# Past Computational Mathematics and Applications Seminar

Prof Ivan Graham
Abstract
When steady solutions of complex physical problems are computed numerically it is often crucial to compute their <em>stability</em> in order to, for example, check that the computed solution is "physical", or carry out a sensitivity analysis, or help understand complex nonlinear phenomena near a bifurcation point. Usually a stability analysis requires the solution of an <em>eigenvalue problem</em> which may arise in its own right or as an appropriate linearisation. In the case of discretized PDEs the corresponding matrix eigenvalue problem will often be of generalised form: \\ $Ax=\lambda Mx$ (1) \\ with $A$ and $M$ large and sparse. In general $A$ is unsymmetric and $M$ is positive semi-definite. Only a small number of "dangerous" eigenvalues are usually required, often those (possibly complex) eigenvalues nearest the imaginary axis. In this context it is usually necessary to perform "shift-invert" iterations, which require repeated solution of systems of the form \\ $(A - \sigma M)y = Mx$, (2) \\ for some <em>shift</em> $\sigma$ (which may be near a spectral point) and for various right-hand sides $x$. In large applications systems (2) have to be solved iteratively, requiring <em>"inner iterations"</em>. \\ \\ In this talk we will describe recent progress in the construction, analysis and implementation of fast algorithms for finding such eigenvalues, utilising algebraic domain decomposition techniques for the inner iterations. \\ \\ In the first part we will describe an analysis of inverse iteration techniques for (1) for a model problem in the presence of errors arising from inexact solves of (2). The delicate interplay between the convergence of the (outer) inverse iteration and the choice of tolerance for the inner solves can be used to determine an efficient iterative method provided a good preconditioner for $A$ is available. \\ \\ In the second part we describe an application to the computation of bifurcations in Navier-Stokes problems discretised by mixed finite elements applied to the velocity-pressure formulation. We describe the construction of appropriate preconditioners for the corresponding (3 x 3 block) version of (2). These use additive Schwarz methods and can be applied to any unstructured mesh in 2D or 3D and for any selected elements. An important part of the preconditioner is the adaptive coarsening strategy. At the heart of this are recent extensions of the Bath domain decomposition code DOUG, carried out by Eero Vainikko. \\ \\ An application to the computation of a Hopf bifurcation of planar flow around a cylinder will be given. \\ \\ This is joint work with Jörg Berns-Müller, Andrew Cliffe, Alastair Spence and Eero Vainikko and is supported by EPSRC Grant GR/M59075.
• Computational Mathematics and Applications Seminar
24 January 2002
14:00
Prof Lloyd N Trefethen
Abstract
• Computational Mathematics and Applications Seminar
29 November 2001
14:00
Prof Alastair Spence
Abstract
• Computational Mathematics and Applications Seminar
Dr Milan Mihajlovic
Abstract
In this presentation we examine the convergence characteristics of a Krylov subspace solver preconditioned by a new indefinite constraint-type preconditioner, when applied to discrete systems arising from low-order mixed finite element approximation of the classical biharmonic problem. The preconditioning operator leads to preconditioned systems having an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. We compare the convergence characteristics of a new approach with the convergence characteristics of a standard block-diagonal Schur complement preconditioner that has proved to be extremely effective in the context of mixed approximation methods. \\ \\ In the second part of the presentation we are concerned with the efficient parallel implementation of proposed algorithm on modern shared memory architectures. We consider use of the efficient parallel "black-box'' solvers for the Dirichlet Laplacian problems based on sparse Cholesky factorisation and multigrid, and for this purpose we use publicly available codes from the HSL library and MGNet collection. We compare the performance of our algorithm with sparse direct solvers from the HSL library and discuss some implementation related issues.
• Computational Mathematics and Applications Seminar
15 November 2001
14:00
Dr Raphael Hauser
Abstract
(Joint work with Felipe Cucker and Dennis Cheung, City University of Hong Kong.) \\ \\ Condition numbers are important complexity-theoretic tools to capture a "distillation" of the input aspects of a computational problem that determine the running time of algorithms for its solution and the sensitivity of the computed output. The motivation for our work is the desire to understand the average case behaviour of linear programming algorithms for a large class of randomly generated input data in the computational model of a machine that computes with real numbers. In this model it is not known whether linear programming is polynomial time solvable, or so-called "strongly polynomial". Closely related to linear programming is the problem of either proving non-existence of or finding an explicit example of a point in a polyhedral cone defined in terms of certain input data. A natural condition number for this computational problem was developed by Cheung and Cucker, and we analyse its distributions under a rather general family of input distributions. We distinguish random sampling of primal and dual constraints respectively, two cases that necessitate completely different techniques of analysis. We derive the exact exponents of the decay rates of the distribution tails and prove various limit theorems of complexity theoretic importance. An interesting result is that the existence of the k-th moment of Cheung-Cucker's condition number depends only very mildly on the distribution of the input data. Our results also form the basis for a second paper in which we analyse the distributions of Renegar's condition number for the randomly generated linear programming problem.
