# Past Computational Mathematics and Applications Seminar

11 May 2000
14:00
Abstract
Interval arithmetic is a way to produce guaranteed enclosures of the results of numerical calculations. Suppose $f(x)$ is a real expression in real variables $x= (x_1, \ldots, x_n)$, built up from the 4 basic arithmetic operations and other 'standard functions'. Let $X_1, \ldots, X_n$ be (compact) real intervals. The process of {\em interval evaluation} of $f(X_1, ..., X_n)$ replaces each real operation by the corresponding interval operation wherever it occurs in $f$, e.g. $A \times B$ is the smallest interval containing $\{a \times b \mid a \in A, b \in B\}$. \\ \\ As is well known, it yields a guaranteed enclosure for the true range $\{f(x_1, \ldots, x_n) \mid x_1 \in X_1, \ldots, x_n \in X_n\}$, provided no exceptions such as division by (an interval containing) zero occur during evaluation. \\ \\ Interval arithmetic takes set inputs and produces set outputs. Noting this, we show there is a consistent way to extend arithmetic to $R^* = R \cup \{-\infty, +\infty\}$, such that interval evaluation continues to give enclosures, and there are {\em no exceptions}. The basic ideas are: the usual set-theory meaning of evaluating a relation at a set; and taking topological closure of the graph of a function in a suitable $(R^{*})^n$. It gives rigorous meaning to intuitively sensible statements like $1/0 = \{-\infty, +\infty\}$, $0/0 = R^*$ (but $(x/x)_{|x=0} = 1$), $\sin(+\infty) = [-1,1]$, etc. \\ \\ A practical consequence is that an exception-free floating-point interval arithmetic system is possible. Such a system is implemented at hardware level in the new Sun Fortran compiler, currently on beta-release.
• Computational Mathematics and Applications Seminar
Prof Nick Higham
Abstract
The Cholesky factorization approach to solving the symmetric definite generalized eigenvalue problem $Ax = \lambda Bx$, where $A$ is symmetric and $B$ is symmetric positive definite, computes a Cholesky factorization $B = LL^T$ and solves the equivalent standard symmetric eigenvalue problem $C y = \l y$ where $C = L^{-1} A L^{-T}$. Provided that a stable eigensolver is used, standard error analysis says that the computed eigenvalues are exact for $A+\dA$ and $B+\dB$ with $\max( \normt{\dA}/\normt{A}, \normt{\dB}/\normt{B} )$ bounded by a multiple of $\kappa_2(B)u$, where $u$ is the unit roundoff. We take the Jacobi method as the eigensolver and explain how backward error bounds potentially much smaller than $\kappa_2(B)u$ can be obtained. To show the practical utility of our bounds we describe a vibration problem from structural engineering in which $B$ is ill conditioned yet the error bounds are small. We show how, in cases of instability, iterative refinement based on Newton's method can be used to produce eigenpairs with small backward errors. Our analysis and experiments also give insight into the popular Cholesky--QR method used in LAPACK, in which the QR method is used as the eigensolver.
• Computational Mathematics and Applications Seminar
15 March 2000
14:00
Prof Albrecht Böttcher
Abstract
In contrast to spectra, pseudospectra of large Toeplitz matrices behave as nicely as one could ever expect. We demonstrate some basic phenomena of the asymptotic distribution of the spectra and pseudospectra of Toeplitz matrices and show how by employing a few simple $C^*$-algebra arguments one can prove rigorous convergence results for the pseudospectra. The talk is a survey of the development since a 1992 paper by Reichel and Trefethen and is not addressed to specialists, but rather to a general mathematically interested audience.
