This is less important than some of what follows, but is worth saying. Much of this work will be "mathematical" but will not be "mathematics" as the term is currently understood. In that sense, it is not what postgraduate research mathematicians would have expected to do. On the other hand, it is exactly the sort of work that one would expect bachelor's and master's level mathematicians to do. This should be viewed as an opportunity for research mathematicians, not a hindrance. If mathematics involves studying the abstract structure and logical interrelationships of things, surely it is good for mathematicians to be involved in work that is mathematical and exposes them to entirely new universes of "things" that must have structure? There may be a delay before it becomes possible to raise the level of abstraction to the point that a "mathematics" of these structures is synthesized from experiential information, but there isn't really anything fundamentally wrong with applying familiar logical processes and skills in trying to understand the complexities of new things.

There are some things that are different from normal academic mathematics. Not better, or worse, but just different in ways that should be kept in mind.
These are large scale projects.
These are real projects, i.e., there are people who care not about studying the problem, or proving theorems about the problem, or studying the abstract versions of the problem, but about the actual outcome of computations in this particular problem.
The details involved in the specific projects are vitally important and one cannot (at least not at present) abstract away the details without becoming vacuous. For example, we saw on Wednesday two examples of what resembles a transitive closure:

1. Suppose one is constructing or using a thesaurus. In one sense, this consists of a graph with a node for each term and (directed?) arcs with weights (normalized from 0 to 1?) representing the extent to which one word is "connected" to another. One way of looking at a thesaurus is as the transitive closure (with some sort of thresholding and/or attenuation functions based on distance?) under "connectedness."

2. Suppose one is trying to determine which phone numbers have fax machines connected. The same approach as above is basically what could be done.

But I maintain that beyond the brief and almost-too-obvious description above, there probably isn't much real commonality between what gets computed to solve the two different problems. The details don't just matter, they matter almost more than anything else. The connectedness functions will differ, being based on the nature of language, the statistics of the 130,000 words in common English usage, the statistics of telephone calling patterns, etc. The attenuation functions will matter. The size of the data bases will matter-things one might get away with in processing 130,000 English words might be infeasible in dealing with (how many millions a day was it?) phone calls.

Therefore:

As Grossman more or less said but didn't quite say this way, if you don't know how to measure the scale of these projects based on the number of seconds in a year, you probably aren't thinking about them in the right way. Similarly, to follow up on what Wegman implied but didn't say outright, it is important to measure the cost of these computations in appropriate units. These units are wall clock times-seconds for an interactive response, CPU hours/days/weeks/months for a massive preprocessing step, etc. The units are NOT O(n^1.5) steps in an algorithm. Although an O(n^1.5) algorithm might be considered "quite nice" by the J. Random Scientist who is really interested in getting the problem solved, what would really stimulate J. Random's endorphins is an O(n^1.5) algorithm that when implemented dropped the execution time on a 10⁸ size data set from the 10⁷ seconds (4 months) on a 1 gigaops machine with an O(n²) algorithm down to 1000 seconds (17 minutes) on the implemented O(n^1.5) algorithm.

It is also very important to remember that the details of each application are probably vital to the eventual successful solutions. One of the hardest things for me to understand when, after having been trained as a pure mathematician, I suddenly became in 1977 a computer science professor, is that there didn't seem to be any theorems anymore, only examples. The "theorems" or general truths about how to compute things were vague or contradictory, and in spite of this vagueness there were good things being done based on the phenomenology of each specific problem.

One can easily produce all the reasons why virtual memory should and shouldn't work (a) If a page in memory has not been used recently, it is from a previous part of the program and won't be executed again and can be swapped out...or else (b) the page not recently referenced is from the top of a loop whose bottom part is just finishing up being executed so the page in question is just about to referenced again and better not be swapped out.) But virtual memory does work, because the operating system as written is tailored to the specifics of the computer architecture as designed and because, in spite of the fact that it is easy to write an individual program that will cause any given machine to thrash, the phenomenology of the execution characteristics of "most" real programs is such that virtual memory, with some caveats, does in fact work.

I believe this whole area should be approached from the bottom up and not from the top down. That is to say, DMS should attempt to clone successful massive data set activities with mathematicians as important members of the research teams rather than trying to create, ab initio, a massive data sets initiative in the mathematics community. Over time no doubt the math community would learn inductively from its experience, and some more traditional mathematics might follow, but this work is problem driven, and mathematics per se simply doesn't have the problems. The purpose of the MACHO experiment was not to collect and analyze lots of data. The purpose was to do astronomy. The only reason we heard about the problem is that the computational problem on the data has become the bottleneck preventing further work in astronomy. Similarly, the growing problem on the net is making some sense out of all ~the "stuff" that's out there. The stuff that people want to get to is out there, but not being able to compute a semantic organization to all that stuff is getting in people's way.

So I think the effort has to be problem driven, and probably therefore has to be done in collaboration with the other areas that "own" the problems.

Margaret Cheney from Rensselaer Polytechnic Institute (Mathematical Sciences) was a visitor from January 1 - June 30, 1997. Her report follows:

I certainly had a wonderful visit there at the IMA. I hope you're doing well!

Papers:

My first order of business was to finish revisions to a paper [1] I had begun during my previous IMA visit. This paper is joint work with Paul Bigeleisen, an anesthesiologist who at the time was working at St. Paul - Ramsey Medical Center in St. Paul. This paper develops a mathematical model of the breathing circuit used to administer anesthetic gases to patients. We then used the model to answer some open questions about the best way to replace lung nitrogen by oxygen before surgery. I spent quite a bit of time working on the problem of proving that measurements on a plane uniquely determine the medium parameters of an inhomogeneous half-space. This work is joint work with Matti Lassas, whom I had met at the IMA during my 1994-1995 visit. We finished the proof [2] when we were together at a conference in July 1997. I wrote a book review for Inverse Problems on Andreas Kirsch's new book. I spent a good part of June writing a paper on inverse boundary value problems for American Scientist magazine [3].

People

I especially enjoyed the opportunity to get to know Petter Petter Bjørstad, Ralph Kleinman, and Frank Natterer. Although I haven't actually initiated any specific joint projects with Natterer, I'm particularly interested in the work that he is doing. As a result, I visited him and his group in Muenster in March. One of the students I met there, Oliver Dorn, has been working on inverse transport problems, and is now at Stanford. I recently suggested Dorn's name to a Duke electrical engineer (Larry Carin) who is thinking about millimeter-wave penetration of foliage and wants to use a transport equation model. It's hard to point to specific results of these personal relationships, but they certainly contribute to the health of the scientific enterprise. I also enjoyed getting to know Sheri Moskow. Maybe at some point I'll be able help her career along.

New Ideas

I audited two classes in the Minnesota Electrical Engineering department, namely "Microwave Circuits and Devices" and "Antenna Theory and Design." This is part of my continuing campaign to learn more about radar and the associated inverse problems. I learned about beam-forming techniques in acoustics. This is a key part of the side-scan sonar technology. I read Yu Chen's recent preprints on his stable algorithms for solving multidimensional inverse problems. I studied Frank Natterer's algorithms, and induced him to try his method on Yu Chen's examples. I understand that this led Natterer to some improvements of his algorithm. At Ralph Kleinman's and my urging, Frank also tried his method on some problems with caustics, with good results.

References

[1] P. Bigeleisen and M. Cheney, "Models for an anesthesia breathing circuit," IMA preprint 1476. [2] M. Lassas and M. Cheney, "Uniqueness for a wave propagation inverse problem in a half space," in preparation. [3] M. Cheney, "Inverse boundary value problems," American Scientist, 85 (September-October 1997) 448-455.

Rolf Clack from University of Utah, Medical Imaging Research Lab, Department of Radiology was a long-term participant. He was also a speaker during the March 17-21, 1997 workshop on "Computational Radiology and Imaging: Therapy and Diagnostics." He writes:

I wanted to drop a quick note of thanks for the hospitality I have enjoyed here at the IMA for the month of March. I was very pleased to get the invitation (as you see, I took advantage of the maximum stay!) and I had a very productive time, collaborating with colleagues and pursuing interesting research projects I have long put off.

The facilities here are excellent. The office space (I don't have a window in my university office), the computer systems with full Internet access, the coffee room, all really don't leave any excuse. You HAVE to be productive in this environment!

I am very impressed by the courteous and professional attitude of all the staff. I interacted at least once with each of the groups: office, accounting, computer systems, and TeX experts, and they were all very helpful, especially the office staff.

I will probably be taking further advantage of your staff when I try to TeX (IMA-style) my report for the proceedings.