• Computational Mathematics and Applications Seminar
8 November 2001
14:00
Dr Mark Embree
Abstract
Toeplitz matrices enjoy the dual virtues of ubiquity and beauty. We begin this talk by surveying some of the interesting spectral properties of such matrices, emphasizing the distinctions between infinite-dimensional Toeplitz matrices and the large-dimensional limit of the corresponding finite matrices. These basic results utilize the algebraic Toeplitz structure, but in many applications, one is forced to spoil this structure with some perturbations (e.g., by imposing boundary conditions upon a finite difference discretization of an initial-boundary value problem). How do such perturbations affect the eigenvalues? This talk will address this question for "localized" perturbations, by which we mean perturbations that are restricted to a single entry, or a block of entries whose size remains fixed as the matrix dimension grows. One finds, for a broad class of matrices, that sufficiently small perturbations fail to alter the spectrum, though the spectrum is exponentially sensitive to other perturbations. For larger real single-entry perturbations, one observes the perturbed eigenvalues trace out curves in the complex plane. We'll show a number of illustrations of this phenomenon for tridiagonal Toeplitz matrices. \\ \\ This talk describes collaborative work with Albrecht Boettcher, Marko Lindner, and Viatcheslav Sokolov of TU Chemnitz.
• Computational Mathematics and Applications Seminar
1 November 2001
14:00
Dr Michael Ferris
Abstract
We investigate the use of interior-point and semismooth methods for solving quadratic programming problems with a small number of linear constraints, where the quadratic term consists of a low-rank update to a positive semi-definite matrix. Several formulations of the support vector machine fit into this category. An interesting feature of these particular problems is the volume of data, which can lead to quadratic programs with between 10 and 100 million variables and, if written explicitly, a dense $Q$ matrix. Our codes are based on OOQP, an object-oriented interior-point code, with the linear algebra specialized for the support vector machine application. For the targeted massive problems, all of the data is stored out of core and we overlap computation and I/O to reduce overhead. Results are reported for several linear support vector machine formulations demonstrating that the methods are reliable and scalable and comparing the two approaches.
• Computational Mathematics and Applications Seminar
18 October 2001
14:00
Dr Marco Marletta
Abstract
Non-selfadjoint singular differential equation eigenproblems arise in a number of contexts, including scattering theory, the study of quantum-mechanical resonances, and hydrodynamic and magnetohydrodynamic stability theory. \\ \\ It is well known that the spectra of non-selfadjoint operators can be pathologically sensitive to perturbation of the operator. Wilkinson provides matrix examples in his classical text, while Trefethen has studied the phenomenon extensively through pseudospectra, which he argues are often of more physical relevance than the spectrum itself. E.B. Davies has studied the phenomenon particularly in the context of Sturm-Liouville operators and has shown that the eigenfunctions and associated functions of non-selfadjoint singular Sturm-Liouville operators may not even form a complete set in $L^2$. \\ \\ In this work we ask the question: under what conditions can one expect the regularization process used for selfadjoint singular Sturm-Liouville operators to be successful for non-selfadjoint operators? The answer turns out to depend in part on the so-called Sims Classification of the problem. For Sims Case I the process is not guaranteed to work, and indeed Davies has very recently described the way in which spurious eigenvalues may be generated and converge to certain curves in the complex plane. \\ \\ Using the Titchmarsh-Weyl theory we develop a very simple numerical procedure which can be used a-posteriori to distinguish genuine eigenvalues from spurious ones. Numerical results indicate that it is able to detect not only the spurious eigenvalues due to the regularization process, but also spurious eigenvalues due to the numerics on an already-regular problem. We present applications to quantum mechanical resonance calculations and to the Orr-Sommerfeld equation. \\ \\ This work, in collaboration with B.M. Brown in Cardiff, has recently been generalized to Hamiltonian systems.
• Computational Mathematics and Applications Seminar