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DavidYANGGAO 姓 名: David YANG GAO
系 别: Science, Information Technology & Engineering
职 称: Alexander Rubinov Professor of Mathematics
办 公 室: T137
办公电话: +61 3 5327 9791
传 真: +61 3 5327 9077 
E-mail: d.gao@ballarat.edu.au, Personal Web

个人简介

Professor David Gao

Research Leader

Professor David Gao has an impressive reputation in the world of global optimization, applied math and mechanics. His name is writ large among some of the most famous researchers in his multi-disciplinary fields, after he solved a 50-year non-convex variational problem in nonlinear mechanics, conceiving the Gao principal.

This principle lays a foundation for his canonical duality theory, which can be used to model complex systems within a unified framework.

In his groundbreaking work, Professor Gao improved on mathematical and engineering theories relating to large deformed elastic beams, which play a crucial role in engineering structures.

However, despite this achievement, Professor Gao remains modest about his work.

“It is just a small step for understanding and solving challenging real-world problems and we hope that through the CRN, working with our collaborators like Professor Saeid Nahavandi and Professor Peter Hodgson at Deakin,  Professor Kate Smith-Miles at Monash, we can have many more.”

In his research with the CRN, Professor Gao, who is heading up a project on regional science and technological innovation, would like to boost the University of Ballarat’s profile to attract more funding from ARC, National ICT Australia Ltd (NICTA), Australians Information and Communications Technology Research Centre of Excellence.

“We [the University of Ballarat] should be in a position where we are able to apply for serious funding from NICTA and the Australian Research Council, those major funding bodies,” he said.

Professor Gao sees the CRN as part of a global trend towards multi-displinary research, after he saw a similar model, the Multi-Disciplinary Research Initiative (MURI), set up in the United States.

讲授课程

Teaching Areas

Applied mathematics

Engineering mechanics

Optimization and control

Numerical analysis and simulation

Computational methods

Nonsmooth and nonconvex analysis and mechanics

Modeling

Simulation

Analysis of complex systems

Global optimization and canonical duality theory

Advanced primal-dual algorithms

Numerical methods 

Teaching Schedule 

Math 2224: Multivariable Calculus (93957 & 93965) 

Students’ Comments, Presentations, and Poem (see below)

Teaching record for ALL of my 2000-level undergraduate courses 

(The history of the common-time finals at Virginia Tech)

 2008 Fall Math 2214 (total 15 multiple-choice problems)

 

Department: average (mean): .10.82 

My session (42 students): 11.83

2008 Spring Math 2214 (total 14 multiple-choice problems)

 

Department: average: .9.79 

My session (44 students): 10.84

 2007 Fall, Math 2224 (total 15 multiple-choice problems)

 

Department: average: .6.83 (total 711 examinees)  

My session 93754 (25 students): 7.08

My session 93756 (32 students): 7.16

 2006 Spring, Math 2214 (total 12 multiple-choice problems)

 

Department: average: .7.5  

My session (35 students): 7.78   (Median: 8.0)

2005 Spring, Math 2224 (total 15 multiple-choice problems)

Department: MEAN NO. to the RIGHT: 8.45 (Median: 8.0)

My session (25 students): MEAN NO. to the RIGHT: 9.32   (Median: 9.0)

2004, Fall, Math 2214 (total 12 multiple-choice problems)

·Department: MEAN NO. RIGHT: 4.39 (Median: 4.0)

·My session (40 students): MEAN NO. RIGHT: 5.26   (Median: 5.0)

2003, Fall, Math 2224 (total 14 multiple-choice problems)

 

·Department: MEAN NO. RIGHT: 7.51

·My 2:30 session: MEAN NO. RIGHT: 8.96

My 4:00 session: MEAN NO. RIGHT: 8.52

2002, Spring,  Math 2214 (total 12 multiple-choice problems)

 

·Department: MEAN NO. RIGHT: 7.08 (MEDIAN: 7.0) 

My 11:00 session: MEAN NO. RIGHT: 8.28  (MEDIAN: 9.0) Total 64 students 

My 12:30 session: MEAN NO. RIGHT: 7.85  (MEDIAN: 8.0)

2001, Fall, Math 2214 (total 12 multiple-choice problems)

 

·Department: MEAN NO. RIGHT: 8.35 (MEDIAN: 9.0) 

My session: MEAN NO. RIGHT: 9.68  (MEDIAN: 10.0)

2001, Spring, Math 2214 (total 11 multiple-choice problems)

 

·Department: MEAN NO. RIGHT: 6.53 (MEDIAN: 6.0) 

My session: MEAN NO. RIGHT: 6.67 (MEDIAN: 7.0) (33 students)

2000, Fall, Math 2214 (total 12 multiple-choice problems)

 

·Department: MEAN NO. RIGHT: 9.84 (MEDIAN: 10.0)