Bernardo Cockburn from the University of Minnesota (School of Mathematics) reports:

During year 1996-1997, the IMA activities focused on the topic `Mathematics in High Performance Computing.' The reduced teaching load I was awarded by the School of Mathematics allowed me to strongly interact with several visitors. In particular, I had the opportunity to strongly interact with J. Flaherty, Pierre A. Gremaud, Mirko Rokyta, Chi-Wang Shu, and Endre Suli:

(1) With J. Flaherty, we started a project centered around the first mathematically sound adaptive methods for nonlinear convection-diffusion equations.
(2) With Pierre A. Gremaud, we were able to finish a paper devoted to a priori error estimates for nonlinear conservation laws.
(3) With Mirko Rokita and Endre Suli, we started a very promising project whose aim is to devise error estimates for convection-dominated problems in bounded domains.
(4) With Chi-Wang Shu, we finished two papers. One concerning a computationally intense study of discontinuous Galerkin methods for nonlinear hyperbolic systems, a project we started five years ago; and the other, concerning the devising of discontinuous Galerkin methods for convection-dominated flows. This second paper is the first in a series in which we will study these methods for convection-diffusion flows.
(5) With Chi-Wang Shu and Endre Suli, we started a long-term project devoted to the study of new a priori and a posteriori error estimates for finite element methods for hyperbolic systems. This project strongly links the research groups at Providence (Shu), Oxford (Suli) and Minnesota.

As you can see, my participation in the IMA activities was greatly enhanced by the reduced teaching load I received from the School of Mathematics.

Laurent Desbat of Universite Joseph Fourier (d'Informatique) was a visitor from March 1-29, 1997. He participated in the workshop " Computational Radiology and Imaging: Therapy and Diagnostics." His report follows:

I was very happy to have the opportunity to work with David Walnut on sampling problems and to have very nice discussions with T.E. Quinto on the rotational invariant generalized Radon transform ant its application in astronomy and Rolf Clack on 3D tomography. I had also some time to start to write my habilitation thesis, and it was very good to be far from France for such activity. The IMA is really a wonderful place to do research with very nice people. I really enjoyed to spend more time with them than just the few days of a congress. Once again, thank you very much for the invitation. It was simply great!

In tomography a function is reconstructed from its projection. It is mainly a medical imaging techniques and a non destructive control industrial method. Because of the star rotation tomography is also encountered in astronomy. Sampling conditions of the measure are in practice necessary. In tomography, the essential support of the Fourier transform of the Radon transform (in 2D) and the X-ray transform (in 3D) have particular geometries that can be used in order to produce regular efficient sampling schemes generated by a non singular matrix W satisfying the non overlapping Shannon conditions. In 2D the interlaced scheme has been proposed by Cormack, Rattey et Lindgreen, Natterer (in particular for fan-beam geometries), in 3D we have proposed in [3] the sampling condition of the X-ray transform with in the unit circle and , more precisely of . This is the geometry of nuclear imaging when the gamma-camera turns around the patient with a parallel collimation. A. Faridani and D. Walnut have proposed new sampling theorems for non regular schemes, that yields new sampling schemes useful for industrial tomography[1,2].

Our recent work thus concerns 3D [3]. During our stay in IMA we could prepare with Rolf Clack the post-doc project in Salt Lake city of a very good french student, Catherine Mennessier, on reconstruction from incomplete projections. Its PhD in astronomy has allowed to establish the link between Doppler imaging and the rotational invariant generalized Radon transform. I had a very helpful discussion with E.T. Quinto at IMA: its invertibility proof can be partly adapted to Doppler imaging. When the rotational invariant weigh function is polynomial (with low degree) the interlaced sampling is shown to be efficient. This can be applied in certain geometries of Doppler imaging[4]. I could also work with David Walnut on multidimensional sampling on unions of lattices, which could have very nice practical applications in tomography.

REFERENCES

[1] L. Desbat. Efficient sampling on coarse grids in tomography. Inverse problems, 9:251-269, 1993.

[2] L. Desbat. 3D efficient parallel sampling perturbation in tomography. In International Meeting on Fully 3D Image Reconstruction in Radiology and Nuclear Medicine, 1997.accepted.

[3] L. Desbat. Echantillonnage parallhle efficace en tomographie 3D. C.R. Acad. Sci. Paris, Séerie I, t. 324, pages 1193-1199, 1997.

[4] L. Desbat and C. Mennessier. Efficient sampling in Doppler imaging. Proceedings of SampTA, 1997. accepted.

Keneth A. Dill of the University of California (Pharmaceutical Chemistry) gave a presentation at the workshop on "Mathematical and Computational Issues in Drug Design held on April 7-11 1997. Below is report on his successful collaboration with other mathematicians and computer scientists.

About 2 years ago, I have a talk at an IMA on Protein Folding. Two computer scientists from the University of Minnesota were in the audience - Professor Ben Rosen and his former student Andy Phillips, now Associate Professor at the Naval Academy. They were working on a global optimization method they felt might help us with protein folding. We stated a collaboration. About 3 papers have resulted from this collaboration so far and we are now beginning to prepare a grant proposal. It's been great!

Xiaobing Feng an Assistant Professor at Mathematics Department of the University of Tennessee at Knoxville writes:

From April 1 to June 30, 1997 I was a resident of the IMA for participating 1996-97 IMA program: Mathematics in High-Performance Computing. This was my fisrt visit of the IMA. To summarize my three month experience at the IMA using one word, I would like to say "wonderful." Indeed, the three months I spent at the IMA was the most memorable and the most productive period of time for me in my five year academic career. I enjoyed very much the IMA's unique environment, conferences, workshops, short courses, seminars, and especially, meeting and communicating with other IMA visitors.

My research areas are computational and applied mathematics with applications in fluid and solid mechanics and geosciences. Mathematically, most problems I am interested in are described by a set of linear or nonlinear partial differential equations, hence, the core of my research involves analytical analysis of solutions of the differential systems and developing efficient (sequential and/or parallel) numerical methods to compute the solutions on high-preference computers, which is the theme of the IMA program in Spring of 1997.

While I was at the IMA, I had been worked on two research projects. The first project is to finish a ongoing project on computing wave propagation in unbounded domain using artificial boundary condition approach. In this approach the unbounded domain is truncated into computationally feasible finite domain, so finding the right boundary conditions on the artificial boundary is the key step of the method. The work I did at the IMA is to test numerically a class of artificial boundary conditions for electromagnetic waves proposed by myself earlier, the final results was reported in the IMA preprint #1482: "Absorbing Boundary Conditions for Electromagnetic Wave Propagation." The second project is to develop efficient parallel numerical methods based on the domain decomposition (divide-and-conquer) technique for solving plate bending problems from structure mechanics. This project has been worked jointly with three other IMA participants, Petter Bjørstad and Talal Rahman (University of Bergen, Norway), and Maksymilian Dryja (University of Warsaw, Poland). In this project we construct and analyze some so-called Additive Schwarz Methods for the plate bending problems based on non-overlapping domain decomposition. We have finished the theoretical analysis of our new methods and currently we are doing numerical tests to check the efficiency of the methods. The final resluts of this project will be written into a joint paper entitled: "Non-overlapping Additive Schwarz Method for the Biharmomic Equation." Finally, in addition to working on the above two projects, the IMA support enables me to have time to make the following my three earlier papers in their final versions while I was at the IMA. The three papers are: "Analysis of a Domain Decomposition Method for the Nearly Elastic Wave Equations Based on Mixed Finite Element Methods" (IMA preprint #1491); "Formulation and Mathematical Analysis of a Fluid-Solid Interaction Problem" (IMA preprint #1495); "Finite Element Methods and Domain Decomposition Algorithms for a Fluid-Solid Interaction Problem" (IMA preprint #1496).

Once again I like thank you, Professor Friedman and all IMA people for all your help to me. The three month stay in IMA was very very memorable for me. I enjoyed every moment while I was a member of the IMA family. In fact, I am looking forward to revisit IMA in year 2000.

Ronald H.W. Hoppe from the University of Augsburg (Lehrstuhl fur Angew Mathematics) was a visitor in June'97. His report follows:

My visit at the Institute of Mathematics and its Applications, University of Minnesota at Minneapolis from June 1 - June 30, 1997 was within the IMA Annual Program "Mathematics in High Performance Computing. During my stay I participated in the workshop "Parallel Solution of PDEs" which took part from June 9 - June 13, 1997. I contributed by a talk entitled "Adaptive finite element methods for domain decomposition on nonmatching grids."

That workshop offered the opportunity to intensify already existing cooperations with colleagues as, for instance, Olof Widlund (Courant Institute of Mathematical Sciences) and to initiate new contacts with, e.g., Yvon Maday (Université Pierre et Marie Curie, Paris) in the area of domain decomposition on nonmatching grids which is a specific topic in domain decomposition methods that attracted considerable attention during the last couple of years. During my stay at the IMA I prepared two papers: one will appear in the proceedings of the above mentioned workshop and the other one entitled "Efficient parallel solution of fully potential flow problems by domain decomposition methods on nonmatching grids" reflecting some joint work with the same coauthors over the last two years is scheduled to appear in the "East-West Journal of Numerical Mathematics."

