Past Seminars › Seminar Series

Computational Mathematics and Applications (past)

Thu, 22/05/2008
14:00 		Dr Michiel Hochstenbach (Technical University Eindhoven) Computational Mathematics and Applications Add to calendar Rutherford Appleton Laboratory, nr Didcot
An overview of the Jacobi-Davidson method
The Jacobi-Davidson method, proposed by Sleijpen and Van der Vorst more than a decade ago, has been successfully used to numerically solve large matrix eigenvalue problems. In this talk we will give an introduction to and an overview of this method, and also point out some recent developments.
Thu, 08/05/2008
14:00 		Prof Beresford Parlett (UC Berkeley) Computational Mathematics and Applications Add to calendar Comlab
The Envelope Method
The task is to compute orthogonal eigenvectors (without Gram-Schmidt) of symmetric tridiagonals for isolated clusters of close eigenvalues. We review an "old" method, the Submatrix method, and describe an extension which significantly enlarges the scope to include several mini-clusters within the given cluster. An essential feature is to find the envelope of the associated invariant subspace.
Thu, 01/05/2008
14:00 		Prof Nick Trefethen (Computing Laboratory, Oxford) Computational Mathematics and Applications Add to calendar Comlab
Eigenvalue avoidance
"Eigenvalue avoidance" or "level repulsion" refers to the tendency of eigenvalues of matrices or operators to be distinct rather than degenerate. The mathematics goes back to von Neumann and Wigner in 1929 and touches many subjects including numerical linear algebra, random matrix theory, chaotic dynamics, and number theory. This talk will be an informal illustrated discussion of various aspects of this phenomenon.
Thu, 17/04/2008
14:00 		Dr Nicolas Smith (Computing Laboratory, Oxford) Computational Mathematics and Applications Add to calendar Comlab
Developing multi-scale and multi-physics computational models of the heart
Thu, 06/03/2008
14:00 		Prof Volker Mehrmann (Technical University of Berlin) Computational Mathematics and Applications Add to calendar Rutherford Appleton Laboratory, nr Didcot
Nonlinear eigenvalue problems with structure. A challenge for current computational methods.
We discuss general and structured matrix polynomials which may be singular and may have eigenvalues at infinity. We discuss several real industrial applications ranging from acoustic field computations to optimal control problems. We discuss linearization and first order formulations and their relationship to the corresponding techniques used in the treatment of systems of higher order differential equations. In order to deal with structure preservation, we derive condensed/canonical forms that allow (partial) deflation of critical eigenvalues and the singular structure of the matrix polynomial. The remaining reduced order staircase form leads to new types of linearizations which determine the finite eigenvalues and corresponding eigenvectors. Based on these new linearization techniques we discuss new structure preserving eigenvalue methods and present several real world numerical examples.
Thu, 28/02/2008
14:00 		Prof Folkmar Bornemann (Technical University of Munich) Computational Mathematics and Applications Add to calendar Comlab
Painlevé Numerics: From Operator Determinants to the Chebfun System
Thu, 21/02/2008
14:00 		Prof Holger Wendland (University of Sussex) Computational Mathematics and Applications Add to calendar Comlab
Meshfree Methods: Theory and Applications
Meshfree methods become more and more important for the numerical simulation of complex real-world processes. Compared to classical, mesh-based methods they have the advantage of being more flexible, in particular for higher dimensional problems and for problems, where the underlying geometry is changing. However, often, they are also combined with classical methods to form hybrid methods. In this talk, I will discuss meshfree, kernel based methods. After a short introduction along the lines of optimal recovery, I will concentrate on results concerning convergence orders and stability. After that I will address efficient numerical algorithms. Finally, I will present some examples, including one from fluid-structure-interaction, which will demonstrate why these methods are currently becoming Airbus's preferred solution in Aeroelasticity.
Thu, 14/02/2008
14:00 		Prof Ya-xiang Yuan (Chinese Academy of Sciences, Beijing) Computational Mathematics and Applications Add to calendar Comlab
Distance Geometry Problem for Protein Modeling via Geometric Buildup
