------- Seminar on Numerical Analysis and Applied Mathematics Date: May 23rd(Thursday), 24th(Friday) and 27th(Monday) Place: Room 1212-1213, 12th Floor, National Center of Sciences, Tokyo (http://www.nii.ac.jp/help-e.html) Sponsors: Japan Society for Industrial and Applied Mathematics (JSIAM), National Institute of Informatics (NII) Admission: Free Organizers: Taketomo Mitsui (Nagoya University) Ken Hayami (National Institute of Informatics) Contact: Taketomo Mitsui (mitsui@malt.math.human.nagoya-u.ac.jp) Ken Hayami ( hayami@nii.ac.jp) Schedule May 23rd (Thursday) 1:30-1:40 pm Opening (Professor Taketomo Mitsui) 1:40 - 2:40 pm Professsor Olavi Nevanlinna (Helsinki University of Technology) Meromorphic Resolvents and Iterative Methods. 2:50 - 3:50 pm Ian H. Sloan (The University of New South Wales) Numerical Integration with Many, Many Variables. 4:00 - 5:00 pm Professor Alain Damlamian (University of Paris, ) Multi-Scale Homogenization and Periodic Unfolding. May 24th (Friday) 2:00 - 3:00 pm Professor Reinhard Mennicken (University of Regensburg) Spectral Theory for Systems of Differential Operators of Mixed Order and Applications 3:10 - 4:10 pm Professor Rolf Jeltsch (ETH Zuerich) The Method of Transport for Systems of Hyperbolic Conservation Laws. May 27th (Monday) 2:00 - 3:00 pm @ Professor Bob Russel (Simon Fraser University) Adaptive Algorithms for Solving Time-Dependent PDEs. 3:10 - 4:10 pm Professor Robert OfMalley (University of Washington): Shock Motion for Singularly Perturbed Partial Differential Equations. Abstracts: Professsor Olavi Nevanlinna (Helsinki University of Technology) Meromorphic Resolvents and Iterative Methods. Scalar meromorphic functions can have Picard special values, that is, values which are not taken at all. More generally, certain values can be defective but the sum of all defects is bounded. We have studied operator valued meromorphic functions, developed a perturbation theory for them (Ann. Acad. Sci.Fenn. Mathematica 2000, Vol 25, 3-30), which can be used to show for example that growth of resolvents as meromorphic functions is robust under small rank perturbations. We have published earlier that the growth of resolvent can be linked to speed of iterations (together with Saara Hyv"onen, BIT, 2000, vol 40, 267-290). In this talk we link the Picard defect values to "nonnormality", by showing that if the operator is normal and the growth of the resolvent is of finite order, then the resolvent is not defective in the sense of value distribution. We also give a result of the following general nature. Suppose f(z) is a scalar meromorphic function and A is a bounded operator with a meromorphic resolvent, then f(zA) is an operator valued meromorphic function and its growth is is on the level of the maximum of the that of f and the resolvent. The new results are intended to appear in a monograph in the AMS Fields Institute monograph series. Ian H. Sloan (The University of New South Wales) Numerical Integration with Many, Many Variables. Nowadays integration problems with very large numbers of variables arise in many applications, including mathematical finance. While Monte Carlo methods are available for problems with any number of integration variables, increasingly attention has turned to quasi-Monte Carlo methods. These have the appearance of Monte Carlo methods, but with the integration points chosen deterministically instead of randomly. This talk will review the history of quasi-Monte Carlo methods, finishing with recent step-by-step constructions of shifted lattice rules with very large numbers of points and hundreds of variables. Professor Alain Damlamian (University of Paris, ) Multi-Scale Homogenization and Periodic Unfolding. We present the method of periodic unfolding and its applications to periodic homogenization, including the multi-scale case. This sheds new light on the so called two-scale convergence but is much more elementary and straightforward. It also applies to cases which two-scale convergence has not been able to handle. Some examples will be also given. Professor Reinhard Mennicken (University of Regensburg) Spectral Theory for Systems of Differential Operators of Mixed Order and Applications (Abstract to be announced.) Professor Rolf Jeltsch (ETH Zuerich): The Method of Transport for Systems of Hyperbolic Conservation Laws. In 1992 M. Fey introduced the method of transport for the Euler equations of gas dynamics. We shall derive this method for the Euler equations using the advection form. In this form one sees more easily the characteristic propagation directions inherent in the differential equation. A straightforward decomposition and linearisation of this advection form leads to a genuinely multi-dimensional method. This method is robust, but of first order only. It is indicated how one can obtain a second order scheme. It is then shown that the method can be applied not only to the Euler equations of gas dynamics but also to the shallow water equation, MHD, Navier-Stokes and equations to model elasto-plastic waves in solids. We discuss briefly the treatment of boundary conditions and mesh adaptations as well as parallelisations for MIMD computers. Professor Bob Russel (Simon Fraser University) Adaptive Algorithms for Solving Time-Dependent PDEs. Of fundamental concern in the development of numerical algorithms for solving time dependent PDEs is the construction of the numerical grid. It must adjust to extreme features of the physical solution while conforming to boundary behaviour and maintaining smoothness. This problem has proven to be extremely challenging and has motivated extensive mathematical analysis (e.g., for harmonic maps) and numerical experimentation. We shall review recent progress in the development of moving grid algorithms and outline some of the outstanding problems which remain. Numerical software will be illustrated for a variety of challenging problems. Professor Robert OfMalley (University of Washington): Shock Motion for Singularly Perturbed Partial Differential Equations. We consider the solution of some special initial boundary value problems for parabolic PDEs in a bounded spatial domain and show how shocks can move on very long time intervals. Results follow from an analysis of the tail behavior for the appropriate shock profile function. -------