Besides the activities in the field of domain decomposition methods I had very interesting discussions with Professor Friedman and Professor Reitich about mathematical modelling and numerical simulation of electro- and magnetorheological fluids. The modelling and simulation of electrorheological fluids and applications in the automobile industry (dampers, clutches, and mounts) is currently one of the main research topics in my group. Actually, it is part of an interdisciplinary national research project (Sonderforschungsbereich 438: Mathematische Modellierung, Simulation und Verifikation in materialorientierten Prozessen und intelligenten Systemen [Mathematical modelling, simulation, and verification in material oriented processes and intelligent systems]). I will keep in touch with Professors Friedman and Reitich concerning this issue and hope that they will have the opportunity to be guests of the "Sonderforschungsbereich" in 1998.

I would like to take the opportunity to thank you, Professor Friedman and the staff of the IMA again for the kind hospitality and for providing me with such excellent working conditions. I wish you and the IMA all the best for the future and hope that the contacts that have initiated will result in a fruitful co-operation.

I think that it was a fascinating and fruitful experience for me and I also hope that some direct co-operation with the IMA will result in the area of "Mathematical Modelling and Numerical Simulation of Electro- and Magnetorheological Fluids" as already discussed during my stay with Professors Friedman and Reitich.

Guido Kanschat from Universität Heidelberg (Institut für Angewandte Mathematik) was a visitor from May 29 - June 29, 1997. has the following report:

The aim of my present work is to provide algorithms and code for astrophysicists to simulate radiative transfer problems. Simulation by the linear Boltzmann equation

requires enormous computing resources, memory as well as time. The monochromous model is five-dimensional for three space dimensions. The typical data (, and f in (1)) vary over several orders of magnitude with concentration to small areas. Accordingly, solutions may have large gradients in parts of the domain which require a high resolution to avoid pollution effects.

These "exterior" conditions called for the use of most advanced solution techniques and I decided for parallelization and adaptive grid generation. Since the latter relies on Galerkin discretizations to be efficient and reliable, a finite element discretization was developed. During my month at the IMA I had the time and concentration -- away from daily university business -- to do a thorough stability analysis for this discretization. The resulting article will soon be ready for publication.

The opportunity to meet other scientists for discussion was of great value: Endre Süli gave a lot of hints according to his experience with the streamline diffusion method and I could talk with Ronald Hoppe about the numerical treatment of Maxwell's equations, which will be my future field of interest.

I also used the time to port my code to newly available machines, the T3E and the SGI/Cray Origin 2000.

Finally, the invitation to the IMA gave me the opportunity to visit the NCSA in Urbana/Champaign and to start a collaboration on radiative transfer simulation with Mike Norman there.

Ralph E. Kleinman of the University of Delaware (Mathematics and Science) was a visitor from February 1 - March 31, 1997. He writes:

While at the IMA in February and March my main activity was concerned with a problem of uniqueness for scattering by obstacles in waveguides. The boundary conditions on the wave guide walls as well on the boundary of the scatterer play a crucial role in establishing uniqueness. Moreover it is not altogether obvious how best to incorporate a condition of propagation, a radiation condition. I spent a lot of time trying to revise a paper on this subject. It turns out that uniqueness can be establish for all but a discrete number of frequencies however we were only able to accomplish this for Dirichlet conditions on the scatterer and then only by imposing substantial geometric restrictions. I interacted with many people at the IMA on this subject including Fadil Santosa,Walter Littman and Hans Weinberger. After giving a talk on this subject at the PDE seminar Hans suggested an alternate method of proof which might help reduce the geometric constraints. I am still working on this and of course will gratefully cite the IMA in the forthcoming paper. If this idea with Hans works out I would hope it will result in another paper. A second area of activity at the IMA dealt with inverse problems. I had many stimulating discussions with Frank Natterer and Margaret Cheney which may result in a collaboration with Margaret. I also presented a talk on this subject at the post doc seminar. After my talk in the analysis seminar I had some very interesting discussions with John Lowengrub on preconditioners for first kind integral equations on planar boundaries. I described a method we found which resulted in startling improvement in the convergence rate of conjugate gradient solutions. John suggested that a generalization to non-planar surfaces might be possible using some of his results. I am carrying the papers he gave me trying to find some time to study them. If this idea bears fruit it could have significant impact in accelerating convergence of a large class of first kind integral equations. I attended the workshop on medical imaging and found it most illuminating. All in all I found the stay very useful for me in providing some new ideas and interaction with a large number of interested and interesting colleagues. I am grateful for the chance to be there.

Martin Kruzík of the Czech Academy of Sciences (Decision-making Theory) reports:

My ten month stay (September 1996 - June 1997) with the IMA was supported by the Fulbright fellowship and by the IMA. During that time I defended my Ph.D. thesis [1] and have been working on several projects.

First, I finished the final version of my paper [2] which has already been accepted for publication in the SIAM Journal on Numerical Analysis. This paper was also published as the IMA preprint No. 1485. The paper deals with numerical solutions to two-dimensional problems in nonlinear elasticity when the stored energy density of a material attains its minimum on two mutually different rotationally invariant sets (e.g. two crystallographic states - phases of the material). In many cases gradients of elements of minimizing sequences oscillate between these two sets. The alternation of these phases makes a microstructure. The oscillations cause the energy functional being not weakly lower semicontinuous and its minimum is not generally attained. Therefore, some kind of relaxation is needed. We use relaxation based on Young measures representation.

The second paper written during my stay in the IMA is [3]. This paper is devoted to investigation of geometric properties of sets of (generalized) Young functionals. Roughly speaking, taking a bounded sequence , where and bounded we define the Young functional as (after possible extraction of a subsequence) , where a separable subspace of the space of Carathéodory functions. Of course, the set of Young functionals depends on how large we take the subspace H. Under certain assumptions it can be shown that this set is convex and -compact; cf. [8]. In the paper we study geometric properties of these sets, especially, rays, extreme points and extreme rays. Applications to optimal control problems and the Choquet theory are given.

The third paper [4] which has appeared as the IMA preprint No. 1494 studies a relation between a special kind of Young functionals called DiPerna-Majda measures and uniform integrability of L¹-bounded sequences. We apply our results to Fatou's lemma and study when the weak convergence in L^p implies the strong one.

Further, I have been working on three ongoing papers. The first one is devoted to studying of a finite element approximation of the DiPerna-Majda measures [5]. Especially, we are about to establish results concerning convergence and error estimates. Further, we will apply our methods to examples from the optimal control theory.

In the second paper [6] we numerically solve tasks appearing in micromagnetics. Mathematically, the problem reduces to minimization of not weakly lower semicontinuous integral functional and to its relaxation.

The last ongoing paper [7] (together with M. Luskin) contains three-dimensional problems solved in [2] in two dimensions.

I had discussions (but without common papers) especially with V. Sverak, S. Mueller, T. Iwaniec, P. Plechac and also with A. Prohl and M. Gobbert.

REFERENCES

[1] M. Kruzík: Oscillations, Concentrations and Microstructure Modeling., Ph.D. thesis, Charles University, Prague, 1996.

[2] M. Kruzík: Numerical approach to double well problems., accepted.

[3] M. Kruzík, T. Roubícek: Geometric properties of generalized Young functionals., submitted.

[4] M. Kruzík: DiPerna-Majda measures and uniform integrability., submitted.

[5] M. Kruzík, T. Roubícek, M. Vítková: Optimal control problems admitting concentrations and oscillations effects and their numerical treatment., in preparation.

[6] M. Kruzík: Numerical solution to relaxed problems in micromagnetics., in preparation.

[7] M. Kruzík, M. Luskin: Numerical solutions to three-dimensional double well problems., in preparation.

[8] T. Roubícek: Relaxation in Optimization Theory and Variational Calculus., W. de Gruyter, Berlin, 1997.

Jon C. Luke of Indiana University/Purdue University IUPUI(Mathematical Sciences) writes:

Thank you for the opportunity to be in residence for two months during the past academic year. The stay at the IMA, during the months of September and April, was a crucial part of my sabbatical leave from IUPUI during this past year.

Here is a brief summary of my activities at the IMA, which I hope will be useful for your records as well as my own:

I was in residence at the IMA for Sept. 1 - Sept. 30 in 1996 and April 1 - April 30 in 1997. That period afforded an excellent opportunity for me to work intensively on my research, which at present has to do with nonlinear wave equations (e.g., nonlinear complexvalued Klein-Gordon equations and related systems). The IMA's computer facilities and the University's library facilities were both very useful in this regard.

During my stay in September, 1996, I also attended the tutorials on Message Passing Interface (MPI) and High Performance Fortran (HPF) (Sept. 9-20) as well as the workshop on Algorithms for Parallel Processing (Sept. 16-20). I found the work with MPI (with Rusty Lusk and Bill Gropp) particularly informative. In my work during the rest of the month, discussions with Matthias Gobbert and Qing Nie were very helpful.

