NSF Award
1115384
The PI proposes to develop Computational Theory and Methods for Solving Multiple Solution Problems representing four types of natures: (1) a general problem, where a usual continuation transform used in a Newton continuation method fails to do so, a singular transform is proposed to solve an augmented problem for a New solution; (2) nonlinear eigen problems (NEP) in the nonlinear Schrodinger equation, a basic model in physics, for both focusing and defocusing cases, an implicit variational method is developed to solve NEP on its energy profile. After discovering their mathematical structure, new local maxmin methods are designed to solve the defocusing cases; (3) singularly perturbed problems from math biology and other reaction-diffusion systems for both focusing/defocusing cases, an adaptive local refinement method is developed to numerically solve the problem. (4) Steklov nonlinear boundary-value problems in corrosion engineering and scattering applications, a boundary integral equation approach is used to develop a local minmax/maxmin-boundary element method for finding multiple solutions. All the proposed problems have strong and wide application background but are not yet solvable in the literature. New methods will be developed by exploring the mathematical structure of the problems, deriving solution characterizations, implementation tests, convergence and instability analysis. The PI has conducted investigations on the projects. Preliminary analysis/numerical results are very promising. Due to its unprecedented and complex nature, the research in this proposal has to be creative and original; multi-disciplinary knowledge and collaboration on advanced nonlinear analysis, PDE, multi-level optimization, numerical algorithm design/implementation/analysis, are required.<br/><br/>Multiple (unstable) solutions to many systems have been observed and mathematically proved to have a variety of configurations, instability/maneuverability, but are not applicable with conventional technology. So traditional analysis/computation focus on stable solutions. Scientists are now able to induce and control unstable solutions with NEW advanced (synchrotron, laser, etc.) technologies and search for new applications for higher performance index, in particular, for MISSION CRITICAL SITUATIONS. So far people's knowledge on such solutions is still very limited and unstable solutions are too elusive to traditional numerical methods. The PI proposes to develop efficient/reliable numerical methods to solve such problems and establish their related math justification. The outcome of this proposal will (a) provide efficient/reliable numerical algorithms for people to use and promote new application; (b) lay a solid math foundation for solving such problems; (c) significantly enhance people?s knowledge on the nature and properties of such problems and can be used in computational math education due to their general setting. (d) The proposed projects provide an excellent opportunity for multi-disciplinary collaboration and to give Ph.D. students a balanced training on creative thinking, advanced analysis, numerical computation and problem solving. Three Ph.D. students are working the projects for their theses.
