Research Projects and Publications

Many projects are listed below, some from the past and some I'm pursuing now in group sizes varying from one to a dozen. Collaboration is welcomed, or you might suggest new projects.

Research projects for students: you should read this if you are interested in working with me.

Research areas are organized as follows:

Here is a listing of my books, papers, etc., organized by research area, and here is a standard academic vita.

Parallel Computing

Our work in parallel computing has involved developing efficient algorithms for large problems in computational science and engineering, and more theoretical algorithms for a variety of problems and models. Other work has involved understanding and improving the performance of parallel systems, and proposing new parallel architectures.

Most of the work on using large machines involves the research mentioned in scientific computing and computational science and some of that described in adaptive sampling. Research involving more theoretical algorithms includes algorithms for graph theory, geometry, image processing, sorting, routing, and message-passing. Russ Miller and I wrote Parallel Algorithms for Regular Architectures: Meshes and Pyramids which has many algorithms for these machines.

Some of the papers we've written for various machine models are For fundamental problems such as sorting, finding a minimal spanning tree, etc., all of the above models are faster than quantum computers. Few people are aware that while quantum is extremely good for factoring, for many other problems it is slower than classical mechanics.

Some projects I'm involved with, or would like to start:

The parallel computing topics I'm interested often change, and, as I said, I'm interested in collaboration and new problems. I'm particularly interested in problems where being able to exploit a very large number of processors helps one solve problems that otherwise are impossible to solve in a useful amount of time.

Here is a somewhat whimsical explanation of parallel computing.

Additional publications

Adaptive Sampling, Learning, Statistics, Probability

In standard clinical trials, one may allocate equal numbers of patients to one of two treatments, and then after all of the outcomes have been observed make a statistical decision as to which is the better treatment. With an adaptive design, however, one may modify the allocation as the outcomes are being observed, so that more patients in the trial can be allocated to the better treatment. This can simultaneously allow one to achieve a desired statistical goal (such as determining which is the better treatment) while also achieving ethical or cost goals (such as reducing patient suffering during the trial, or reducing costs in an industrial setting).

Adaptive designs have many natural uses in resource allocation problems or choosing parameter settings for computer simulations, since the computer can both collect the information and run a program to decide what to do next. They can be viewed as a dynamic learning process, and there are numerous ties to optimization.

Here is a more extensive discussion of adaptive sampling and bandit problems.

Additional publications

Scientific Computing, Computational Science

Here the emphasis is on developing efficient algorithms and data structures that are motivated by needs in computational science, and on ways of putting large scientific codes together. Most of my work in this area started with a scientist coming to me requesting help on a research project. I've worked with social scientists, environmental engineers, space scientists, statisticians, etc. Some of the scientists are internationally famous, while others are graduate students just starting to explore a topic. I married one of these scientists - you never know where exciting research will lead.

For adaptive sampling I'm involved in both the science and the computing, while within the Center for Space Environment Modeling (CSEM) and Center For Radiative Shock Hydrodynamics (CRASH) I concentrate on the computer science aspects. CRASH is a Dept. of Energy Center of Excellence in Predictive Science, where we are focusing not just on predicting the results of an experiment, but also quantifying the uncertainty of our predictions, and why we think we know the uncertainty. ``Uncertainty quantification'' is a research area which is attracting a lot of attention.

Because several of these applications involve large problems, much of our work in this area involves parallel computing, including some of the largest supercomputers in the world, though some involves serial algorithms. Projects include

Additional publications in modeling aspects of space and Earth and more general scientific computing

Algorithms, Data Structures, Theoretical Computer Science

Most of the work here concentrates on trying to develop optimal algorithms, optimal at least as measured by O-notation. The majority of my work on algorithms has been in parallel computing, adaptive allocation, or graph theory, but there are several other topics that my students or collaborators and I have looked at. Some of the work we did involves

Some projects I'm currently involved with, or would like to start (in addition to the topics listed in parallel computing)

Additional publications in serial algorithms and data structures and other topics in theoretical computer science.

Discrete Mathematics: Graph Theory, Combinatorics, Coding

Many questions about resource allocation, fault tolerance, routing, scheduling, synchronization, etc., for distributed memory machines can be modeled via graphs, and one can exploit the graph structure to find solutions. We have particularly emphasized hypercube, mesh, and torus models, since many parallel computers are of this form. Example of this work are a rather comprehensive paper on the subcube fault tolerance of the hypercube and one on embedding graphs into hypercubes.

One set of graph theory questions Marilynn Livingston and I pursued concerns the determination of various properties of families such as k x n meshes, with k fixed. We have shown that values such as domination numbers, packing numbers, number of code words, number of tilings, etc., can be computationally determined in closed form, using dynamic programming and properties of finite state automata. This was a rather dramatic improvement upon earlier results. There are several interesting open questions and conjectures related to this.

The work with Julia Lipman mentioned above, concerning repeated synchronization of multiple agents, is also research in discrete mathematics.

Related publications

Operator Theory, Analysis

I don't work in this area any more. Most of the work in operator theory concerned Schur multiplication of matrices, where the Schur product is formed by multiplying matrices termwise (much as addition is performed). For infinite dimensional Hilbert spaces or lp spaces, this forms an interesting Banach algebra, which has a many ties to normal multiplication. Investigating the Schur product led to questions about the essential numerical range and majorization of diagonal entries. Slightly related results concerned determining the numerical range of finite dimensional weighted shift operators.

One paper analyzed conditionally convergent series and their rearrangements. There is a natural duality between sets of conditionally convergent series and sets of permutations which leave their sums unchanged, and this duality has some unusual properties.

Related publications


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