During the PDE Software workshop that I attended in April (April 21-25, 1997) I interacted especially with Olivier Pironneau, Petter Bjørstad, and Klaus Johannsen. At other times during the month of April, I had extensive talks with Xiaobing Feng and Martin Kruzik.

A special objective completed during my sabbatical was to have some discussions with Fadil Santosa, so as to learn more about the Minnesota Center for Industrial Mathematics (MCIM). Based on those discussions I will be able to act as a resource person in the event that a somewhat similar program is initiated in the Indianapolis area at some time in the future.

In addition it was especially enjoyable to have the opportunity to renew my acquaintance with a number of people after many years, in particular, with David Sattinger, Walter Littman, Peter Rejto, George Sell, and Hans Weinberger.

Thanks again for making my stay so productive and enjoyable.

I want to thank you again for the invitation to participate in the conference this past weekend. All of the honorees except for Willard were at Minnesota when I was a student, and Willard came just as I left. So I greatly appreciated the opportunity to honor their long and distinguished careers. And, of course, it was a special pleasure to honor Han's career.

Andreas Prohl from Christian-Albrechts Universität Kiel (Mathematisches Seminar II) reports:

The primary goal of my stay at the IMA of the University of Minnesota during the academic year August 1, '96 - July 31, '97 was to get acquainted with the modeling and analysis of martensite crystals exhibiting microstructure, with emphasis on numerical methods. This project was supported by the german research association (DFG) through a scholarship.

I gratefully acknowledge the support granted by Prof. M. Luskin from the School of Mathematics (University of Minnesota) who introduced me to this field and focused my attention on relevant literature, current talks or courses, like the course offered by Prof. James (Department of Aerospace Engineering and Mechanics) during the winter quarter 96/97, dealing with martensite crystals and the physical as well as mathematical background.

In particular, I read articles of the authors B. Li, C. Collins, P. Gremaud, D. Kinderlehrer, P. Kloucek and M. Luskin that have been published throughout the last decade (see [3] for references), dealing with the approximation of (simply laminated) microstructures. These authors propose and analyze conforming and nonconforming finite element models to compute microstructures through finite dimensional models. It is well-known, that these numerical methods perform well in the context of minimizing convex problems.
But, if we are concerned with the minimization of nonconvex functionals on a given set as it is the case here for the Ericksen-James energy density allowing microstructure, we need the triangulation of the domain to be aligned to the laminates in order to be able to numerically resolve the microstructure. This is clearly seen in computational experiments and is reflected by the fact that the related numerical analysis gives suboptimal rates of convergence for quantities of interest in this context.

The conclusion from this is that the standard finite element methods enforce too many constraints of continuity to the computed solution, thus polluting it with (non-aligned) mesh information. This was the starting point of Dr. M. Gobbert (Postdoc at the IMA during the same period of time) and me to propose new numerical ansatzes that are more flexible, thus being capable to resolve microstructure in the case of acting forces or for evolutionary models.

In our joint work, we proposed a new finite element method, based on discontinuous elements, in order to increase the flexibility of the approximation for general triangulations of the domain. The (weakened) continuity constraint of the computed deformation is taken into account through a penalty term added to the bulk energy. In the same sense, the prescription of the given boundary data is relaxed to avoid pollution effects on the approximation coming from the averaged boundary data.

Despite of the short time, we succeeded in obtaining the following results:

1. Proposal of a new finite element method suitable for resolving simply laminated microstructure on general grid topologies,

2. Presentation of a mathematical convergence analysis, qualifying the approximation of several quantities describing microstructure. We emphasize that we obtained results of convergence that are better by a factor of 4 compared to those obtained by the previously named authors,

3. Computational Experiments - based on the finite-element package FEAT2D -- have been performed. In particular, they show the superiority of this ansatz in comparison with 'classical' (non-)conforming methods. The resolution of microstructure is highly independent on the underlying mesh.

We are now preparing publication of these results, see [1] and [2]. It is further intended to extend our collaboration to test this new algorithm for evolutionary models, e.g., describing movement of interfaces between different phases.

A second subject of research was the construction of time-splitting schemes for computing chemically reacting flows. This project is part of a collaboration with Prof. Sell (School of Mathematics of the University of Minnesota) and his student D. Norman. I proposed and analyzed first and second order time-discretization schemes that allow computation of velocity, temperature, and species in a decoupled manner without loosing accuracy, [6], [7]. These new ansatzes give way to parallelization ansatzes to effectively compute chemically reacting flows in a reliable way. In these studies, I assume homogeneous Dirichlet type boundary data, and it is intended in the ongoing collaboration with Prof. Sell and D. Norman to construct numerical models applicable to cases with physically more relevant boundary conditions.

The latter project is closely related to fluid flows governed by the incompressible Navier-Stokes equations, where I worked on time-discretization schemes (projection methods) during my Ph.D. studies in Heidelberg. During my stay at the IMA, I finished a monograph on this topic that is recently published, [4].

Finally, I would like to thank the IMA for the hospitality during my stay there. I enjoyed the nice and stimulating atmosphere: the broad range of workshops was a very good opportunity for me to get introduced to new topics and the acquaintance with researchers in various fields of (numerical) analysis gave me the chance of further collaboration.

REFERENCES

[1] Matthias Gobbert and Andreas Prohl, On a new finite element method for solving a multi-well problem, in preparation

[2] Matthias Gobbert and Andreas Prohl, A comparison of classical and new methods for approximating laminated microstructure, in preparation

[3] Mitchell Luskin, On the computation of crystalline microstructure, Acta Numerica (1996).

[4] Andreas Prohl, Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations, Teubner (1997).

[5] Andreas Prohl, An adaptive method for computing laminated microstructure, in preparation

[6] Andreas Prohl, Proposal and analysis of a first-order projection-based time-splitting scheme for solving a model for chemically reacting flows, in preparation

[7] Andreas Prohl, Proposal and analysis of a second-order projection-based time-splitting scheme for solving a model for chemically reacting flows, in preparation

Rolf Rannacher from the University of Heidelberg (fur Angewandte Mathematik) during his April - June 1997 visit gave a presentation on Special Short Course: Computational Methods for the Incompressible Navier-Stokes Equations held on April 14-15, May 5-6, 1997. He also delivered a lecture on the following workshops: "Grid Generation and Adaptive Algorithms, April 28 - May 2, 1997 and "Parallel Methods for Partial Differential Equations," June 9-13, 1997; Below is his a research report:

The area of my current research is the efficient numerical solution of partial differential equations. Topics of particular interest are:

1. Adaptive Finite Element Methods: A general concept for a posteriori error estimation and adaptive mesh design has been developed which is applicable to almost any problem posed in variational form and yields control for arbitrary functionals of the error. The potential of this approach is currently investigated for various problems in computational science, e.g., fluid mechanics, hyperbolic conservation laws, elasticity and elasto-plasticity, radiative transfer, reactive flows, dynamical systems, shape optimization.

Activities:

- Lecture "A general concept for adaptivity in finite element methods" (IMA Workshop on Grid Generation and Adaptive Algorithms, April 28 - May 2, 1997)

- Report "A general concept of adaptivity in finite element methods with applications to problems in fluid and structural mechanics" (with R. Becker, Proc. IMA Workshop on Grid Generation and Adaptive Algorithms, April 28 - May 2, 1997)

- Lecture "How to control the error in the numerical approximation of PDE" (Colloquium, School of Mathematics, University of Minnesota, April 17, 1997)

- Current project "A posteriori error control and mesh adaptation for FE models in elasticity and elasto plasticity" (with F.-T. Suttmeier, University of Heidelberg)

- New project "A posteriori error control and mesh optimization in the h-p finite element method" (with R. L. Scott, Visitor/Organization at IMA)

2. Domain-Decomposition and Parallel Multigrid Methods: Large scale simulations particularly in fluid mechanics require the use of parallel computers. It is investigated how the recently developed highly efficient sequential multigrid solvers for the Navier-Stokes equations can be implemented on distributed memory machines. The goal is to find the best compromise between the concept of domain-decomposition on the discretization level and simple grid-decomposition on the algebra level.

Activities:

- Lecture "A parallel multigrid solver for the incompressible Navier-Stokes equations" (IMA Workshop on Parallel Methods for Partial Differential Equations, June 9-13, 1997)

- Report "Parallel solution of the incompressible Navier-Stokes equations by domain decomposition and multigrid methods" (Proc. IMA Workshop on Parallel Methods for Partial Differential Equations, June 9-13, 1997)

3. Incompressible Navier-Stokes Equations:The numerical simulation of the incompressible Navier-Stokes equations on complex geometries requires highly robust and efficient methods. Recently such methods have been devised by combining the flexibility of finite element discretizations with the efficiency of multigrid methods.

