Past Event: Oden Institute Seminar

Canonical Duality-Triality: Unified Understanding on Bifurcation, Chaos and NP-Hard Problems in Complex Systems & Computational Science. Part I: Complementary-dual extremum principle and Analytic Solutions to Nonlinear Elasticity and Nonconvex Variational Problems.

David Gao, Professor, Australian National University

Abstract

Duality is a beautiful, fundamental, and inspiring concept that underlies all natural phenomena. According to Sir M.F. Atiyah, duality in mathematics is not a theorem, but a “principle”. Therefore, duality gap is not allowed. For convex systems, mathematical theory of duality has been well developed with extensive applications in engineering and sciences. However, in nonconvex systems, the classical Lagrange multiplier method and Fenchel-Moreau-Rockafellar dualities have been misused for solving nonconvex/ discrete problems, and produced different duality gaps. It turns out that many nonconvex variational/optimization problems are considered to be NP-hard. The canonical duality-triality [1] is a powerful methodological theory, which can be used not only to model complicated phenomena with in a unified framework, but also for solving a wide class of nonconvex/nonsmooth/discrete problems inmultidisciplinary fields. The associated triality theory reveals an interesting multi-scale duality pattern in complex systems, which can be used to identify both global and local extrema and to design efficient algorithms for solving challenging problems. This lecture will provide an introduction to the canonical duality-triality theory with a purpose toward a unified understanding the fundamental difficulties in large deformation mechanics, bifurcation/chaos and PDEs in nonlinear science, and NP-hard problems in global optimization and computer science. This talk will bring some fundamental new insights into modern mechanics, complex systems, and computational science Lecture I: Analytical Solutions to Certain Nonlinear PDEs in Nonconvex Analysis and Mechanics Beginning with some interesting phase transitions in our daily life, the speaker will explain why the symmetry breaking leads to phenomena and what is the fundamental reason that leads to difficulties, such as bifurcation, big-band theory, chaos in nonlinear dynamics, NP-hard problems in computer science, and the paradox of Buridan’s donkey in decision making. Based on oriental philosophy and some fundamental principles in systems theory, he will show how nonlinear phenomena can be modelled as a unified nonconvex variational problem and the canonical duality theory can be naturally, precisely developed. Using a simple nonconvex variational problem of Ericksen’s bar in phase transitions, the speaker will demonstrate that a class of nonlinear partial differential equations can be transformed to certain algebraic (tensor) equations and can be solved completely to obtain all possible solutions [2]. A movie will show that both global and local minimal solutions are usually nonsmooth and can’t be captured by any Newton-type numerical approaches, which is one of reasons for NP-hardness. Some open problems proposed by Sir J. Ball will be address and the speaker will show that for St. Venant_Kirchhoff material in finite elasticity, the nonlinear PDE could have at most 27 solutions at each coordinate, but only one global minimizer [5], BIO: Professor David Y. Gao received his Ph.D. in Engineering Mechanics and Applied Mathfrom Tsinghua University. Since then, he has held research and teaching positions in different institutes including MIT, Yale, Harvard, and Virginia Tech. He moved to Australia in 2010 for his current position as the Alexander Rubinov Chair Professor in School of Science and Engineering at the Federation University. He is also a research professor in the Research School of Engineering at Australian National University. He is the author of 14monograph/handbook/special volumes and about 180 research papers on applied math, theoretical and computational mechanics, global optimization and operations research, etc. He is an editor and associate editor for 2 book series & about 8 international journals. Currently, he serves as the Secretary-General and Vice President of the International Society of Global Optimization. Detailed information can be found at his web pages: https://researchers.anu.edu.au/researchers/gao,http://sitevm1.ballarat.edu.au/dgao/