Year 4

1. Tackle a challenging application problem that uses all the tools of the MathBus. Here we give an example of the kind of problem that might be suitable. The electrical impedance tomography (EIT) problem may be stated as follows. An electrical field is induced on a domain; it is known that the electrical potential u obeys the law . Here, c is a scalar field that denotes conductivity. However, neither c nor u is known a priori. Instead, what is known is a sequence of Neumann/Dirichlet boundary data specifications. The problem is to reconstruct c from this data. This problem, which has been the subject of a fair amount of recent interest (see the 7/94 issue of SIAM News and [98]), arises in medical imaging. The medical imaging application is to determine the internal structure of organs using electrical field measurements. EIT is a very inexpensive way to carry out medical imaging, but is currently far less precise than more expensive imaging technology such as CAT scans and MRI scans. The lack of precision is partly due to the need for better numerical algorithms.

   To design algorithms for the EIT problem, one must have finite element solvers, mesh generators that can specify and handle internal boundaries, and sophisticated optimization tools. Furthermore, to develop algorithms, one must be able to rewrite parts of the various packages and specifications ``on the fly'' as algorithmic ideas are refined. Our plan is to have by Year 4 an environment sufficiently powerful to make the design of these algorithms much easier than the current environments.

2. Generation of parallel code for sparse matrix applications. We will generate parallel implementations of direct methods for solving large, sparse systems of equations. In particular, we will produce code for multifrontal, supernodal sparse Cholesky factorization, as well as multifrontal sparse QR factorization, starting from the corresponding dense matrix programs. Combined with the iterative solvers produced by Year 2, this will give us a large arsenal of sparse parallel solvers. The Bernoulli project will be the consumer of this technology, and it will drive the development of the technology.