Mathematical Sciences: Variational Methods and Applications

[501148]这个项目将涉及经典场论、量子化学和旋转自引力流体理论中的一些问题的研究。这些问题中的每一个都涉及到非标准的变分原理，它的目的是发展必要的变分分析，在某些情况下，还会研究涉及模拟解决方案的数值问题。这些问题通常是非凸的，因此需要专门研究这些原理的某些对偶理论和对偶问题。在本项目下，研究者计划开发描述各种经典连续介质理论中的场的新方法。这将回答流体力学，电磁场理论和材料科学的一些领域的一些基本问题。研究了量子化学的基本模型Hartree-Fock模型理论，以及一些行星和恒星的流体动力学模型。对于这些问题，主要的兴趣是开发好的数值算法来模拟解。所提出的算法基于控制理论思想，似乎比目前用于模拟这些问题的方法更稳定，更有效。

9501148 Auchmuty This project will involve the investigation of a number of problems in classical field theories, quantum chemistry, and the theory of rotating self- gravitating fluids. Each of these problems involves non-standard variational principles and it is intended to develop the requisite variational analysis and in some cases also look at numerical problems involved in the simulation of solutions. These problems often are non-convex and special efforts will be devoted to studying certain duality theories and dual problems for these principles. Under this project, the investigator plans to develop new methods for describing fields in various classical continuum theories. This will answer some foundational questions in fluid mechanics, electromagnetic field theory and some areas of materials science. It is also planned to investigate the theory of Hartree-Fock models which are basic models in quantum chemistry and also some hydrodynamical models of planets and stars. For these problems, the major interest is in developing good numerical algorithms for simulating solutions. The proposed algorithms are based on control theoretic ideas and appear to be both more stable and much more efficient than the methods currently being used for simulation of these problems.