstandard method and for standard GMRES.Journal of Computational and Applied Mathematics 09/2013; 249:57–68. · 0.99 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In this manuscript, we describe effective solvers for the optimal control of stabilized convectiondiffusion control problems. We employ the local projection stabilization, which results in the same matrix system whether the discretizethenoptimize or optimizethendiscretize 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 BramblePasciak 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 stepsize h and the regularization parameter β for a range of problems.Electronic transactions on numerical analysis ETNA 01/2013; 40. · 1.26 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The fast iterative solution of optimal control problems, and in particular PDEconstrained 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 timedependent PDEconstrained 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 stepsize and regularization parameter. (© 2012 WileyVCH Verlag GmbH & Co. KGaA, Weinheim)PAMM 12/2012; 12(1). 
Article: REGULARIZATIONROBUST PRECONDITIONERS FOR TIMEDEPENDENT PDE CONSTRAINED OPTIMIZATION PROBLEMS
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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 timesteps.SIAM Journal on Matrix Analysis and Applications 10/2012; 33(4). · 1.34 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Iterative methods of Krylovsubspace 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 NavierStokes equations in streamfunction and vorticity formulation. By using a Picard iteration, we use this technology to solve fully nonlinear NavierStokes problems. The solvers which result scale very well with problem parameters. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011Numerical Methods for Partial Differential Equations 04/2012; 28(3):888  898. · 1.21 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Solving efficiently the incompressible Navier–Stokes equations is a major challenge, especially in the threedimensional 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 blockpreconditioner that guarantees convergence in 2 or 3 iterations. In this paper, we investigate the numerical behavior for the threedimensional liddriven 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 steadystate 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. · 1.35 Impact Factor 
Article: A new approximation of the Schur complement in preconditioners for PDE‐constrained optimization
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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 PDEconstrained 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 nonstandard 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 10/2011; · 1.20 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Solving problems regarding the optimal control of partial differential equations (PDEs)—also known as PDEconstrained 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 lowertriangular 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 J. Scientific Computing. 01/2011; 33:29032926.  [Show abstract] [Hide abstract]
ABSTRACT: This paper proposes a new preconditioning scheme for a linear system with a saddlepoint structure arising from a hybrid approximation scheme on the sphere, an approximation scheme that combines (local) spherical radial basis functions and (global) spherical polynomials. Making use of a recently derived infsup condition [13] and the Brezzi stability and convergence theorem for this approximation scheme, we show that the linear system can be optimally preconditioned with a suitable blockdiagonal preconditioner. Numerical experiments with a nonuniform distribution of data points support the theoretical conclusions. Comment: 14 pages, 1 figure, submittedNumerische Mathematik 09/2010; · 1.33 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Existing convergence bounds for Krylov subspace methods such as GMRES for nonsymmetric linear systems give little mathematical guidance for the choice of preconditioner. Here, we establish a desirable mathematical property of a preconditioner which guarantees that convergence of a minimum residual method will essentially depend only on the eigenvalues of the preconditioned system, as is true in the symmetric case. Our theory covers only a subset of nonsymmetric coefficient matrices but computations indicate that it might be more generally applicable.09/2010;  [Show abstract] [Hide abstract]
ABSTRACT: Optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering, particularly in problems of design. The solution of such PDEconstrained optimization problems is usually a major computational task. Here we consider simple problems of this type: distributed control problems in which the 2 and 3dimensional Poisson problem is the PDE. The largedimensional linear systems which result from discretization and which need to be solved are of saddlepoint type. We introduce two optimal preconditioners for these systems, which lead to convergence of symmetric Krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem. These preconditioners are block structured and involve standard multigrid cycles. The optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases. The theoretical proof indicates that these approaches may have much broader applicability for other PDEs.SIAM J. Scientific Computing. 01/2010; 32:271298.  [Show abstract] [Hide abstract]
ABSTRACT: Saddlepoint systems arise in many applications areas, in fact in any situation where an extremum principle arises with constraints. The Stokes problem describing slow viscous flow of an incompressible fluid is a classic example coming from PDEs and in the area of optimization such problems are ubiquitous. In this paper we present a framework into which many wellknown methods for solving saddlepoint systems fit. Based on this description we show how new approaches for the solution of saddlepoint systems arising in optimization can be derived from the BramblePasciak 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 nonstandard 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.SIAM J. Scientific Computing. 01/2010; 32:249270.  [Show abstract] [Hide abstract]
ABSTRACT: We analyze pressure stabilized finite element methods for the solution of the generalized Stokes problem and investigate their stability and convergence properties. An important feature of the method is that the pressure gradient unknowns can be eliminated locally thus leading to a decoupled system of equations. Although stability of the method has been established, for the homogeneous Stokes equations, the proof given here is based on the existence of a special interpolant with additional orthogonal property with respect to the projection space. This, makes it a lot simpler and more attractive. The resulting stabilized method is shown to lead to optimal rates of convergence for both velocity and pressure approximations.Computer Methods in Applied Mechanics and Engineering 01/2009; 198(5):877883. · 2.62 Impact Factor  01/2008;
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ABSTRACT: Saddle point problems arise frequently in many applications in science and engineering, including constrained optimization, mixed finite element formulations of partial differential equations, circuit analysis, and so forth. Indeed the formulation of most problems with constraints gives rise to saddle point systems. This paper provides a concise overview of iterative approaches for the solution of such systems which are of particular importance in the context of large scale computation. In particular we describe some of the most useful preconditioning techniques for Krylov subspace solvers applied to saddle point problems, including block and constraint preconditioners.12/2007: pages 195211;
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2k  Citations  
50.76  Total Impact Points  
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Institutions

1970–2013

University of Oxford
 Mathematical Institute
Oxford, ENG, United Kingdom 
University of Strathclyde
 Department of Mathematics and Statistics
Glasgow, SCT, United Kingdom


2007

Technische Universiteit Eindhoven
Eindhoven, North Brabant, Netherlands


2003

University of Maryland, College Park
Maryland, United States 
Duke University
 Department of Biology
Durham, NC, United States


2001

Uppsala University
 Department of Information Technology
Uppsala, Uppsala, Sweden


1999

University of Sussex
Brighton, England, United Kingdom 
The University of Manchester
 School of Mathematics
Manchester, England, United Kingdom


1995

University of Bristol
Bristol, England, United Kingdom
