The Research Of Algorithms With Global Convergence For Solving Multiple Solutions Of Nonlinear PDEs And Their Applications

This paper is aimed to investigate the algorithms with global convergence for multiple solutions of the nonlinear partial differential equations and their applica-tions. The multiplicity and instability of the solutions to the nonlinear partial differ-ential equations bring many significant difficulties to the algorithm implementation and the relevant theoretical analysis. Especially, the studies on the algorithms with global convergence devoted directly to the nonlinear partial differential equations are at the primary stage. How to design a stable numerical algorithm to approximate the instable solutions, reduce the dependence on the initial values for the multiple solutions of the nonlinear partial differential equations in order to gain global con-vergence and make sure that the solution obtained each time is broadly new, etc. These are very important and challenging scientific problems.This paper consists of two parts. The first part will present the basic concept-s and ideas of the local Minimax method (LMM) based on the normalized search rules for a class of nonlinear partial differential equation with mountain pass vari-ational structure. It tries to answer the question could the Goldstein line search strategy in optimization theory be applied to find multiple solutions to the nonlinear partial differential equations in the infinite dimensional Hilbert space. Based on the relationship between the gradient of the energy functional J and the variation of a peak selection p(v), a normalized Goldstein search rule is proposed to overcome the shortcoming of the normalized Armijo search rule in which the minimum iteration step size has to be set artificially. It is noted that the local Lipschitz continuity of the local peak selection p(v) is a necessary condition to the analysis on feasibility for the original LMM algorithm. Borrowing the definition of the superlinearity for the peak selection p(v), which is introduced by X.D. Yao in [114], the feasibility of the LMM algorithm based on the normalized Goldstein search rule is proved under the regularity condition that the peak selection p(v) is just continuous. Obviously, it relaxes the assumption that p(v) has to be locally Lipschitz continuous for the o- riginal LMM algorithm. Furthermore, under the weak assumption above, the global convergence of both the normalized Armijo and Goldstein search rule is verified.The second part is related to the augmented partial Newton method (APNM), which is aimed to find new solutions. By using the information of the previously found solutions, a suitable augmented singular transform (AST) is constructed and the so-called APNM algorithm is implemented to solve the corresponding augment-ed singular equation. This approach restricts the iterations in a class of generalized Nehari manifold MG. It changes the structure of the singular lines of the classic Newton method and breaks the symmetric invariance. As a result, it is advanta-geous over other Newton-type algorithms. It is worthwhile to point out that the APNM does not care about whether the problem is variational or non-variational at all and also guarantees that the solutions obtained are broadly new. Actually its main ingredient is to construct an appropriate augmented singular transform. Based on our previous works [115], a new skillful augmented singular transform G is in-troduced. Although the new augmented singular transforms G seems to change slightly at the first glimpse in comparison with the augmented singular transforms G in [115], its mathematical structure changes greatly. In fact, the introduction of the new augmented singular transforms G relaxes the conditions for finding new so-lutions, which is quite easy to be checked. Moreover, the idea of the new augmented singular transforms G is suitable for inhomogeneous cases. Consequently it extend-s the application range of APNM. Further, mathematical justification of our new approach will be provided. Moreover, it will be applied to find multiple solutions to several classes of nonlinear partial differential equations including Henon equa-tion, Gross-Pitaevskii equation and one class of inhomogeneous nonlinear partial differential equation.