Forthcoming Seminars › Seminar Series

Computational Mathematics and Applications

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, 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, 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, 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, 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, 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, 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.

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

Syndicate content