Dr. Zhou, the principal investigator proposes to develop theory and methods which can be implemented by numerical algorithms for efficiently and stablely solving multiple saddle point problems. The PI proposes to (1) establish local characterizations of Unconstrained or Constrained saddle points in Hilbert, Banach spaces for various multiple solution problems; (2) develop numerical algorithms for finding multiple saddle points; to carry out numerical tests on various model problems and make codes available to other researchers; (3) do convergence and error analysis of the algorithms; develop techniques to enhance stability, efficiency and convergence rate of the algorithms; (4) develop tools that can be numerically carried out for investigating instability and maneuverability of multiple saddle points; (5) study important multiple saddle point problems in applications, such as, an eigen-pair problem in nonlinear optics and the elliptic sine-Gordon equation in condensed matter physics for multiple solutions and their instabilities and maneuverabilities.
Stability is one of the main concerns for system design and control. Conventional theory and numerical methods are designed to find stable solutions. However, in many applications, PERFORMANCE or MANEUVERABILITY is more desirable, especially in system design and control of EMERGENCY machineries for MISSION CRITICAL SITUATIONS. On the other hand, many problems in applications possess multiple (unstable) solutions with different performance, maneuverabilities (an ability to shift from one solution to others) and instabilities. An unstable solution is a transitional solution which exists only in a certain time period. An unstable solution may have much higher performance and maneuverability than others. Finding multiple solutions enables one to select the best solution. However, due to unstable nature, those solutions are very elusive to numerical computation. There is virtually no theory in the literature to devise an efficient and stable numerical algorithm for this purpose until recently (2001) when the PI and his student developed a New Method. With which a numerical algorithm is devised for finding multiple unstable solutions and has been tested on many multiple solution problems. Preliminary results are quite promising and show great potential. The projects described in this proposal are further developments of the preliminary results in several important directions. The outcome of the projects will greatly enhance people's understanding on unstable solutions, provide numerical algorithms for finding multiple solutions and promote more applications.
