Computational Mathematics and Applications (past)
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Thu, 17/06/2010 14:00 |
Prof Joseph Ward (Texas A&M University) |
Computational Mathematics and Applications  |
3WS SR |
| This talk will focus on highly localized basis functions which exist for certain kernels and spaces associated with these kernels. Such kernels include certain radial basis functions (RBFs), their restrictions to spheres (SBFs), and their restrictions to more general manifolds embeddable in Rd. The first part of the talk will be of an introductory nature. It will discuss radial basis functions and their restriction to manifolds which give rise to various kernels on these manifolds. The talk will then focus on the development (for certain kernels) of highly localized Lagrange functions which serve as effective bases: i.e., bases which are stable and local. Scaled versions of these bases will then be used to establish the stability of the L2 minimization operator in Lp, 1 ≤ p ≤ ∞, thus obtaining a multivariate analogue of a result of de Boor. Since these bases are scalable with the data, they have potential uses beyond approximation including meshless methods and, more generally, computations of a multiresolution nature. The talk is primarily based on joint work with T. Hangelbroek, F. J. Narcowich and X. Sun. | |||
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Thu, 10/06/2010 14:00 |
Prof Gil Strang (MIT) |
Computational Mathematics and Applications |
3WS SR |
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Thu, 03/06/2010 14:00 |
Dr Garth Wells (University of Cambridge) |
Computational Mathematics and Applications |
3WS SR |
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Thu, 27/05/2010 14:00 |
Prof Mahadevan Ganesh (Colorado School of Mines) |
Computational Mathematics and Applications |
3WS SR |
| We discuss a class of high-order spectral-Galerkin surface integral algorithms with specific focus on simulating the scattering of electromagnetic waves by a collection of three dimensional deterministic and stochastic particles. | |||
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Thu, 20/05/2010 14:00 |
Dr Jan Van lent (UWE Bristol) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
| In the eighteenth century Gaspard Monge considered the problem of finding the best way of moving a pile of material from one site to another. This optimal transport problem has many applications such as mesh generation, moving mesh methods, image registration, image morphing, optical design, cartograms, probability theory, etc. The solution to an optimal transport problem can be found by solving the Monge-Ampère equation, a highly nonlinear second order elliptic partial differential equation. Leonid Kantorovich, however, showed that it is possible to analyse optimal transport problems in a framework that naturally leads to a linear programming formulation. In recent years several efficient methods have been proposed for solving the Monge-Ampère equation. For the linear programming problem, standard methods do not exploit the special properties of the solution and require a number of operations that is quadratic or even cubic in the number of points in the discretisation. In this talk I will discuss techniques that can be used to obtain more efficient methods. Joint work with Chris Budd (University of Bath). | |||
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Thu, 13/05/2010 14:00 |
Dr Francisco Bernal (OCCAM, University of Oxford) |
Computational Mathematics and Applications |
3WS SR |
| Meshless (or meshfree) methods are a relatively new numerical approach for the solution of ordinary- and partial differential equations. They offer the geometrical flexibility of finite elements but without requiring connectivity from the discretization support (ie a mesh). Meshless methods based on the collocation of radial basis functions (RBF methods) are particularly easy to code, and have a number of theoretical advantages as well as practical drawbacks. In this talk, an adaptive RBF scheme is presented for a novel application, namely the solution of (a rather broad class of) delayed- and neutral differential equations. | |||
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Thu, 06/05/2010 14:00 |
Prof Roland Herzog (Chemnitz University of Technology) |
Computational Mathematics and Applications |
3WS SR |
| We consider saddle point problems arising as (linearized) optimality conditions in elliptic optimal control problems. The efficient solution of such systems is a core ingredient in second-order optimization algorithms. In the spirit of Bramble and Pasciak, the preconditioned systems are symmetric and positive definite with respect to a suitable scalar product. We extend previous work by Schoeberl and Zulehner and consider problems with control and state constraints. It stands out as a particular feature of this approach that an appropriate symmetric indefinite preconditioner can be constructed from standard preconditioners for those matrices which represent the inner products, such as multigrid cycles. Numerical examples in 2D and 3D are given which illustrate the performance of the method, and limitations and open questions are addressed. | |||
