Computational Mathematics and Applications
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Thu, 13/10/2005 14:00 |
Prof Chris Budd (University of Bath) |
Computational Mathematics and Applications  |
Comlab |
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Thu, 20/10/2005 14:00 |
Prof Jacek Gondzio (University of Edinburgh) |
Computational Mathematics and Applications |
Comlab |
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We discuss a method for solving very large structured symmetric indefinite equation systems arising in optimization with interior point methods. Many real-life economic models involve system dynamics, spatial distribution or uncertainty and lead to large-scale optimization problems. Such problems usually have a hidden structure: they are constructed by replication of some small generic block. The linear algebra subproblems which arise in optimization algorithms for such problems involve matrices which are not only sparse, but they additionally display a block-structure with many smaller blocks sparsely distributed in the large matrix. We have developed a structure-exploiting parallel interior point solver for optimization problems. Its design uses object-orientated programming techniques. The progress OOPS (Object-Orientated Parallel Solver: http://www.maths.ed.ac.uk/~gondzio/parallel/solver.html) on a number of different computing platforms and achieves scalability on a number of different computing platforms. We illustrate its performance on a collection of problems with sizes reaching 109 variables arising from asset liability management and portfolio optimization. This is a joint work with Andreas Grothey. |
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Thu, 27/10/2005 14:00 |
Dr Pierre-Antoine Absil (University of Cambridge) |
Computational Mathematics and Applications |
Comlab |
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It is well known that the computation of a few extreme eigenvalues, and the corresponding eigenvectors, of a symmetric matrix A can be rewritten as computing extrema of the Rayleigh quotient of A. However, since the Rayleigh quotient is a homogeneous function of degree zero, its extremizers are not isolated. This difficulty can be remedied by restricting the search space to a well-chosen manifold, which brings the extreme eigenvalue problem into the realm of optimization on manifolds. In this presentation, I will show how a recently-proposed generalization of trust-region methods to Riemannian manifolds applies to this problem, and how the resulting algorithms compare with existing ones. I will also show how the Joint Diagonalization problem (that is, approximately diagonalizing a collection of symmetric matrices via a congruence transformation) can be tackled by a differential geometric approach. This problem has an important application in Independent Component Analysis. |
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Thu, 03/11/2005 14:00 |
Dr John Reid (Rutherford Appleton Laboratory) |
Computational Mathematics and Applications |
Comlab |
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Direct methods for solving large sparse linear systems of equations are popular because of their generality and robustness. As the requirements of computational scientists for more accurate models increases, so inevitably do the sizes of the systems that must be solved and the amount of memory needed by direct solvers. For many users, the option of using a computer with a sufficiently large memory is either not available or is too expensive. Using a preconditioned iterative solver may be possible but for the "tough" systems that arise from many practical applications, the difficulties involved in finding and computing a good preconditioner can make iterative methods infeasible. An alternative is to use a direct solver that is able to hold its data structures on disk, that is, an out-of-core solver. In this talk, we will explain the multifrontal algorithm and discuss the design and development of a new HSL sparse symmetric out-of-core solver that uses it. Both the system matrix A and its factors are stored externally. For the indefinite case, numerical pivoting using 1x1 and 2x2 pivots is incorporated. To minimise storage for the system data, a reverse communication interface is used. Input of A is either by rows or by elements. An important feature of the package is that all input and output to disk is performed through a set of Fortran subroutines that manage a virtual memory system so that actual i/o occurs only when really necessary. Also important is to design in-core data structures that avoid expensive searches. All these aspects will be discussed. At the time of writing, we are tuning the code for the positive-definite case and have performance figures for real problems. By the time of the seminar, we hope to have developed the indefinite case, too. |
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Thu, 10/11/2005 14:00 |
Dr Serge Gratton (Toulouse) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
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Alan Turing introduced the sensitivity of the solution of a numerical problem to changes in its data as a way to measure the difficulty of solving the problem accurately. Condition numbers are now considered fundamental to sensitivity analysis. They have been used to measure the mathematical difficulty of many linear algebra problems, including linear systems, linear least-squares, and eigenvalue problems. By definition, unless exact arithmetic is used, it is expected to be difficultto accurately solve an ill-conditioned problem. In this talk we focus on least-squares problems. After a historical overview of condition number for least-squares, we introduce two related condition numbers. The first is the partial condition number, which measures the sensitivity of a linear combination of the components of the solution. The second is related the truncated SVD solution of the problem, which is often used when the matrix is nearly rank deficient. Throughout the talk we are interested in three types of results :closed formulas for condition numbers, sharp bounds and statistical estimates. |
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Thu, 17/11/2005 14:00 |
Prof Folkmar Bornemann (Technical University of Munich) |
Computational Mathematics and Applications |
Comlab |
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Image Inpainting turns the art of image restoration, retouching, and disocclusion into a computer-automated trade. Mathematically, it may be viewed as an interpolation problem in BV, SBV, or other fancy function spaces thought suitable for digital images. It has recently drawn the attention of the numerical PDE community, which has led to some impressive results. However, stability restrictions of the suggested explicit schemes so far yield computing times that are next to prohibitive in the realm of interactive digital image processing. We address this issue by constructing an appropriatecontinuous energy functional that combines the possibility of a fast discrete minimization with high perceptible quality of the resulting inpainted images. The talk will survey the background of the inpainting problem and prominent PDE-based methods before entering the discussion of the suggested new energy functional. Many images will be shown along the way, in parts with online demonstrations. This is joint work with my student Thomas März. |
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Thu, 24/11/2005 14:00 |
Dr Spencer Sherwin (Imperial College London) |
Computational Mathematics and Applications |
Comlab |
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Through the advent of enhanced medical imaging computational modelling can now be applied to anatomically correct arterial geometries. However many flow feautures under physiological and pathological flow paraemeters, even in idealised problems, are relatively poorly understood. A commonly studied idealisation of an arterial blockage or stenosis, potentially generated by atherosclerosis, is a sinusoidally varying constricted tube. Although most physiological flow conditions are typically laminar, in this configuration turbulent flow states can arise due to the local increase in sectional Reynolds number. To examine the onset of turbulence in this geometry, under a simple non-reversing pulsatile flows, we have applied Floquet stability analysis and direct |
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Thu, 01/12/2005 14:00 |
Dr Jean-Yves L'Excellent (ENS Lyon) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
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Parallel sparse direct solvers are an interesting alternative to iterative methods for some classes of large sparse systems of linear equations. In the context of a parallel sparse multifrontal solver (MUMPS), we describe a new dynamic scheduling strategy aiming at balancing both the workload and the memory usage. More precisely, this hybrid approach balances the workload under memory constraints. We show that the peak of memory can be significantly reduced, while we have also improved the performance of the solver. Then, we present preliminary work concerning a parallel out-of-core extension of the solver MUMPS, enabling to solve increasingly large simulation problems. This is joint work with P.Amestoy, A.Guermouche, S.Pralet and E.Agullo. |
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