Fast Multiresolution Methods and Nonlinear Approximations for Multidimensional Problems

本项目旨在发展一类新的数值方法，其基础是有效的算子核的非线性逼近以及积分和微分方程解。该方法将用于求解三维Lippmann-Schwinger散射理论方程。我们将展示基于多分辨率结构和非线性近似的数值方法的灵活性和通用性。许多现代技术的发展要求对物理过程进行准确的模拟，从而导致计算困难的多维问题。例如，凝聚态物理和材料科学中出现了一些难题，这些难题对新材料的设计至关重要。研究人员开发的数学工具，以及它们最近在量子化学中的成功应用，指向了一类新的高维问题的数值算法。这些算法具有非常理想的特点：它们速度快，它们提供有保证的(用户可控制的)精度，并且它们足够灵活，可以应用于许多不同的问题领域。这个项目打算在计算数学和应用领域产生广泛的影响，并为解决一些目前无法解决的问题开辟一条道路。这项建议的一个组成部分是对研究生和博士后研究人员进行这些新开发的分析和数值技术的培训，并广泛传播结果。

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 ofintegral and differential equations. The approach will be used to solve thethree-dimensional Lippmann-Schwinger equation of scattering theory. We intend to demonstrate flexibility and generality of numerical methods based onmultiresolution structure and nonlinear approximations.The development of many modern technologies requires accurate simulation ofphysical processes leading to computationally difficult multidimensionalproblems. 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 forproblems 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 currentlyinaccessible. An integral part of this proposal is training of graduate students and postdoctoral researchers in these newly developed analytic and numericaltechniques, and a broad dissemination of the results.