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Thu, 29/04/2010 14:00 |
Prof Dominique Orban (Ecole Polytechnique de Montréal and GERAD) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
| Interior-point methods for linear and convex quadratic programming require the solution of a sequence of symmetric indefinite linear systems which are used to derive search directions. Safeguards are typically required in order to handle free variables or rank-deficient Jacobians. We propose a consistent framework and accompanying theoretical justification for regularizing these linear systems. Our approach is akin to the proximal method of multipliers and can be interpreted as a simultaneous proximal-point regularization of the primal and dual problems. The regularization is termed "exact" to emphasize that, although the problems are regularized, the algorithm recovers a solution of the original problem. Numerical results will be presented. If time permits we will illustrate current research on a matrix-free implementation. This is joint work with Michael Friedlander, University of British Columbia, Canada | |||
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Thu, 22/04/2010 14:00 |
Dr Martin van Gijzen (Delft University of Technology) |
Computational Mathematics and Applications |
3WS SR |
| Shifted Laplace preconditioners have attracted considerable attention as a technique to speed up convergence of iterative solution methods for the Helmholtz equation. In the talk we present a comprehensive spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian. Our analysis is valid under general conditions. The propagating medium can be heterogeneous, and the analysis also holds for different types of damping, including a radiation condition for the boundary of the computational domain. By combining the results of the spectral analysis of the preconditioned Helmholtz operator with an upper bound on the GMRES-residual norm we are able to derive an optimal value for the shift, and to explain the mesh-depency of the convergence of GMRES preconditioned with a shifted Laplacian. We will illustrate our results with a seismic test problem. Joint work with: Yogi Erlangga (University of British Columbia) and Kees Vuik (TU Delft) | |||
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Thu, 11/03/2010 14:00 |
Prof. Yangfeng Su (Fudan University Shanghai) |
Computational Mathematics and Applications |
3WS SR |
| Nonlinear eigenvalue problem (NEP) is a class of eigenvalue problems where the matrix depends on the eigenvalue. We will first introduce some NEPs in real applications and some algorithms for general NEPs. Then we introduce our recent advances in NEPs, including second order Arnoldi algorithms for large scale quadratic eigenvalue problem (QEP), analysis and algorithms for symmetric eigenvalue problem with nonlinear rank-one updating, a new linearization for rational eigenvalue problem (REP). | |||
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Thu, 04/03/2010 14:00 |
Mr. Thomas Goldstein (University of California, Los Angeles) |
Computational Mathematics and Applications |
3WS SR |
| This talk will introduce L1-regularized optimization problems that arise in image processing, and numerical methods for their solution. In particular, we will focus on methods of the split-Bregman type, which very efficiently solve large scale problems without regularization or time stepping. Applications include image denoising, segmentation, non-local filters, and compressed sensing. | |||
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Thu, 25/02/2010 14:00 |
Prof. Ekkehard Sachs (University of Trier) |
Computational Mathematics and Applications |
3WS SR |
| There is a widespread use of mathematical tools in finance and its importance has grown over the last two decades. In this talk we concentrate on optimization problems in finance, in particular on numerical aspects. In this talk, we put emphasis on the mathematical problems and aspects, whereas all the applications are connected to the pricing of derivatives and are the outcome of a cooperation with an international finance institution. As one example, we take an in-depth look at the problem of hedging barrier options. We review approaches from the literature and illustrate advantages and shortcomings. Then we rephrase the problem as an optimization problem and point out that it leads to a semi-infinite programming problem. We give numerical results and put them in relation to known results from other approaches. As an extension, we consider the robustness of this approach, since it is known that the optimality is lost, if the market data change too much. To avoid this effect, one can formulate a robust version of the hedging problem, again by the use of semi-infinite programming. The numerical results presented illustrate the robustness of this approach and its advantages. As a further aspect, we address the calibration of models being used in finance through optimization. This may lead to PDE-constrained optimization problems and their solution through SQP-type or interior-point methods. An important issue in this context are preconditioning techniques, like preconditioning of KKT systems, a very active research area. Another aspect is the preconditioning aspect through the use of implicit volatilities. We also take a look at the numerical effects of non-smooth data for certain models in derivative pricing. Finally, we discuss how to speed up the optimization for calibration problems by using reduced order models. | |||
