A new augmented singular transform and partial newton-correction method for finding more solutions


主讲人:Zhou Jianxin,美国德克萨斯州A&M大学数学系教授




主讲人先容:Biographical Sketches---Dr. Jianxin Zhou(周建新) (Email:jzhou@math.tamu.edu,  Webpage:www.math.tamu.edu/~jzhou) Education Background: Pennsylvania State  University, University Park, PA, USA, Optimal Control, Postdoc, 1987.  Pennsylvania State University, University Park, PA, USA, Mathematics, Ph. D.,  1986. Shanghai University of Science & Tech., Shanghai, China, Applied  Mathematics, M.S., 1982. Shanghai University of Science & Tech., Shanghai,  China, Computational Mathematics, 1977. Professional Working Experience.  6/2015-5/2016: Oversea Distinguished Professor, Hunan Normal University, China  5/2004-6/2004: Feng Kang Professor, Shanghai University, Shanghai, China  1999-present: Full Professor, Texas A&M University, College Station, TX, USA  1993-1999: Associate Professor, Texas A&M University, College Station, TX.  USA 1987-1993: Assistant Professor, Texas A&M University, College Station,  TX. USA Research Interests: Applied Analysis and Scientific Computation in  differential multiple solution problems, multi-level optimization, control and  games theory. Research Grants: supported by NSF and other funding agencies.  Publications (published more than 80 papers and 5 advanced books) Most  significant contribution to mathematics: Pioneer work on developing  computational theory and methods for finding multiple (unstable) solutions in  various differential systems. His results have been used in books, in research  and graduate education by other researchers around the world and have  established him as a respected international leader in this research area. Five  Most Closely Related Publications: 1. A minimax method for finding multiple  critical points and its applications to semilinear PDE, SIAM J. Sci. Comp.,  23(2001) 840-865. (with Y. Li) 2. A local min-orthogonal method for finding  multiple saddle points, JMAA, 291(2004) 66-81.

内容先容:In this talk, in order to find more solutions to a nonvariational quasilinear  PDE, a new augmented singular transform (AST) is developed to form a barrier  surrounding previously found solutions so that an algorithm search from outside  cannot pass the barrier and penetrate into the inside to reach a previously  found solution. Thus a solution found by the algorithm must be new. Mathematical  justifications of AST are established. A partial Newton-correction method is  designed accordingly to solve the augmented problem and to satisfy a constraint  in AST. The new method is applied to numerically investigate bifurcation,  symmetry-breaking phenomena to a nonvariational quasilinear elliptic equation  through finding multiple solutions.

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