Activities:

- Lecture Series on "Computational Methods for the incompressible Navier-Stokes Equations" Part I: Discrete models Part II: Multigrid solution Part III: A priori error analysis Part IV: Adaptive methods

(Special IMA Short Course, April 14, 15, May 5, 6, 1997)

- Current project "An analysis of Chorin's projection method for the incompressible Navier-Stokes equations" (with A. Prohl, Postdoc at the IMA)

4. Transport-Dominated Problems: The reliable numerical solution of transport-dominated equations poses new problems. There are significant far-field effects in the error propagation which cannot be captured by the usual local error estimators. The discretization by the finite element Galerkin method with streamline diffusion stabilization opens the way for rigorously based adaptive error control by means of global "duality arguments."

Activities: - Current project "An adaptive streamline-diffusion finite element method for hyperbolic conservation laws" (with Chr. Führer, University of Heidelberg)

- New project "A posteriori error analysis of the streamline diffusion finite element method" (with E. Suli, Visitor at IMA)

Thanks again for this pleasant time at the IMA.

Rosemary Renaut of the University of Arizona State University (Mathematics) was a visitor from May 11 - June 14, 1997. He attended the workshop on "Parallel Solution of PDE" held on June 9 - 13, 1997. She writes:

I am not convinced that my contributions are immediately very stunning. For myself the contacts I developed, although not immediately collaborative, have led to several new directions for both of the research projects I was involved in. In particular the workshop on finite elements has broader impact on my general research activities. I expect that I will take some discussions on additive Schwarz methods into a draft of another paper, which I will then send to some of the other participants. At that point some collaborations I hope the collaborations will materialize.

I think you know also that I had a severe distraction from ASU during my time at IMA. I want to thank you though for the opportunity to be at the IMA. Even with the distraction of ASU via email, I was able to make some substantial progress in my own research.

Thanks again for the hospitality shown by all the staff of the IMA.

Research Projects

Numerical Methods for Third Order Partial Differential Equations

Partial differential equations containing terms that are third order spatial derivatives arise in the modeling of many practical situations involving wave propagation in nonlinear dispersive media. One well-known and well-studied example is the Korteweg-de Vries equation which accounts for dissipative effects in addition to non-linearity and dispersion. Numerical solutions of such equations must accurately predict all dissipative, dispersive and nonlinear effects of the wave propagation. The design of suitable solvers is therefore not straightforward.

My research has focused on the use of pseudospectral methods for the solution of equations with third order derivatives in space on non-periodic domains. Pseudospectral methods offer the advantage of high accuracy, in fact spectral accuracy, if the underlying approximations are of sufficiently high order. Previous approaches have considered the use of either i) approximations based on Fourier expansions of the wave on a periodic domain or ii) finite difference approaches for non-periodic domains. The former approach has the clear disadvantage of requiring periodicity, which is not always realistic physically, and hence leads to the use of numerical domains which are artificially large with respect to the physical domain under consideration. In the latter case the design of schemes which are both of high accuracy and allow for adequate representation of boundary conditions and nonlinear effects is cumbersome. To obtain approximations of high order to third-order derivatives the stencils needed are not compact, which requires that sets of stencils are required to model the regions near boundaries. Moreover, high accuracy with respect to local error control does not ensure minimal artificial dispersion and dissipation of the method.

Pseudospectral methods offer the advantage of globality and spectral accuracy, and hence avoid the aforementioned problems. On the other hand the resulting operators are notoriously ill conditioned with respect to stability and are computationally intensive with respect to advancement in time.

During my visit to the IMA I was able to complete my study of the use of a modified approach in pseudospectral methods for solution of equations with third order derivatives. The methods developed were compared with standard finite difference and periodic pseudospectral methods. Results demonstrate the effectiveness of the approach and are described in the first listed paper below. The research was enhanced by the short term visit of Jerry Bona to the IMA. Jerry presented some thought-provoking results of his research in this area at one of the weekly seminars.

Multisplitting Methods for Solution of Least Squares Problems

The other portion of my time at the IMA was spent studying domain decomposition approaches for the solution of over-determined sets of linear equations. Parallel solution of large-scale problems can be achieved by the application of decomposition in space. The approach follows ideas adopted in recent years for finite-element solution of elliptic equations by additive Schwarz methods.

During my visit I presented this work at the IMA postdoctoral seminar. Because of the anticipated audience I concentrated on the connections between the multisplitting methods, which have been proposed for linear systems, with additive schwarz techniques. The reformulation of development of not only the multisplitting approach, but also of additive Schwarz for finite-element problems. One of the drawbacks of the domain decomposition approach is the effective incorporation of coupling effects between subdomains. In multisplitting a secondary iterative stage has been incorporated to couple the information from local solutions, hence facilitating the faster solution of the global solution. Although no collaborative work has yet arisen out of this discussion the resulting discussions were very helpful and it is still possible that some collaborations could arise if the initial research is promising. Some of this research is described in the second report listed.

Publications and Reports

• R.A. Renaut , "Evaluation of Chebychev Pseudospectral Methods for Third-Order Differential Equations." Numerical Algorithms, submitted .

• A. Frommer and R. A. Renaut, "Parallel Subspace Decomposition Methods for Quadratic Functionals," in preparation.

Chi-Wang Shu from Brown University (Applied Mathematics) was a visitor from April 1 - June 30, 1997.

Here is my report for the IMA visit.
I have enjoyed my IMA visit very much. Thank you for all the arrangements.

Joint research with Professor Bernardo Cockburn, who is also an IMA visitor, on discontinuous Galerkin method for convection dominated problems has been performed. Discontinuous Galerkin method combines the advantages of finite element methods in treating complicated geometry and of finite difference methods in non-oscillatory high resolution for shocks. The progress during my visit is two-fold:

1. We have studied the Local Discontinuous Galerkin methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge-Kutta Discontinuous Galerkin methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, their high-order formal accuracy, and their easy handling of complicated geometries, for convection dominated problems. It is proven that for scalar equations, the Local Discontinuous Galerkin methods are L²-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are k-th order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.

2. We have extended the method to multidimensional nonlinear systems of conservation laws. The algorithms are described and discussed, including algorithm formulation and practical implementation issues such as the numerical fluxes, quadrature rules, degrees of freedom, and the slope limiters, both in the triangular and the rectangular element cases. Numerical experiments for two dimensional Euler equations of compressible gas dynamics are presented that show the effect of the (formal) order of accuracy and the use of triangles or rectangles, on the quality of the approximation.

The following two IMA preprints summarize the results:

1. B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, IMA Preprint 1478. Submitted to SIAM Journal on Numerical Analysis.

2. B. Cockburn and C.-W. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, IMA Preprint 1492. Submitted to Journal of Computational Physics.

Joint research with Professors Bernardo Cockburn and Endre Suli (both are IMA visitors) about accuracy enhancements for discontinuous Galerkin methods is also initiated and is still ongoing.

Paul Siegel from University of California, San Diego (Electrical and Computer Engineering) was a visitor from November 10 - 15, 1997. He expresses:

Thank you very much for arranging for my participation in the recent IMA Workshop on the Mathematics of Information Coding, Extraction, and Distribution. The meeting provided a nice opportunity to hear about an unusually broad range of technical activities, and to interact informally with a wonderful group of researchers. Thanks, also, to you and the IMA staff for providing the comfortable office, convenient (and user-friendly!) computing facilities, useful local information, enjoyable social functions, and the generous travel support.

Vladimir Sverak from the University of Minnesota (Mathematics) reports on his activities at the IMA during academic year 1996-1997:

During the 1996-1997 program my activities mostly concentrated on the following areas:

1. using computational methods to attack certain open problems in the Calculus of Variations and harmonic analysis (in collaboration with P. Plechac, an IMA visitor, and also Tadeusz Iwaniec, and Stefan Muller, short-term IMA visitors)

2. "supervising" (or at least providing some guidance to) Martin Kruzik, a Sloan Fellow, who stayed at the IMA for most of the year

3. learning about recent advances in CFM (computational fluid mechanics) from the world's leading experts who visited the IMA during the year (R. Rannacher, O. Pirroneau)

I will now expand on (1): this work mostly concentrated on trying to obtain a "computational" answer to the long-standing open question whether rank-one convexity implies quasi-convexity for function on two by two matrices. Recently there has been a new surge of interest in this question, since it turned out that it is closely related to the problem of the best constants in the L^p estimates for the complex Hilbert transformation. Together with Plechac, we tried a new computational approach to answer this question. Pleachac wrote the codes and by now the calculations has been going on for about 9 months. So far the evidence we gathered supports the conjectures. There is several numerical problems one encounters in the calculation. The most difficult is the following: Given a smooth function f on two by two matrices, how can we compute its rank-one convex envelope Rf (which is the sup of all rank-one convex function which are less or equal to f)? This problem seems to be interesting not only from the numerical point of view, but also from the point of view of general computability questions.

Plechac and I will write a joint paper explaining our approach and our conclusions.

