Bengt Fornberg - US grants

9706916 B. Fornberg Pseudospectral (PS) methods - a high-accuracy alternative to finite difference (FD) and finite elemnt methods - are particularly effective for solving convection-dominated PDEs over long times and in relatively simple geometries. Both the algorithms themselves and their analysis have usually been closely tied to expansions in different classes of orthogonal functions. The recent book "A Practical Guide to Pseudospectral Methods" (by the present investigator) notes that a large body of generalizations, enhancements and insights can be gained by viewing PS methods instead as special cases of FD methods. These opportunities will now be explored further, in particular with the aim of combining the PS approach with domain decomposition for time-dependent computational electromagnetics. Pseudospectral (PS) methods were first proposed in the early 1970's (in connection with meteorology and turbulence modeling). They are now routinely used in many fields such as nonlinear wave motions, weather forecasting, fluid mechanics, computational chemistry, and with time-domain computational electromagnetics (TD CEM) emerging as a new and potentially major application area. This present investigation will explore how a series of new leads for PS methods can be developed further and then brought to bear on applications, in particular in the field of TD CEM. Such applications include simulations of the radar visibility of objects and of how electromagnetic fields are generated by/ interact with components on integrated circuits.

9810751 Meiss The Department of Applied Mathematics of the University of Colorado at Boulder's VIGRE program is focussed in the areas of computational and nonlinear mathematics. The program creates tetrahedral research groups consisting of a faculty mentor, a postdoctoral fellow, several graduate students, and undergraduate majors. Tetrahedra will focus on particular interdisciplinary research problems, hold seminars, write research proposals, and participate in the curriculum reform program. The curriculum will be integrated with computation using multi-layered case study modules. Affiliated faculty serve as co-advisors for Ph.D. theses of our students; the affiliated faculty program will be extended to government laboratories and technology companies. The program will create four tetrahedral groups in the areas of Dynamical Systems, Nonlinear Waves, Multilevel Computation, and Fast Algorithms and Modeling. The four facets of interaction within the groups include: teaching--as a seminar, and in the development of case study projects; learning--to develop mathematical, computational, and communication skills; discovering--to develop the techniques for formulation of useful and solvable research problems; and communicating--to collaborate in the joint production of research papers, grant proposals, and interim research reports. Computational mathematics provides the unifying theme for vertical integration of our training program. Computation will be integrated into lower division courses through Case Study Modules. Vertical integration will be implemented through the development of a multi-layered modeling course that has both lower and upper division undergraduate components as well as a graduate component. The VIGRE grant will support four Postdoctoral Fellows. They will receive mentoring from a faculty advisor and teach one course each semester for the Department. Twelve Graduate Trainees will be funded by the proposal. They will participate in the tetrahedra, research pro posal development, and the development and implementation of case study modules and receive teacher training through the Teaching and Learning Seminar and the Graduate Teacher Program. Four Undergraduate Research Experiences will be funded each year. Students will participate in one of the tetrahedra part-time during the academic year and for two months during the summer. These initiatives will be sustainable and lead to a number of permanent structural changes: successful tetrahedral research groups will endure and be emulated, case study modules will form an essential part of our curriculum, and the extended affiliated faculty will result in a continuing option for training of our students in application areas. Funding for this activity will be provided by the Division of Mathematical Sciences and the MPS Office of Multidisciplinary Activities.

Both pseudospectral (PS) methods and radial basis functions (RBFs) were first proposed in the early 1970's, in connection with turbulence modeling and cartography respectively. Only in the last few years have the two topics begun to merge, in the area of high-accuracy solutions of partial differential equations (PDEs). Superior performance of PS methods was demonstrated early on in applications such as accurate long term integrations of wave- type equations in very simple geometries. In contrast to usual spectral basis functions, different RBFs are not distinguished by how rapidly they oscillate, but instead by being different translates of one single function. Although certain orthogonalities are lost, spectral accuracy not only remains, but can now also be reached on complex domains with arbitrary node distributions. A new (and counterintuitive) result tells that, in the limit of RBFs becoming increasingly flat, the classical PS methods are recovered. Recently great progress has been made in overcoming the high computer cost and numerical ill-conditioning that earlier were thought to severely limit this approach. The observations above point towards extensions and generalizations of almost all numerical methods which, by tradition, have been polynomial-based. The challenge has changed from exploring the potential of RBFs for arbitrary-geometry spectral methods for PDEs, into exploiting it.

