This project aims to develop a new class of numerical methods based on efficient nonlinear approximations of kernels of operators as well as the solutions of integral and differential equations. The approach will be used to solve the three-dimensional Lippmann-Schwinger equation of scattering theory. We intend to demonstrate flexibility and generality of numerical methods based on multiresolution structure and nonlinear approximations.
The development of many modern technologies requires accurate simulation of physical processes leading to computationally difficult multidimensional problems. For example, a number of difficult problems arise in condensed matter physics and materials science and are critical to the design of new materials. Mathematical tools developed by the investigators, and their recent, successful use in quantum chemistry, point to a new class of numerical algorithms for problems in high dimensions. These algorithms have very desirable features: they are fast, they provide guaranteed (user-controllable) precision, and they are flexible enough to be applicable in many different problem domains. This project intends to have broad impact in computational mathematics and applied fields, and to open a way for tackling a number of problems which are currently inaccessible. An integral part of this proposal is training of graduate students and postdoctoral researchers in these newly developed analytic and numerical techniques, and a broad dissemination of the results.
