Publications (111)126 Total impact

Article: Preconditioning
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ABSTRACT: The computational solution of problems can be restricted by the availability of solution methods for linear(ized) systems of equations. In conjunction with iterative methods, preconditioning is often the vital component in enabling the solution of such systems when the dimension is large. We attempt a broad review of preconditioning methods.  [Show abstract] [Hide abstract]
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 RBFQR method (Fornberg et al, SIAM J. Sci. Comput. (33), 2011) which can stably compute Gaussian radial basis function interpolants.  [Show abstract] [Hide abstract]
ABSTRACT: The solution of quadratic or locally quadratic extremum problems subject to linear(ized) constraints gives rise to linear systems in saddle point form. This is true whether in the continuous or the discrete setting, so saddle point systems arising from the discretization of partial differential equation problems, such as those describing electromagnetic problems or incompressible flow, lead to equations with this structure, as do, for example, interior point methods and the sequential quadratic programming approach to nonlinear optimization. This survey concerns iterative solution methods for these problems and, in particular, shows how the problem formulation leads to natural preconditioners which guarantee a fast rate of convergence of the relevant iterative methods. These preconditioners are related to the original extremum problem and their effectiveness  in terms of rapidity of convergence  is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matrix on which iteration convergence depends.  [Show abstract] [Hide abstract]
ABSTRACT: Circulant preconditioning for symmetric Toeplitz linear systems is well established; theoretical guarantees of fast convergence for the conjugate gradient method are descriptive of the convergence seen in computations. This has led to robust and highly efficient solvers based on use of the fast Fourier transform exactly as originally envisaged in [G. Strang, Stud. Appl. Math., 74 (1986), pp. 171176]. For nonsymmetric systems, the lack of generally descriptive convergence theory for most iterative methods of Krylov type has provided a barrier to such a comprehensive guarantee, though several methods have been proposed and some analysis of performance with the normal equations is available. In this paper, by the simple device of reordering, we rigorously establish a circulant preconditioned short recurrence Krylov subspace iterative method of minimum residual type for nonsymmetric (and possibly highly nonnormal) Toeplitz systems. Convergence estimates similar to those in the symmetric case are established. 
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.  [Show abstract] [Hide abstract]
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 pressurelike field. For the threefield problem, we develop and analyse new preconditioners and we show numerically that the new threefield preconditioners are optimal in terms of problem size and less sensitive to model parameters compared to the twofield preconditioner. This extends the applicability of optimal preconditioners for coupled mantle/magma dynamics into parameter regimes of important physical interest. 
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. 
Chapter: Optimization with PDE Constraints
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ABSTRACT: We consider the convergence of the algorithm GMRES of Saad and Schultz for solving linear equations Bx=b, where B ∈ Cn × n is nonsingular and diagonalizable, and b ∈ Cn. Our analysis explicitly includes the initial residual vector r0. We show that the GMRES residual norm satisfies a weighted polynomial leastsquares problem on the spectrum of B, and that GMRES convergence reduces to an ideal GMRES problem on a rank1 modification of the diagonal matrix of eigenvalues of B. Numerical experiments show that the new bounds can accurately describe GMRES convergence.  [Show abstract] [Hide abstract]
ABSTRACT: Mastronardi and Van Dooren [SIAM J. Matrix Anal. Appl., 34 ( 2013), pp. 173196] 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 rank1 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.  [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. 
Article: Preconditioners for stateconstrained optimal control problems with MoreauYosida penalty function
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ABSTRACT: Optimal control problems with partial differential equations as constraints play an important role in many applications. The inclusion of bound constraints for the state variable poses a significant challenge for optimization methods. Our focus here is on the incorporation of the constraints via the Moreau–Yosida regularization technique. This method has been studied recently and has proven to be advantageous compared with other approaches. In this paper, we develop robust 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. Copyright © 2012 John Wiley & Sons, Ltd.  [Show abstract] [Hide abstract]
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 Wathentype 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 threedimensional numerical examples on unstructured grids, from simple illustrate problems through to large problems on subduction zonelike geometries. The computer code to reproduce all numerical examples is freely available as supporting material.  [Show abstract] [Hide abstract]
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.  [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.  [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.  [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) 
Article: RegularizationRobust Preconditioners for TimeDependent PDEConstrained 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. 
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.  [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, 2011
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4k  Citations  
126.00  Total Impact Points  
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Institutions

19702015

University of Oxford
 Mathematical Institute
Oxford, England, United Kingdom


1999

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


19871995

University of Bristol
 School of Mathematics
Bristol, England, United Kingdom


19861991

Stanford University
Stanford, California, United States