Most phenomena in science, engineering, sciences, and society can be approximated by some kinds of mathematical models. These often involve a construct known as partial differential equations (PDEs). Equations of this kind can only rarely be solved without resorting to computational methods. During the last century, a few main classes of such solution methods have evolved. A quite new class - pseudospectral (PS) methods - emerged some 30 years ago. Around the same time, radial basis functions (RBFs) were invented in a quite different context. The latter have still seen only exploratory use for PDEs. However, during the last three-year period, the PI demonstrated that PS methods can be seen as a special case of RBFs, with the latter approach greatly extending the PS methods' scope and generality. Continuing research will expand on this fundamental result, with the long-term goal of producing practical algorithms which are capable of becoming routine tools for high-precision solutions of PDEs in irregular multidimensional domains.

Radial basis functions (RBF) represent powerful mathematical techniques for interpolation and smoothing in multidimensional data space. Their use in solving time-dependent partial differential equations (PDEs) for modeling is to be explored by a multidisciplinary group of mathematicians and geoscientists from Arizona State, University of Colorado Boulder, University of Michigan, and NCAR. Attractive attributes of this new methodology for use in problems ranging from climate science, to shallow water equations in spherical geometry to solar corona dynamics include: i) the ability to achieve spectral accuracy and local mesh refinement at arbitrary node locations including resolution in steep-gradient events, ii) grid geometry independence allowing application to irregular geometries, iii) algorithmic simplicity, and iv) higher accuracy than competing spectral methods.

Of interest to many geoscientists, applications of RBFs in spherical coordinate systems will be investigated, with initial applications to climate and solar modeling. Educational outreach will feature an interactive web-site with instructional and applications modules.

Numerical computations in multiple space dimensions have traditionally been
based on structured grids (e.g. finite differences or spectral methods) or
on unstructured grids (e.g. finite elements). Even in the latter case,
there is structure in the sense that one needs to work out which subsets of
nodes should be connected into local triangles, tetrahedra, etc. Aspects
associated with such grid generations can at times consume as much or even
more computer resources than do the subsequent computations on the grid.
Furthermore, it has been all but impossible to achieve high (spectral)
computational accuracy on such grids without resorting to extensive use of
domain decompositions. Numerous mesh-free methods have been proposed
recently. The Radial Basis Function (RBF) approach stands out in several
respects, most notably in that it generalizes traditional spectral methods
to entirely mesh-free settings. Furthermore, implementations are usually
remarkably simple. For example, 20-30 lines of Matlab typically suffices
for solving an elliptic PDE on an irregular 3-D domain, to spectral
accuracy. Both pseudospectral (PS) methods and RBFs were first proposed in
the early 1970's. The superior performance of PS methods for solving many
PDEs was demonstrated early on. A NSF proposal by the present investigator
about 5 years ago was the first time RBFs were presented in terms of being
a direct replacement for the traditional basis functions in PS methods.
Recently, significant progress has been made both on overcoming the high
computer cost and the numerical ill-conditioning that earlier were thought
to severely limit the RBF approach. Recent NSF-DMS supported work by the
present investigator has opened up numerous further opportunities in this
area, which will now be pursued. These include combining spectral accuracy
with local node clustering in a fully stable way, a new stable algorithm in
the extremely high accuracy flat basis function limit, developments towards
faster algorithms, and the application of RBFs to PDEs in the geometries
that are most relevant in astro/geophysics, etc.

In the last several decades, computational methods have become an
increasingly essential part of how science and engineering are conducted,
partly because of a rapid evolution of computer hardware, but equally much
thanks to improvements in computational algorithms. Very often, the task to
be solved, when formulated in mathematical terms, require the solution of
partial differential equations (PDEs), often in irregularly shaped regions
in several space dimensions. The Radial Basis Function (RBF) methodology,
which is the subject of the present grant, opens entirely new opportunities
in this regard, combining very high accuracy with unsurpassed geometric
flexibility. One of several application areas of particular interest
(pursued in collaboration with scientists at NCAR and NOAA) concerns
geophysical and astrophysical modeling in spherical geometries, which are
critical issues for effective climate modeling and for studies of solar
dynamics.