From: Hans Munthe-Kaas Date: Tue, 9 Apr 1996 10:25:07 +0200 Subject: Report on "State of the Art in Numerical Analysis" Conference State of the art in Numerical Analysis? What has happened in Numerical Analysis during the last 10 years, and what are the most interesting future developments in the field? These questions were the topics in the conference 'State of the art in Numerical Analysis' held April 1-4 in York, England. The conference has been arranged every tenth year since 1966 by IMA (The Institute of Mathematics and its Applications), and this year it gathered 112 mathematicians from about 16 countries. Gene Golub asked me to summarize the conference for the NA-Net readers: The conference centered around the central themes in numerical analysis: - Linear algebra - Ordinary differential equations - Integral equations - Approximation - Optimization - Partial differential equations And in addition there was a session on new applications. I will here give a brief personal summary of the various topics, which of course to some extent is colored by my own personal interests. All the talks will appear in the conference proceedings, published by IMA. Judged from the quality of the talks, this will be a very valuable reference source. Linear Algebra: Talks were given by Nick Higham (dense linear algebra), Iain Duff (sparse direct methods), Gene Golub (iterative methods for linear systems) and Henk van der Vorst (sparse eigenproblems). The main developments in dense linear algebra during the last 10 years has been centered around all the work with the LAPACK project for dense linear algebra. Parallel computers have been around only for about a decade, so most of the work on parallel linear algebra is done in this period. This is now seen through the organization of algorithms around block formulations via the BLAS 2 and 3 routines. Interestingly, also sequential computers gain speed by this organization. Also in sparse computations, much of the activity has been inspired by parallel computers. For iterative methods, the main contributions during the last decade is perhaps the development of Krylov subspace techniques for unsymmetric systems (GMRES, CGS, Bi-CGSTAB, QMR). And in eigenproblems there have been a significant development of Lanczos and Arnoldi type methods. Some old methods have gained new significance (Jacobi) and some new ideas have been introduced due to parallel computers (divide and conquer algorithms). Even some new basal mathematical tools have gained significant importance (pseudospectra). Where are we going now? It seems as parallelism per se is not a topic of major popularity, but it will of course remain constantly in our heads when we contemplate over new algorithms. Since the 'black box' software packages in linear algebra is now so excellent, much work in the future will be centered around exploiting structures which arise in various application areas. This was pointed out by Gene, who said that he 'just late in life' realized the importance of exploiting all the information which comes from knowing the structure of the underlying problems. A lot of this information is lost if we regard our problems as being 'purely' linear algebra. Ordinary differential equations: In this area, the need for alternatives to the 'black box' software was even more emphasized than in linear algebra. All the three speakers; Chus Sanz-Serna (geometric integrators), Andrew Stuart (dynamical systems) and Arieh Iserles (beyond the classical theory of ODEs), pointed out that there is a major need for understanding how to conserve various properties of equations that are essential mathematically, and which has not been given enough consideration numerically. Chus summarized the work done during the last 10 years on preserving symplecticity and Andrew Stuart talked about recovering the correct asymptotic properties of dissipative dynamical systems (limit sets and attractors). Here the classical notion of measuring quality by the global error is not relevant. Ariehs talk summarized the work done on delay differential equations and differential algebraic systems during the last decade. He pointed out some areas of significant current research, where we may gain major insight in the next decade. This includes the work currently undertaken to understand the integration of equations where the solution is known to sit on a specific manifold or on a Lie group. In the discussion someone pointed out that "Whereas one 20 years ago didn't need to know much about differential equations to work with numerical solutions of them, this is no longer the case". Integral equations: Two talks were given about integral equations; Christopher Baker (Volterra functional and integral equations), Kendall Atkinson (Boundary integral equations). Also in these areas the last decade has been very fruitful. For boundary integral equations much of the understanding of the numerical analysis of corner singularities have been gained in this period. For me as an outside viewer in this field, the most fascinating developments have perhaps been the various fast algorithms for solving the dense matrix problems arising in these fields. (Fast multipole algorithms and algorithms based on wavelet compression and multiresolution analysis). Now the solution techniques for these dense linear algebra problems have become so fast that it is important not to form the coefficient matrix explicitly. (The complexity of solving the linear systems is smaller than the complexity of assembling the coefficient matrix!) Approximation: Talks were given by Alistair Watson (emphasis on the univariate case), Mike Powell (multivariate interpolation), David Broomhead (neural net approximations). The most important development in approximation has probably been the field of wavelets, briefly summarized in Watsons talk. Optimization: Three talks in this field: Jorge Nochedal (unconstrained optimization), David Shanno (interior point methods), Nick Gould (nonlinear constraints). There has been a tremendous amount of work on interior point methods this decade, and Shanno referred to large industrial optimization problems where interior point methods beat simplex by a factor 50 in speed. Partial diff. eqn: The talks on PDEs were: Franco Brezzi (Stabilization techniques and subgrid scales capturing), Charlie Elliott (Large approximation of curvature dependent interface motion), Endre Suli (Finite element methods for hyperbolic problems: stability, accuracy, adaptivity), Bill Morton (Approximation of multi-dimensional hyperbolic PDEs). Some keywords from these talks are error control and adaptivity. New applications: There were two talks on applications; Frank Natterer (Tomography) and Jean-Michel Morel (nonlinear filtering and PDEs). Morels talk about the connection between filtering techniques in computer vision and partial differential evolution equations was highly inspiring. The idea is to classify various families of discrete image filters via the PDEs they approximate. In some sense, the work in this field resembles the early work on statistical mechanics/ transport theory/ continuum mechanics in the last century. This is an area in its infancy, where the basic understanding of the processes involved is being developed in the language of PDEs. Concluding remarks: It is hard to summarize all the developments that has been going on in numerical analysis during the last decade. It has been an immensely fruitful period, and the subject is truly alive and developing. It is also a pleasure to remark that the numerical analysis community consists of a bunch of cheerful people, and that the friendly spirit of the 'late hours' is also a part of the 'State of the Art' in our field. This was evident in the hilarious dinner speech by John C. Mason. Thanks to the organizing committee chaired by Alistair Watson, and to Pamela Bye for arranging all the practical details. Info is also found at: http://www.amtp.cam.ac.uk/user/na/SotANA/SotANA.html Hans Munthe-Kaas University of Bergen Norway