Computational Linear Algebra for Partial Differential Equations: An LMS Durham Research Symposium

Many areas of mathematical research are constrained only by a limitation on the ideas of researchers in the subject. Numerical analysis and scientific computing are altogether different, since research in these disciplines is also constrained by the limits of the computers currently available. This technological barrier is, however, rapidly receding and as a result, the numerical analysis research community is having to continually update its aims in response to the inexorable increases in processor speed and memory capability. The pressure from engineering applications is always to ``solve a bigger problem''. This is true not only in terms of a desire to use finer meshes for greater accuracy, but also in terms of the increasingly realistic possibility of solving fully coupled systems of partial differential equations (PDEs), as opposed to crudely decoupling the problem into component subproblems that are successively solved in an iterative framework.Research is now moving towards the study of inverse or ``design'' problems, where some optimality criterion is desired to be satisfied within the restriction of constraints which are themselves PDE problems linked to the design parameters. This cutting edge mathematical research area, commonly known as PDE optimization, is inspired by the use of computers as an engineering design tool as opposed to a simulation tool. For example, aeronautical engineers can now use computational fluid dynamics (for solving the Navier-Stokes equations) to design turbine blades with desired/optimal properties. This paradigm shift provides the primary motivation for this proposal to hold a numerical analysis symposium with PDE optimization as its central theme. Broadly speaking, the UK standing in the numerical analysis of PDEs is very high, with significant research groups at various universities. In particular, the UK numerical analysis community is internationally regarded for the research of individuals and groups working on important aspects of computational linear algebra which are applicable in this regard. It seems to be timely and appropriate to bring the top UK and international researchers in these fields together in the informal atmosphere provided by an LMS Durham Symposium. This will give the opportunity for the exchange of ideas and cross-fertilisation of research expertise. We are confident that exciting new international collaborations will be realised if this proposal comes to fruition. The symposium will have an uncluttered lecture programme, with ample opportunities for discussions, collaborations and informal talks. Our aim is to ensure that all are aware of new techniques and the new challenges that are arising for our areas of focus. By bringing together a broad range of experts, this meeting will have a significant impact on the development of these rich areas of research. The results of the meeting will be distributed initially via an actively-managed and easily accessible website, which will contain a list of participants, abstracts of talks and other relevant information. The main talks will be recorded (with permission of the speakers) with a camcorder, and will be placed on the website within one month of the finish of the meeting. Lecture notes and/or slides will also be put on the site, and all this material will be made freely available to all.A more detailed record of the meeting will be provided by the publication in research journals of cutting-edge articles inspired by the subjects covered in the meeting. The conference is expected to provide a major lasting fillip to world research in the theory of PDE optimization.