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Durham Cathedral

London Mathematical Society Durham Symposium
Computational Linear Algebra for Partial Differential Equations
Monday 14th July - Thursday 24th July 2008

Research themes
The meeting will be based around three main themes:
  • Optimization;
  • Saddle-point systems;
  • Eigenvalues and model reduction.
A brief description of each of these themes is given below. As indicated in the title, it is the consideration of these closely-related themes in the important context of partial differential equations problems which will be the focus of the proposed symposium.
In almost all formulations, constrained optimization problems give rise to so-called saddle-point systems in linear algebra. Note that even though the mathematics is the same, the nomenclature used is often different in the two areas: saddle-point systems are widely called KKT systems in optimization, since they derive from expression of the Karush-Kuhn-Tucker conditions for optimality.
For very large scale systems, iterative methods and preconditioning become key components in solution strategies. Furthermore, because most saddle-point problems are symmetric, issues regarding the speed of iterative convergence are usually expressed in terms of eigenvalues.
Within each theme, we have identified two international experts (one from a PDE background and one an expert in linear algebra) who will each present a series of three keynote one-hour lectures. We have been especially careful to suggest participants with research interests which span the linear algebra/PDE boundary. Clearly, the themes are very closely related and most of the participants will be interested in more than one of the topics: this will ensure that the symposium does not degenerate into three mini-conferences.

The keynote speakers on this topic will be Max Gunzburger (Florida State) and Eldad Haber (Emory). As mentioned above, there is an ongoing conjoining of PDE simulation and optimization to enable design calculations.
Many different methods and much excellent software already exists for numerical optimization, however, in the context of large scale problems coming from PDE applications in which matrix factorisation (even sparse factorisations) are not conceivable because of memory and time complexity considerations, a re-evaluation of appropriate approaches is underway. Ideas such as multigrid, which emanated in the context of solving "forward" PDE problems, are now being considered in the context of optimization by Haber and others. Model reduction provides an alternative approach to generating cost-effective methods, and Gunzburger is a leading expert in this area. From the PDE side, problems of flow control and the like have until recently been considered with simple descent algorithms. However, the real need to combine knowledge from both optimization and PDEs for devising general and effective approaches is now appreciated and this workshop should provide the ideal circumstances for the strong UK communities in these two subjects to learn about and build upon what has been done in this area to date (mostly in Austria, Germany and the USA).
Overall, this research field is very fertile and there are important unanswered questions regarding the best approaches to adopt.

Saddle-point systems
The two keynote speakers on this topic will be Valeria Simoncini (Bologna) and Ragnar Winther (Oslo). There has been a huge resurgence in research interest in this topic within the linear algebra community over the last decade. This is driven by the increasing need to solve problems expressed in terms of equilibria with constraints (in for example optimization, contact problems, computer graphics etc.). In the area of sparse direct methods, there have been new methods and software from, for example, the Rutherford Appleton group, and for even larger scale problems, new preconditioned iterative approaches which are specifically designed for the saddle-point structure have come from several research groups worldwide. Successful techniques for specific saddle-point problems (such as the incompressible Navier-Stokes equations) are available through the work of the proposers and others but many other important problems have still to be tackled. Particularly attractive from the optimization stand-point are methods which mimic geometric considerations: for example, the class of "constraint preconditioners" which ensure iterates remain on the constraint manifold, thus preserving feasibility, throughout an iteration process (and not just at convergence).
All standard approaches to PDE optimization lead to saddle-point systems, so there is a close link with the theme above.

Eigenvalues and model reduction
The two keynote speakers on this topic will be Volker Mehrmann (Berlin) and Alastair Spence (Bath). The aim of model reduction is to provide lower-order models that contain the dominant effects of the original system and lead to reliable results when used in their place. There is a tight connection with numerical linear algebra since state-of-the-art techniques are typically based on Krylov subspace methods. Mehrmann is one of the leading experts in the linear algebra aspects of model reduction. An important aspect of this is the calculation of dominant eigenvalues and singular values. A completely different but very important application of eigenvalues is that of determining PDE stability, for example, sensitivity analysis of flows. Such problems are becoming increasingly tractable using shifted Arnoldi methods because of the advent of effective preconditioners coupled with the increase in available computing power. Spence is at the forefront of developments in this area. Eigenvalues are also a vital component of PDE optimization since nonlinear and linearized stabilty of computed states does not simply follow from their optimality. This theme of eigenvalues also underlies the analysis of the iteration methods used for solving the saddle-point systems in the second theme above.

Organising Committee:
Alison Ramage (Strathclyde University) David Silvester (University of Manchester) Andy Wathen (Oxford University)