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Thu, 18/02/2010 14:00 |
Dr. Alison Ramage (University of Strathclyde) |
Computational Mathematics and Applications |
3WS SR |
| Saddle-point problems occur frequently in liquid crystal modelling. For example, they arise whenever Lagrange multipliers are used for the pointwise-unit-vector constraints in director modelling, or in both general director and order tensor models when an electric field is present that stems from a constant voltage. Furthermore, in a director model with associated constraints and Lagrange multipliers, together with a coupled electric-field interaction, a particular ”double” saddle-point structure arises. This talk will focus on a simple example of this type and discuss appropriate numerical solution schemes. This is joint work with Eugene C. Gartland, Jr., Department of Mathematical Sciences, Kent State University. | |||
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Thu, 11/02/2010 14:00 |
Dr. Melina Freitag (University of Bath) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
We show that data assimilation using four-dimensional variation
(4DVar) can be interpreted as a form of Tikhonov regularisation, a
familiar method for solving ill-posed inverse problems. It is known from
image restoration problems that  -norm penalty regularisation recovers
sharp edges in the image better than the  -norm penalty
regularisation. We apply this idea to 4DVar for problems where shocks are
present and give some examples where the -norm penalty approach
performs much better than the standard -norm regularisation in 4DVar. |
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Thu, 04/02/2010 14:00 |
Dr Peter Giesl (University of Sussex) |
Computational Mathematics and Applications |
3WS SR |
| In dynamical systems given by an ODE, one is interested in the basin of attraction of invariant sets, such as equilibria or periodic orbits. The basin of attraction consists of solutions which converge towards the invariant set. To determine the basin of attraction, one can use a solution of a certain linear PDE which can be approximated by meshless collocation. The basin of attraction of an equilibrium can be determined through sublevel sets of a Lyapunov function, i.e. a scalar-valued function which is decreasing along solutions of the dynamical system. One method to construct such a Lyapunov function is to solve a certain linear PDE approximately using Meshless Collocation. Error estimates ensure that the approximation is a Lyapunov function. The basin of attraction of a periodic orbit can be analysed by Borg’s criterion measuring the time evolution of the distance between adjacent trajectories with respect to a certain Riemannian metric. The sufficiency and necessity of this criterion will be discussed, and methods how to compute a suitable Riemannian metric using Meshless Collocation will be presented in this talk. | |||
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Thu, 28/01/2010 14:00 |
Dr. Catherine Powell (University of Manchester) |
Computational Mathematics and Applications |
3WS SR |
| In the last few years, there has been renewed interest in stochastic finite element methods (SFEMs), which facilitate the approximation of statistics of solutions to PDEs with random data. SFEMs based on sampling, such as stochastic collocation schemes, lead to decoupled problems requiring only deterministic solvers. SFEMs based on Galerkin approximation satisfy an optimality condition but require the solution of a single linear system of equations that couples deterministic and stochastic degrees of freedom. This is regarded as a serious bottleneck in computations and the difficulty is even more pronounced when we attempt to solve systems of PDEs with random data via stochastic mixed FEMs. In this talk, we give an overview of solution strategies for the saddle-point systems that arise when the mixed form of the Darcy flow problem, with correlated random coefficients, is discretised via stochastic Galerkin and stochastic collocation techniques. For the stochastic Galerkin approach, the systems are orders of magnitude larger than those arising for deterministic problems. We report on fast solvers and preconditioners based on multigrid, which have proved successful for deterministic problems. In particular, we examine their robustness with respect to the random diffusion coefficients, which can be either a linear or non-linear function of a finite set of random parameters with a prescribed probability distribution. | |||