• Computational Mathematics and Applications Seminar
9 March 2000
14:00
Abstract
This lecture is about the extension of our CAD-Free platform to the simulation and sensitivity analysis for design and control of multi-model configurations. We present the different ingredients of the platform using simple models for the physic of the problem. This presentation enables for an easy evaluation of coupling, design and control strategies. \\ \\ Sensitivity analysis has been performed using automatic differentiation in direct or reverse modes and finite difference or complex variable methods. This former approach is interesting for cases where only one control parameter is involved as we can evaluate the state and sensitivity in real time with only one evaluation. \\ \\ We show that for some class of applications, incomplete sensitivities can be evaluated dropping the state dependency which leads to a drastic cost reduction. \\ \\ The design concerns the structural characteristic of the model and our control approach is based in perturbating the inflow incidence by a time dependent flap position described by a dynamic system encapsulating a gradient based minimization algorithm expressed as dynamic system. This approach enables for the definition of various control laws for different minimization algorithm. \\ \\ We present dynamic minimization algorithms based on the coupling of several heavy ball method. This approach enables for global minimization at a cost proportional to the number of balls times the cost of one steepest descent minimization. \\ \\ Two and three dimension flutter problem simulation and control are presented and the sensitivity of the different parameters in the model is discussed.
• Computational Mathematics and Applications Seminar
24 February 2000
14:00
--
Abstract
This seminar has been cancelled.
• Computational Mathematics and Applications Seminar
Dr Kurt Lust
Abstract
There is a growing interest in the study of periodic phenomena in large-scale nonlinear dynamical systems. Often the high-dimensional system has only low-dimensional dynamics, e.g., many reaction-diffusion systems or Navier-Stokes flows at low Reynolds number. We present an approach that exploits this property in order to compute branches of periodic solutions of the large system of ordinary differential equations (ODEs) obtained after a space discretisation of the PDE. We call our approach the Newton-Picard method. Our method is based on the recursive projection method (RPM) of Shroff and Keller but extends this method in many different ways. Our technique tries to combine the performance of straightforward time integration with the advantages of solving a nonlinear boundary value problem using Newton's method and a direct solver. Time integration works well for very stable limit cycles. Solving a boundary value problem is expensive, but works also for unstable limit cycles. \\ \\ We will present some background material on RPM. Next we will explain the basic features of the Newton-Picard method for single shooting. The linearised system is solved by a combination of direct and iterative techniques. First, we isolate the low-dimensional subspace of unstable and weakly stable modes (using orthogonal subspace iteration) and project the linearised system on this subspace and on its (high-dimensional) orthogonal complement. In the high-dimensional subspace we use iterative techniques such as Picard iteration or GMRES. In the low-dimensional (but "hard") subspace, direct methods such as Gaussian elimination or a least-squares are used. While computing the projectors, we also obtain good estimates for the dominant, stability-determining Floquet multipliers. We will present a framework that allows us to monitor and steer the convergence behaviour of the method. \\ \\ RPM and the Newton-Picard technique have been developed for PDEs that reduce to large systems of ODEs after space discretisation. In fact, both methods can be applied to any large system of ODEs. We will indicate how these methods can be applied to the discretisation of the Navier-Stokes equations for incompressible flow (which reduce to an index-2 system of differential-algebraic equations after space discretisation when written in terms of velocity and pressure.) \\ \\ The Newton-Picard method has already been extended to the computation of bifurcation points on paths of periodic solutions and to multiple shooting. Extension to certain collocation and finite difference techniques is also possible.
• Computational Mathematics and Applications Seminar
10 February 2000
14:00
Prof Tom Hughes
Abstract
• Computational Mathematics and Applications Seminar
3 February 2000
14:00
Prof Henk van der Vorst
Abstract
Krylov subspace methods offer good possibilities for the solution of large sparse linear systems of equations.For general systems, some of the popular methods often show an irregular type of convergence behavior and one may wonder whether that could be improved or not. Many suggestions have been made for improvement and the question arises whether these corrections are cosmetic or not. There is also the question whether the irregularity shows inherent numerical instability. In such cases one should take extra care in the application of smoothing techniques. We will discuss strategies that work well and strategies that might have been expected to work well.
• Computational Mathematics and Applications Seminar