Paul Thompson of West Group of West Publishing Company (Research) reports:

On 18-20 November 1996, I attended the workshop "Data Mining and Industrial Applications" put on by the Institute for Mathematics and its Applications at the University of Minnesota. I found this workshop to be very useful. It provided the opportunity of hearing many of the leading researchers in the new field of data mining present their research in an informal setting. Unlike at a large conference, it was easy to meet and talk at length with the speakers and other participants. The quality of the presentations was very high. The connections of the new field of data mining to other more mature disciplines such as statistics, pattern recognition, and machine learning was made clear.

My own research interests are in automatic text classification. I had the opportunity of meeting for the first time researchers whose work I had followed in this area. Since the conference, I have continued to have e-mail exchanges with some of the other participants. I am looking forward to meeting with the participants at future conferences and will monitor the literature as new research developments and subsequent commercial products are introduced.

Data mining is currently a very popular field receiving much attention. The workshop did a good job of stimulating my thinking about this field and its ties to the related discipline mentioned above. Should my company decide to pursue research or development in data mining, my having attended the workshop will give us a good starting point.

I found the workshop to be very worthwhile and would be interested in participating in similar workshops in the future. Thank you again for IMA's invitation for me to participate.

5. POSTDOCS

Matthias K. Gobbert from Department of Mathematics and Statistics, University of Maryland, Baltimore County reports:

I spent the academic year 1996/97 at the IMA as part of the special year in High-Performance Computing.

During this year, I participated in many of the workshops and tutorials held at the institute, in particular the ones relating to numerical methods for partial differential equations and their efficient implementation on modern computers. Besides I participated actively in the Post-Doc Seminar. Outside the IMA, I participated actively in the Numerical Analysis seminar, in which I gave a talk on "A Homogenization Technique for the Development of Mesoscopic Scale Models for Chemical Vapor Deposition," as well as the Matlab seminar in the Winter quarter 1997, in which I also gave two talks.

Besides this participation, I continued my dissertation research with collaborators at Arizona State University and Motorola (Mesa, Arizona). To this end, I spent three weeks at Motorola working on extending our results to other physical processes. A paper about the work done in collaboration with Motorola and Arizona State University was published in the Journal of the Electrochemical Society: Matthias K. Gobbert, Tushar P. Merchant, Leonard J. Borucki, and Timothy S. Cale, "A Multiscale Simulator for Chemical Vapor Deposition," J. Electrochem. Soc., vol. 144, no. 11, 1997, pp. 3945-3951.

In addition to continuing my dissertation research, a new project on the computation of crystalline microstructure was begun in collaboration with the long-term visitor Andreas Prohl from the University of Heidelberg (Germany), now at the University of Kiel (Germany). This is work inspired by earlier results of Mitchell Luskin of the School of Mathematics at the University of Minnesota. The problem is the discretization and minimization of a non-convex energy functional arising from modeling the elastic energy of phase transformations in shape-memory alloys, a topic of considerable and growing practical importance. We succeeded in improving the old energy estimates for classical conforming and non-conforming elements (half order in the mesh parameter) by using discontinuous finite elements (second order in the mesh parameter) and a scaled energy functional. This result was both proven by theoretical analyses and demonstrated by numerical calculations for an extensive case study on a test problem. This collaboration represents a great example for the type of collaboration arising in the interdisciplinary environment of the IMA. The Two papers below reports on the theoretical analyses and the numerical case study, respectively, are currently still under preparation and will be submitted for publication in the IMA Preprint Series.

1. Matthias K. Gobbert and Andreas Prohl, On Finite Element Methods for Solving a Multi-Well Problem.

2. Matthias K. Gobbert and Andreas Prohl, A Comparison of Classical and New Finite Elements for the Computation of Crystalline Microstructure.

Michael A. Kouritzin of the University of Alberta, Department of Mathematical Sciences

IMA Industrial Fellowship with Lockheed Martin

I, Michael A. Kouritzin, was fortunate to spend two productive years at the Institute for Mathematics and its Applications. I will always be very grateful to the IMA staff and the personnel at Lockheed Martin Tactical Defense Systems whom I interacted with. Before coming to Minnesota, I had a reasonable background in stochastic convergence and some aspects of applied mathematics as well as a developing interest in parabolic equations. I was asked by my industrial partner to investigate applying non-linear filtering theory to air traffic management and defense industry problems. I was neither familiar with the theory of non-linear filtering nor their modeling and implementation ideas. However, after this became understood by Lockheed, they allowed me the time to learn and the opportunity to develop my own ideas. I suggested several modeling ideas some of which were incorporated into proposals and, more importantly, developed and promoted a practical method of non-linear filtering. The traditional mathematical solutions to the non-linear filtering problem under classical conditions neither are readily convenient for implementation nor apply to the problems which were of interest. Indeed, there is no single best practical method of filtering for nonlinear problems where the Kalman filter does not apply.

My first approach, developed under the encouragement of Avner Friedman, Craig Poling, Brian Leininger, Ron Mahler, and Keith Kastella, was to try to reduce the filtering problem down to the easily implementable problem of convolution (plus certain transformations, scaling etc.). The first subproblem with this approach was to examine the class of problems that could be exactly implemented as a single convolution. In two papers "On exact filters for ..." and "A nonlinear filter for ...", written while at the IMA and listed below, we introduced and studied this concept of exact infinite dimensional filters. Later, near the end of my tenure at the IMA, I discovered additional classes of problems which can be solved as exact infinite dimensional filters. These classes are currently being written into an additional paper. There turned out to be a second interesting subproblem: What type of filtering problems could be approximately solved with a modest number of convolutions? This question lead to some interesting mathematical developments which are also being written up into two papers. Overall, these convolution-based methods performed extremely well in tests, have many inherent advantages, and have gained the support of Lockheed Martin. I also became interested in the branching particle system approach to nonlinear filtering suggested by Dan Crisan and Terry Lyons as well as in the multiple-target filtering problems. Lockheed Martin and myself have agreed to continue collaborating in all these filtering efforts. I am indebted to Professors Avner Friedman, Walter Littman, and Robert Gulliver for their guidance and assistance with these problems.

My other major area of interest while at the IMA was averaging and stochastic averaging for arbitrary-order parabolic equations with random coefficients. I am pleased to acknowledge benefit from discussions with Professors George Sell and Avner Friedman at Minnesota. I wrote four papers in this area while at the IMA and three of them have already been accepted. Furthermore, I also worked on related problems in stochastic partial differential equations which I hope to be able to complete and publish.

REFERENCES

[1] D. Dawson and M. A. Kouritzin. Invariance principles for parabolic equations with random coefficients. J. Functional Analysis, 149:377-414, 1997. IMA Preprint #1433.

[2] M.A. Kouritzin. Averaging for fundamental solutions of parabolic equations. J. Differential Equations, 136:35-75, 1997. IMA Preprint #1379.

[3] M.A. Kouritzin. Approximations for higher order parabolic equations. Submitted. IMA Preprint #1444.

[4] M.A. Kouritzin, "Stochastic processes and perturbation problems defined by parabolic equations with a small parameter," to appear in The Second World Congress of Non-linear Analysts. IMA Preprint #1443.

[5] M.A. Kouritzin, "On exact filters for continuous signals with discrete observations", to appear in the IEEE Transactions on Automatic Control. IMA Preprint #1409.

[6] M.A. Kouritzin, "Arbitrary order parabolic stochastic partial differential equations with general Hilbert-space-valued noise", The IMS Bulletin 26 (1997) pp. 322-323.

[7] K. Kastella, M.A. Kouritzin, and A. Zatezalo, "A nonlinear filter for altitude tracking", October 1996 Proceedings of the Air Traffic Control Association pp. 1-5.

Sergey Lototsky of Massachusetts Institute of Technology (Mathematics)

I. Research. My research during the year was in three areas: nonlinear filtering of diffusion processes, parameter estimation for stochastic parabolic equations, and analytical theory of boundary value problems for stochastic partial differential equations.

1. Nonlinear filtering. The problem in question is estimation of the unobserved component of a partially observed system. This problem arises in many applications (navigation, target recognition, etc.)

The unobserved process (signal) is modeled by a nonlinear diffusion equation, and the observations, by a nonlinear transform of the signal, corrupted by noise. Both continuous and discrete time observations are possible. The objective is to develop a numerical algorithm for efficient real time computation of the approximation of the optimal nonlinear filter. The central idea of the algorithm is to separate the deterministic and stochastic information. By performing off line the computations involving only deterministic information, the on-line performance of the algorithm can be significantly improved. Theoretical issues include the proof of convergence of the approximate filter and the analysis of the complexity of the resulting algorithm.

The following papers were prepared:

The first paper deals with the discrete time observations, while the second paper considers continuous time model.

The results of the investigation were presented at the Workshop on Stochastic Control and Nonlinear Filtering, North Carolina State University, October 11 and 12, 1996.

I also had several fruitful discussions about practical implementation of the algorithm with A. Zatezalo, graduate student at the School of Mathematics, University of Minnesota, Dr. M. Kouritzin, IMA postdoc, and with the representatives of the Lockheed Martin Corporation.