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Tue, 26/01/2010 14:00 |
Dr Konstantinos Zyglakis (OCCAM (Oxford)) |
Computational Mathematics and Applications |
3WS SR |
| In this talk we describe a general framework for deriving modified equations for stochastic differential equations with respect to weak convergence. We will start by quickly recapping of how to derive modified equations in the case of ODEs and describe how these ideas can be generalized in the case of SDEs. Results will be presented for first order methods such as the Euler-Maruyama and the Milstein method. In the case of linear SDEs, using the Gaussianity of the underlying solutions, we will derive a SDE that the numerical method solves exactly in the weak sense. Applications of modified equations in the numerical study of Langevin equations and in the calculation of effective diffusivities will also be discussed. | |||
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Thu, 21/01/2010 14:00 |
Prof. Ernesto Estrada (University of Strathclyde) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
| A brief introduction to the field of complex networks is carried out by means of some examples. Then, we focus on the topics of defining and applying centrality measures to characterise the nodes of complex networks. We combine this approach with methods for detecting communities as well as to identify good expansion properties on graphs. All these concepts are formally defined in the presentation. We introduce the subgraph centrality from a combinatorial point of view and then connect it with the theory of graph spectra. Continuing with this line we introduce some modifications to this measure by considering some known matrix functions, e.g., psi matrix functions, as well as new ones introduced here. Finally, we illustrate some examples of applications in particular the identification of essential proteins in proteomic maps. | |||
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Tue, 19/01/2010 14:00 |
Dr Orly Alter (University of Texas at Austin) |
Computational Mathematics and Applications |
3WS SR |
| Future discovery and control in biology and medicine will come from the mathematical modeling of large-scale molecular biological data, such as DNA microarray data, just as Kepler discovered the laws of planetary motion by using mathematics to describe trends in astronomical data. In this talk, I will demonstrate that mathematical modeling of DNA microarray data can be used to correctly predict previously unknown mechanisms that govern the activities of DNA and RNA. First, I will describe the computational prediction of a mechanism of regulation, by using the pseudoinverse projection and a higher-order singular value decomposition to uncover a genome-wide pattern of correlation between DNA replication initiation and RNA expression during the cell cycle. Then, I will describe the recent experimental verification of this computational prediction, by analyzing global expression in synchronized cultures of yeast under conditions that prevent DNA replication initiation without delaying cell cycle progression. Finally, I will describe the use of the singular value decomposition to uncover "asymmetric Hermite functions," a generalization of the eigenfunctions of the quantum harmonic oscillator, in genome-wide mRNA lengths distribution data. These patterns might be explained by a previously undiscovered asymmetry in RNA gel electrophoresis band broadening and hint at two competing evolutionary forces that determine the lengths of gene transcripts. | |||
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Thu, 14/01/2010 14:00 |
Prof. Zdenek Strakos (Academy of Sciences of the Czech Republic) |
Computational Mathematics and Applications |
3WS SR |
| Regularization techniques based on the Golub-Kahan iterative bidiagonalization belong among popular approaches for solving large discrete ill-posed problems. First, the original problem is projected onto a lower dimensional subspace using the bidiagonalization algorithm, which by itself represents a form of regularization by projection. The projected problem, however, inherits a part of the ill-posedness of the original problem, and therefore some form of inner regularization must be applied. Stopping criteria for the whole process are then based on the regularization of the projected (small) problem. We consider an ill-posed problem with a noisy right-hand side (observation vector), where the noise level is unknown. We show how the information from the Golub-Kahan iterative bidiagonalization can be used for estimating the noise level. Such information can be useful for constructing efficient stopping criteria in solving ill-posed problems. This is joint work by Iveta Hnětynková, Martin Plešinger, and Zdeněk Strakoš (Faculty of Mathematics and Physics, Charles University, and Institute of Computer Science, Academy of Sciences, Prague) | |||