A well-known problem in protein modeling is the determination of the structure of a protein with a given set of inter-atomic or inter-residue distances obtained from either physical experiments or theoretical estimates. A general form of the problem is known as the distance geometry problem in mathematics, the graph embedding problem in computer science, and the multidimensional scaling problem in statistics. The problem has applications in many other scientific and engineering fields as well such as sensor network localization, image recognition, and protein classification. We describe the formulations and complexities of the problem in its various forms, and introduce a so-called geometric buildup approach to the problem. We present the general algorithm and discuss related computational issues including control of numerical errors, determination of rigid vs. unique structures, and tolerance of distance errors. The theoretical basis of the approach is established based on the theory of distance geometry. A group of necessary and sufficient conditions for the determination of a structure with a given set of distances using a geometric buildup algorithm are justified. The applications of the algorithm to model protein problems are demonstrated.
Thu, 07/02/2008
14:00 		Prof Paul Van Dooren (Universite catholique de louvain) Computational Mathematics and Applications Add to calendar Rutherford Appleton Laboratory, nr Didcot
Some graph optimization problems in data mining
Graph-theoretic ideas have become very useful in understanding modern large-scale data-mining techniques. We show in this talk that ideas from optimization are also quite useful to better understand the numerical behavior of the corresponding algorithms. We illustrate this claim by looking at two specific graph theoretic problems and their application in data-mining. The first problem is that of reputation systems where the reputation of objects and voters on the web are estimated; the second problem is that of estimating the similarity of nodes of large graphs. These two problems are also illustrated using concrete applications in data-mining.
Thu, 31/01/2008
14:00 		Prof Tom Melham (Computing Laboratory, Oxford) Computational Mathematics and Applications Add to calendar Comlab
Formal verification of an industrial floating-point adder
Thu, 17/01/2008
14:00 		Prof Zdenek Strakos (Academy of Sciences of the Czech Republic) Computational Mathematics and Applications Add to calendar Comlab
Nonlinear problems in analysis of Krylov subspace methods
Consider a system of linear algebraic equations $ Ax=b $ where $ A $ is an $ n $ by $ n $ real matrix and $ b $ a real vector of length $ n $. Unlike in the linear iterative methods based on the idea of splitting of $ A $, the Krylov subspace methods, which are used in computational kernels of various optimization techniques, look for some optimal approximate solution $ x^n $ in the subspaces $ {\cal K}_n (A, b) = \mbox{span} \{ b, Ab, \dots, A^{n-1}b\}, n = 1, 2, \dots $ (here we assume, with no loss of generality, $ x^0 = 0 $). As a consequence, though the problem $ Ax = b $ is linear, Krylov subspace methods are not. Their convergence behaviour cannot be viewed as an (unimportant) initial transient stage followed by the subsequent convergence stage. Apart from very simple, and from the point of view of Krylov subspace methods uninteresting cases, it cannot be meaningfully characterized by an asymptotic rate of convergence. In Krylov subspace methods such as the conjugate gradient method (CG) or the generalized minimal residual method (GMRES), the optimality at each step over Krylov subspaces of increasing dimensionality makes any linearized description inadequate. CG applied to $ Ax = b $ with a symmetric positive definite $ A $ can be viewed as a method for numerical minimization the quadratic functional $ 1/2 (Ax, x) - (b,x) $. In order to reveal its nonlinear character, we consider CG a matrix formulation of the Gauss-Christoffel quadrature, and show that it essentially solves the classical Stieltjes moment problem. Moreover, though the CG behaviour is fully determined by the spectral decomposition of the problem, the relationship between convergence and spectral information is nothing but simple. We will explain several phenomena where an intuitive commonly used argumentation can lead to wrong conclusions, which can be found in the literature. We also show that rounding error analysis of CG brings fundamental understanding of seemingly unrelated problems in convergence analysis and in theory of the Gauss-Christoffel quadrature. In remaining time we demonstrate that in the unsymmetric case the spectral information is not generally sufficient for description of behaviour of Krylov subspace methods. In particular, given an arbitrary prescribed convergence history of GMRES and an arbitrary prescribed spectrum of the system matrix, there is always a system $ Ax=b $ such that GMRES follows the prescribed convergence while $ A $ has the prescribed spectrum.
Thu, 10/01/2008
14:00 		Prof Ben Leimkuhler (University of Edinburgh) Computational Mathematics and Applications Add to calendar Comlab
Molecular Dynamics and the Accuracy of Averages
Thu, 29/11/2007
14:00 		Dr Rodrigo Platte (University of Oxford) Computational Mathematics and Applications Add to calendar Comlab
Polynomials and potential theory for Gaussian radial basis function interpolation

Radial basis function (RBF) methods have been successfully used to approximate functions in multidimensional complex domains and are increasingly being used in the numerical solution of partial differential equations. These methods are often called meshfree numerical schemes since, in some cases, they are implemented without an underlying grid or mesh.