A discussion with Professor P. Bjørstad helped in the analysis of the complexity of the algorithm.

An interesting idea for the future research -- to use evolutionary programming for generating the coefficients of the optimal filter -- was inspired by the talks at the IMA Workshop on Genetic Algorithms.

2. Parameter Estimation. The objective is to develop a method of estimating unknown parameters from the observations of a random field. One possible application is estimation of the thermodiffusivity and the cooling coefficient in the heat balance equation of physical oceanography.

The mathematical model of the observations is a stochastic evolution equation driven by space-time white noise on a compact smooth manifold. The unknown parameters appear as the coefficients of the (known) partial differential operators in the equation. The novelty of the approach is that no assumptions are made about the spectral properties of the operators.

An estimate based on finite dimensional projections of the observations was suggested and the asymptotic properties of the estimate were studied as the dimension of those projections was increased. Conditions were found for consistency, asymptotic normality, and moment convergence of the estimate. Two papers were prepared:

The first paper gives a detailed treatment of the single parameter case, and the second paper extends some of the results to the multi-parameter setting.

The content of the first paper was presented in February 1997 at the School of Mathematics Probability Seminar.

3. Boundary value problems for SPDEs. This research was in collaboration with Professor N. V. Krylov from the University of Minnesota School of Mathematics. The objective was to establish existence, uniqueness, and the regularity of the solution of the first boundary value problem for SPDEs, without using compatibility conditions. At this point, the research was purely theoretical, although possible applications include optimal nonlinear filtering in a bounded domain and populational genetics.

By introducing a special scale of Banach spaces that allow the derivatives of the functions to blow near the boundary, we proved, in the case of one spatial variable, the solvability of the equation in that scale, and determined the maximal possible rate of explosion of the derivative. A paper was prepared, describing the results:

The content of the paper was presented at the MSRI Workshop on Stochastic Partial Differential Equations, Berkeley, CA, September 15-19, 1997. The collaboration with Professor N. V. Krylov continues after I left the IMA, and we are currently working on extending the results to equations in multi-dimensional domains.

II. Other Activities. I enjoyed the IMA tutorials on MPI/HPF, Mathematics of Medical Imaging, and PDE Software, even though the topics were not directly connected with my research. I also found interesting the talks at the School of Mathematics Probability Seminar, and some talks at the PDE, Analysis, Numerical Analysis, and IMA Postdoc Seminars, and the Colloquium.

During the school year, I was sitting in three classes: Parallel and High Performance Computing (Fall 1996, Professor V. Kumar), Malliavin Calculus (Winter 1997, Professor V. Bogachev), Controlled Diffusion Processes (Spring 1997, Professor N. V. Krylov). Even though not directly connected with the ares of my current research, these courses had a great educational value.

Brian T. Nguyen who is currently affiliated at Medtronic, Inc. was one of the postdocs for the 1996-1997 program on "Mathematics of High-Performance Computing." Below are his activities:

1. Non-linear Dispersive Equations

In collaboration with Peter Olver and Yi Li from the University of Minnesota Mathematics Department, we simulated and studied non-linear dispersive equations in one dimension. I wrote a general code and supporting software for solving equations of the form

using very high-order accurate finite differences in space and a robust trapezoid method (modified for the nonlinear equation) in time.

After testing the method extensively using exact solutions and other numerical solutions for this general equation, we began using the code on the critical surface tension model equation

We developed a method for numerically computing solitary wave solutions in 1D, by starting with a solution that is close to a solitary wave solution and selectively damping dispersed waves in the domain of computation. This method allowed us to obtain two solitary wave solutions for a collision study. In an IMA Preprint, we proved the feasibility of this method for obtaining solitary wave solutions, and showed that to very high accuracies that the solutions obtained have constant propagation speeds and constant wave forms. For the critical surface tension model, we presented a collision solution which shows the level of dispersion induced by the collision. The Figures 1 and 2 show the scale between the full waves and the dispersive waves.

2. A Posteriori Error Analysis for Hyperbolic Equations

This project, a collaboration with Bernardo Cockburn in the University of Minnesota Department of Mathematics, aimed at completing a new method for a posteriori error estimates of a general hyperbolic equation. Cockburn[1] has shown for the equation

that if u(x,t) is an approximate solution with residual R(u), then the error norm ||v-u|| satisfies

Our aim was to derive from this equality an a posteriori error bound not requiring v on the right hand side, then use numerical means to compute it.

Unfortunately, we found this formulation to not be feasible in conjunction with using numerical means to compute it. The computation of estimate is itself is incurring numerical error that is difficult to eliminate. The terms in the formulation must largely cancel out, and the time integration of the whole must be extremely accurate. These error of the estimator grows exponentially in time and quickly overtake the numerical of u. This problem is absent in elliptic equations, because no time variable exists.

3. Other activities

• Chaired fall term postdoc seminars

• Published article entitled "On Comparisons of Time-Domain Scattering Schemes"[4], a critique of methods of comparing numerical methods. The article addressed the shortcomings of current methods and proposed comparison criteria that can be used to assess fairly the differences in efficiency between two different numerical methods.

• Submitted paper entitled "Solitary Waves in the Critical Surface Tension Model"[3]

• Presented a poster on the upwind-leapfrog method for electromagnetic and acoustic scattering predictions at the University of Michigan, Godunov Seminar.

REFERENCES

[1]Bernardo Cockburn. Personal Communication, 1996.

[2]

[3]Yi A. Li, Brian T. Nguyen, and Peter J. Olver. Solitary waves in the critical surface tension model. Will appear in the IMA Preprint Series, February, 1998. Submitted to a special issue of J. Engineering Math, on "Applications of Solitons and Symmetries," M. Ablowitz& P. Clarkson (eds).

[4] Brian T. Nguyen. On comparisons of time-domain scattering schemes. IEEE Antennas and Propagation Magazine, 39(1):99-102, February 1997.

Sanhita Sarkar of Oracle Corporation reports:

As a Post Doctoral Researcher for the program "Mathematics in High Performance Computing, 1996-1997" at IMA, I mainly concentrated on performance issues of Relational Databases and extended it to various applications like internet, data mining, data warehousing, etc. The workshops of my interest were: "High Performance Fortran (HPF);" "The Message Passing Interface Standard (MPI);" "Algorithms on parallel Processing;" "Data Mining and Industrial Applications;" "Mathematics on Information Coding, Extraction and Distribution," etc. During my stay at IMA, I was able to meet several visitors and other IMA postdocs with whom I could exchange my thoughts and was very happy to get the proper suggestions/comments from the pioneers of the field. This led to long-term implementation possibilities of my research ideas mainly in RDBMS performance, data mining and data warehousing here, at Oracle Corporation where I am being able to contribute as the Principal Member of our Performance team. Last but not the least, I want to mention that I am very glad to have had the opportunity to work in an industry oriented research environment of IMA.

Research Reports:

Some of my IMA technical report abstracts are described in the following paragraphs.

* Normal Distribution as a method for data replication in a parallel data server:

Run-time load balancing is always a problem in scalable parallel data servers because of initial data distribution. A data warehouse is a scalable parallel server to operate on high volumes of data for analytical processing. Relational model implementations for analytical processing should incorporate very fast operations like join, scan, sort, etc. Currently, for shared-nothing MPP architecture, data is partioned at load time in a shared-nothing way. The task distribution in a join, scan and sort process is determined by the query optimizer in a relational engine and there is no possibility of load balancing at run-time because of the high network and I/O cost. However, for a large number of nodes in a MPP architecture, there may be a need for load sharing at run-time, unlike the plan generated by the optimizer in relational engines. This paper describes a theory for normal distribution of data from a node to its immediate neighbouring nodes. This normal distribution is implemented as a replication technique for small fractions of data from a node to its adjacent nodes at load time. Since a data warehouse is a read-only data service system, fractional replications can be distributed without worrying about audit or update. During run time, based on mutual agreements, task load can be transferred from a node to its adjacent nodes, for minimizing the overall time involved in relational operations. This normal distribution of fractional replicated data leads to a unique run-time solution for load balancing, not available in current parallel data servers.

* A Graph Theoretic Approach for Parallel View Materialization with Dynamic Load Balancing:

A View Cache is a stored collection of pointers pointing to records of underlying relations needed to materialize a View. This paper deals with the View Cache method of materialization which has been proven to achieve both high performance and efficient incremental updates. Using a bipartite graph representation of a View Cache, the problem of optimal View materialization has been posed as a problem of optimal traversal through the edges of the graph. Next, this paper deals with an efficient method of parallelizing this traversal using normal distribution as a replication technique for small fractions of records and ViewCache partitions.

* Internet and Relational Databases in a Multi-Tier Client/Server Model:

This paper elaborates on the ideas of View Cache creation and maintenance problems. Views are constructors for objects sitting on top of relational databases. View Cache maintenance and manipulations will answer the complex database question for multi-tier client/server applications. A new idea of partial exposition of data values in View Caches has been discussed. Java bytecode along with view records can be transported to other network nodes for materialization and computations. This paradigm may add a completely new feature to object paradigm and relational dbms integration.

