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.
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.
Excited states with different configurations/instabilities exist as unstable local equilibria in many nonlinear differential systems. Nowadays with new (synchrotronic, laser, etc.) technologies, some of them can be induced, reached or controlled as to long-lived, stable for practical purposes, called metastable, and drew great interests from scientists to seek and study for new applications. So far people's knowledge on such unstable solutions is still very limited since traditional math analysis and numerical methods focus on finding a unique stable solution and classify those unstable solutions as ill-posed. Mathematically metastable states are among the first few unstable solutions, which are much more challenging for numerical computation than the stable ones. On the other hand, a metastable state may cause a system failure without knowing its metastability. Thus computational instability analysis of unstable solutions becomes important to avoid such a failure and also to make more excited states metastable in a system design/control. This project is to develop new computational theory and methods for solving various differential multiple solution problems. Due to its unprecedented nature, this research has to be creative and original; multi-disciplinary knowledge and collaboration are required. The research is carried out by the PI's research group and collaborations with Dr. Z.Wang of Utah State University and Dr. Z. Xie of China, involving 3 U.S. (1 female) and 2 China Ph.D. students. This collaboration program was initiated/supported by NSF/NSFC CMR Program in 2008 (Dr. Zhou was the PI) and is still active and efficient. We first develop math approaches to understand /classify their mathematical structures and instability behaviors of different types, then design numerical methods accordingly, do numerical implementation, testing, modification, convergence analysis. We have discovered two classes: M(focusing)-type vs. W(defocusing)-type., each one has wide applications in physics, chemistry and biology, etc. Those two subclasses are very different physically and mathematically as well. The following are the main outcomes from this project (totally 10 papers are written , 6 published, 1 accepted and 3 in review). For the M-type, (1) A global convergence result of the local minimax method (LMM) is finally established after 10 years of persistent study; (2) We study singularly perturbed PDE problems from math biology. A new approach combining the Morse index and bifurcation theory is developed to determine bifurcation and the number of positive solutions. An adaptive local refinement skill is proposed in LMM to numerically find the solutions; (3) To solve PDEs with a nonlinear boundary condition in corrosion/oxidation modeling, metal-insulator/metal-oxide semiconductor systems, we combine a minimax approach with a boundary integral-element method to derive a special expression so that all numerical computation and analysis can be carried out more efficiently only on the boundary. A corresponding LMM is developed. Multiple numerical solutions are successfully obtained; (4) Due to high contrast, non-Darcy/non-Newton flow has to be used to process the data. It leads to solve a new nonlinear threshold problem. By our multiple solution approach, we established a solution characterization and its solvability. A numerical algorithm is designed accordingly. The result is very successful. This is a joint work with Dr. Y. Efendiev and two postdocs; (5) A new implicit approach is developed to solve nonlinear Scrodinger eigensolution problems without using a Lagrange functional. The new approach and method enable people to not only numerically find eigensolutions but also establish some interesting properties, such as wave intensity preserving/control, bifurcation identification, ....; (6) Established instability index estimate of a min-orthogonal method for single PDEs and syetems; (7) For the W-type, there is no method in the literature for finding multiple solutions. We analyze symmetry invariant properties and developed an orthogonal subspace minimization method to find its multiple solutions with some mathematical justifications. Algorithm is implemented and numerical solutions are successfully obtained; We have obtained some break-through results capturing several features of the W-type for algorithm design and instability analysis, and laid a solid foundation for us to move on to the next phase of this long-term research program. Brief Outcome Summary: The results obtained in the NSF funded projects have successfully classified/identified multiple solution problems, which greatly enhanced people's knowledge about unstable solutions. We characterized the mathematical structure and instability behavior of the M-type saddles. Then numerical methods are designed accordingly and successfully implemented/tested on solving many M-type differential problems for multiple solutions. Numerical methods are mathematically justified with stability and convergence analysis. Numerical codes are developed and put on the PI's web-page. Many people have downloaded the codes and consulted the PI for their usages. Some of them have actually used the codes in their education and research projects. The projects provided excellent opportunities for multi-disciplinary collaboration and for giving Ph.D. students a balanced training on creative thinking, advanced analysis, numerical computation and problem solving. Three U.S. Ph.D. students (1 female) are trained in this program. One graduated in 2012 and two remain.