The focus of this talk is on the class of RBFs that allow exponential convergence for smooth problems. We will explore the dependence of accuracy and stability on node locations of RBF interpolants. Because Gaussian RBFs with equally spaced centers are related to polynomials through a change of variable, a number of precise conclusions about convergence rates based on the smoothness of the target function will be presented. Collocation methods for PDEs will also be considered.

Thu, 22/11/2007
14:00 		Prof Stefan Ulbrich (TU Darmstadt) Computational Mathematics and Applications Add to calendar Comlab
Adaptive Multilevel Methods for PDE-Constrained Optimization
Adaptive discretizations and iterative multilevel solvers are nowadays well established techniques for the numerical solution of PDEs. The development of efficient multilevel techniques in the context of PDE-constrained optimization methods is an active research area that offers the potential of reducing the computational costs of the optimization process to an equivalent of only a few PDE solves. We present a general class of inexact adaptive multilevel SQP-methods for PDE-constrained optimization problems. The algorithm starts with a coarse discretization of the underlying optimization problem and provides 1. implementable criteria for an adaptive refinement strategy of the current discretization based on local error estimators and 2. implementable accuracy requirements for iterative solvers of the PDE and adjoint PDE on the current grid such that global convergence to the solution of the infinite-dimensional problem is ensured. We illustrate how the adaptive refinement strategy of the multilevel SQP-method can be implemented by using existing reliable a posteriori error estimators for the state and the adjoint equation. Moreover, we discuss the efficient handling of control constraints and describe how efficent multilevel preconditioners can be constructed for the solution of the arising linear systems. Numerical results are presented that illustrate the potential of the approach. This is joint work with Jan Carsten Ziems.
Thu, 15/11/2007
14:00 		Prof Jan Magnus (Tilburg University) Computational Mathematics and Applications Add to calendar Rutherford Appleton Laboratory, nr Didcot
On the estimation of a large sparse Bayesian system: the Snaer program
The Snaer program calculates the posterior mean and variance of variables on some of which we have data (with precisions), on some we have prior information (with precisions), and on some prior indicator ratios (with precisions) are available. The variables must satisfy a number of exact restrictions. The system is both large and sparse. Two aspects of the statistical and computational development are a practical procedure for solving a linear integer system, and a stable linearization routine for ratios. We test our numerical method for solving large sparse linear least-squares estimation problems, and find that it performs well, even when the $ n \times k $ design matrix is large ( $ nk = O (10^{8}) $ ).
Thu, 08/11/2007
14:00 		Dr Daan Huybrechs (KU Leuven) Computational Mathematics and Applications Add to calendar Comlab
On the benefits of Gaussian quadrature for oscillatory integrals
The evaluation of oscillatory integrals is often considered to be a computationally challenging problem. However, in many cases, the exact opposite is true. Oscillatory integrals are cheaper to evaluate than non-oscillatory ones, even more so in higher dimensions. The simplest strategy that illustrates the general approach is to truncate an asymptotic expansion, where available. We show that an optimal strategy leads to the construction of certain unconventional Gaussian quadrature rules, that converge at twice the rate of asymptotic expansions. We examine a range of one-dimensional and higher dimensional, singular and highly oscillatory integrals.
Thu, 01/11/2007
14:00 		Dr Laura Grigori (INRIA) Computational Mathematics and Applications Add to calendar Rutherford Appleton Laboratory, nr Didcot
Communication avoiding algorithms for dense LU and QR factorizations
We present algorithms for dense LU and QR factorizations that minimize the cost of communication. One of today's challenging technology trends is the increased communication cost. This trend predicts that arithmetic will continue to improve exponentially faster than bandwidth, and bandwidth exponentially faster than latency. The new algorithms for dense QR and LU factorizations greatly reduce the amount of time spent communicating, relative to conventional algorithms. This is joint work with James Demmel, Mark Hoemmen, Julien Langou, and Hua Xiang.

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

Syndicate content