* Views and Data mining in a parallel data server

This paper describes how data mining can be performed on Views in a parallel data server. A decision tree approach is considered and tests are performed on a View Cache generated from relational operations on the base level tables. Quinlan's ID3 algorithm for a proper selection of attributes at a particular level of the decision tree calculates the information gain for all such attributes and considers the one with the maximal gain. This requires multiple View materializations at a single level. When we are considering a parallel data server, the number of records in a View Cache may be skewed to a processor/s and the materialization at any tree-level/s will not exploit efficient parallelism. This paper gives a bipartite representation of a View Cache and poses the problem of optimal view materialization.

* Data Mining on Views in Relational Databases using Bayesian Networks

There are many different ways of representing knowledge, and for each of these ways, there are different discovery algorithms. This paper uses probabilistic graphical models as a unified qualitative and quantitative framework for representing and reasoning with probabilities and independencies and applies such framework on Views in Relational Databases for data mining. It considers a simple form of graphical model, the Bayesian network and incorporates it on a View Cache hierarchy for studying the dependency structure between several attributes in a Relational Database, for knowledge discovery.

* Strategies for Optimal Performance of Relational Databases

High level query languages allow us to write queries that take a great deal of time to execute, and their execution time can be reduced greatly if the query language processor rephrases the query before executing it. Such improvements are called "optimizations" although the rephrased query need not be optimal over all possible ways of implementing the query and the term "improvement" would be more appropriate. Other considerations are appropriate paralleization of query execution, proper choice of indexing, mode of optimization by optimizers and different replication scenarios. This paper has touched upon different strategies of enhancing performance in Relational databases.

Brian J. Suchomel from the University of New Hampshire (Mathematics Department) writes:

I arrived at the IMA a few weeks after completing everything I needed for my PhD. My first order of business was to prepare two papers for publication from my dissertation. These were co-written with Benito Chen and Myron Allen at the University of Wyoming. These are described below:

Network Model of Flow, Transport and Biofilm Effects in Porous Media

We describe how a porous media may be modeled as a set of interconnected cylinders with varying diameters. One system of equations is solved to determine flow rates through individual pores in the medium. We show that the resulting system is symmetric, positive definite. We also show how transport may be modeled on this type of system.

Macroscale Properties of Porous Media from a Network Model of Biofilm Processes

We incorporate a number of species into the model described in the first paper. The goal is to be able to model and predict how flow properties of a porous medium change as a biofilm grows in the pore spaces. Reaction and adsorption terms are added to the advection-diffusion equation modeled in the first paper. Numerical simulations are presented that highlight effects of a few selected parameters.

After completing these two papers, (both have recently been accepted by "Transport in Porous Media.") Petter Bjørstad suggested creating a parallel version of the code I had developed. I created a 2-D and a 3-D version using Fortran and MPI for message passing. We used an Additive Schwarz Domain Decomposition algorithm to solve a pressure equation and an explicit scheme for transport. I have a paper close to completion: "Parallel Implementation of a Network Model for Flow through Porous Media," that I hope to finish late 1997/early 1998. I presented some results at seminars (interviews) at Los Alamos and the University of New Mexico. The paper I am working on will compare performance on the CRAY T3E and Origin 2000. The main focus of the paper is on scaling the problem - keep the size of the problem per processor the same and increase the number of processors. I got a lot of help developing the code from Petter and Barry Rackner.

I have one other paper that I began at the IMA tentatively titled: "Peclet Number as a Function of Pore Radii Variance Computed from a Network Model." It investigates how a network model can be used to examine how varying pore radii influence dispersion. Parallel solution will be included in this paper.

I attended a lot of tutorials, workshops and seminars. I knew absolutely nothing about parallel computing before coming to the IMA, and I consider myself a pretty competent parallel programmer now. I was unaware of many of the issues surrounding high performance computing before arriving at the IMA. Now I feel confident enough to discuss PDE software and solutions, domain decomposition, and scientific visualization software. I also sat in on a class in the Computer Science department on Iterative Methods by Yousef Saad. I hope to include ideas developed while at the IMA in other papers, soon.

I guess I came to the IMA excited about landing an opportunity to work around some pretty well-known mathematicians, and a little intimidated. I know a lot of the other post-docs had much stronger backgrounds than me. I think I learned more last year than I did during any of my years at graduate school. I was not prolific, but I put out a couple of papers last year, and plan on putting out a few more this year. Two of the papers I am currently working on were begun last year. I spent much more time than I wanted to looking for a job for this year. (It's going well, thank you.) I think it's a good idea that future post-docs will be for two years.

Anyway, I enjoyed my time at the University of Minnesota. I feel lucky to have been a part of the IMA.

Lei Wang of Seagate Technology provides a summary of his work at the IMA:

I was an IMA/Honeywell industrial postdoc from August 1995 to May 1997. It is really a great experience for me. Before I came to the IMA, my research work (at the University of Washington) concentrated on the perturbation method. While at the IMA, my work is primarily on numerical PDEs. This transition broadens my knowledge in applied mathematics and greatly enhances my ability in numerical analysis and computation.

During the two years at IMA, I finalized two papers on Hamiltonian system, in collaboration with Prof. J. Kevorkian at the University of Washington. The two papers are published on Physics of Plasmas and Physica D, respectively. I completed a project on the modal analysis of homogeneous optical waveguides using boundary integral formulation and Nystrom method, in collaboration with Prof. Avner Friedman of the IMA and Dr. Allen Cox of Honeywell. We developed an algorithm which can be used to compute the eigenmodes of a multimode optical waveguide very accurately. We also proposed, for the first time, the condition under which the boundary integral formulation of homogeneous optical waveguides has no spurious solutions. This work will be published on the Journal of Optical Society of America A in next January. I would like to thank Prof. Fernando Reitich of North Carolina State University for introducing me the Nystrom algorithm when he was an IMA visitor.

I also completed a project on the perturbation analysis of multimode graded index fiber. I showed that the optical field polarization has very little effect on pulse broadening in the fiber, and the interactions between different modes are very week and can be omitted. My work provides the mathematical justification for the first order perturbation theory for multimode optical fiber, which is widely used in the communication industry. A paper based on this work is almost finished and will be submitted to an optical journal.

I learned a lot on the numerical solution of nonlinear algebraic equations from Prof. Eugene C. Gartland of Kent State University, who is an IMA visitor in 1996. I also learned a great deal on integral equations from Dr. Nie Qing and Dr. Jie Liang, both of them were IMA postdocs in 1996-97. I learned many different topics in applied mathematics from the numerous workshops and seminars in 1995-97. In particular, the weekly industrial mathematics seminar organized by Prof. Avner Friedman will have the greatest impact on me. Now I am in the data storage industry, doing mathematical modeling of magnetic recording head.

Research Papers:

1. L. Wang, J.A. Cox and A. Friedman, "Modal Analysis of Homogeneous Optical Waveguides by the Boundary Integral Formulation and the Nystrom method," IMA preprint #1457, to appear in J. Opt. Soc. Am. A (Jan. 1998).

2. L. Wang and J. Kevorkian, "Asymptotic electron trajectories and an adiabatic invariant for a helical-wiggler free electron lase with weak self-fields," Physics of Plasmas 3(1996) pp. 1162-1175.

3. L. Wang, D.L. Bosley and J. Kevorkian, "An asymptotic analysis of the 1:3:4 Hamiltonian normal form", Physica D 99(1996) pp. 1-17.

4. L.Wang and J. Allen Cox, "Effects of Polarization and Index Profile Deformation on Optical Fiber Spectrum," in preparation.

Daoqi Yang of Wayne State University (Mathematics) reports the following activities:

During my one year stay at the IMA for the Mathematics in High Performance Computing program, I have attended important workshops on Numerical solution of PDES, Numerical Software, Adaptive Grid Generation, MPI, High Perform Fortran, etc, which were very beneficial to me. I have learnt greatly about the current research development of these areas and got to know and talked with the leading figures of my areas.

On the research level, I have had important conversations and information exchange with Mitchell Luskin, Bernardo Cockburn, Petter Bjørstad, Sue Brenner, Chi-Wang Chu, Ridgway Scott, Cai Xiao-Chuan, and many others.

I have also interacted with other postdoc members at the IMA on a daily basis, which provide an active research environment. In particular, I am collaborating with Qing Nie, a former postdoc that I met during my stay at the IMA.

I have written three research papers which later became IMA preprints:

#1472, "Stabilized Schemes for Mixed Finite Element Methods with Applications to Elasticity and Compressible Flow Problems;"

#1489, "Dynamic Finite Element Methods for Second Order Parabolic Equations;" and

#1508, "A Parallel Nonoverlapping Schwarz Domain Decomposition Method for Elliptic Interface Problems."

In summary, my year at the IMA was very fruitful and I wish I could spend more time there.

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