·My session: MEAN NO. RIGHT: 10.50 (MEDIAN: 11.5)

1999, Fall, Math 2214 (total 13 multiple-choice problems)

 

·Department’s average: 9.67

·My 2:00pm session: 10.5

·My 3:30pm session: 10.1

1999, Spring, Math 2214 (total 14 multiple-choice problems)

 

·Department’s average: 7.5

My session: 7.6

 

1998, Fall, Math 2214 (total 14 multiple-choice problems)

 

·Department’s average: 7.19

·My big session (51 students): 8.00 

 

1998, Spring, Math 2224 (Multi-variable Calculus)

 

·Department’s average: 9.29

·My session (41 Students): 9.85 

 

1996, Fall, Math 2214 (total 15 multiple-choice problems)

 

·Department’s average: 7.3

My session: 8.39

 

Poem for Operational Methods (MATH 4564):

 

By Sarah Huffer (Spring, 2005)

Operational methods, ah what a class…

I sit and I listen, but alas,

I learn about even and odd Fourier Series

And separation of variables causing many queries

As Lao Tzu once said: “Trying to understand is like straining through muddy water. Be still and allow the mud to settle.”

Did Lao Tzu know what the heat conduction in a rod with non-homogenous boundary conditions was?  That takes a long time to resolve!

This class, however, goes beyond heat conduction

In fact, this class has many different functions

We learn about what is beautiful and what happens when you eat the Fruit of Knowledge

What we discover in Op. Methods is not a chore but a privilege

With all the time that we spend studying, I conclude

With Lao Tzu words again, “To be worn out is to be renewed." 

Some of Students’ comments: 

  "I would just like to say how much I enjoyed this class. At the beginning of the semester, Professor Gao said he wanted us

to enjoy math and I laughed... but by the end I truly did enjoy it. This is the first time I've had fun in Math". (Math 2224, Spring, 2005) 

 "Great connections of abstracts of mathematics with nature, you make math seem beautiful and fun. It is everywhere! Thank you." (Math 2224, Spring 2005) 

"Really appreciate the visual approach to teaching relevant concepts and materials. Respect his expectations and concern for students". (Math 2224, Spring 2005) 

 "Professor Gao, Thank you for a fantastic class -- I have a new appreciation for math. Your stories have been great in relating complex subject matter to real life situations." (Math 2224, Spring 2005) 

``Stop smiling so much. How can you be so damn happy about math? In the 3 years I've been here, you are the best Math teacher I've had.'' (Math 2214, fall 1993)

研究领域

Research Center:

Center for Numerical Simulation & Modeling

RESEARCH AREAS

(Some of links are closed due to the space limitation)

Dr. Gao studies  modeling, methods, and theories of duality, triality,  as well as  some of closely  related concepts (such as complementarity,  polarity, symmetry and symmetry breaking, etc.) in science, engineering, and computation, with applications to general complex systems, including nonconvex/nonsmooth/discrete and nonconservative problems in database analysis, decision science, nonlinear analysis, finite deformation field theory, engineering  mechanics, global optimization and control, differential equations and geometry,  network flows and communications, energy systems, social systems,  and to large-scale and multi-scale scientific computations. His work on duality theory in convex systems emphasizes how it relates to a unified framework in natural phenomena with symmetry; while the work on triality in non-convex systems aims to understand symmetry breaking, to reveal intrinsic duality, and to discover general pattern of duality in complex systems.  His multi-disciplinary research activities were supported by Divisions of Mathematical Science (DMS), Civil and Structural Engineering (CMS), Operations Research & Production Systems (DMII), and Computer & Information Science & Engineering (CCF) at National Science Foundation. Currently he has two active NSF grants with total $210,000 for 2005-2009.  Some new grants have been approved recently for 2009-2014.

Dr. Gao was trained in the fields of Engineering Science and Applied Mathematics. His research interests range over the following areas:

Theoretic and Applied Mechanics

Nonsmooth/Nonconvex Mechanics

Finite Deformation Modeling and Theory  

Elasto-plastic Structural Limit Analysis

Contact Mechanics

Structural Optimal Design

Phase transitions in solids

Optimization and Control

Global Optimization 

Fractional Programming, mixed integer programming, semi-infinite programming

Canonical duality theory in global optimization

Optimal design and control of complex systems

Dual feedback control of distributed parametrical systems

System Engineering and Management Science

Modeling, analysis, and simulations of complex systems, including man-made (social) systems.

Decision making and management optimization

Duality in optimal energy management

Understand and control chaotic systems

Canonical System theory

Optimization in database analysis, networking and communication theory (including sensor localization, machine learning, max cut, Euclidean distance geometry, etc)

Applied  Mathematics

Mathematical modeling of complex systems

Applied Nonconvex and nonsmooth Analysis, nonlinear PDEs

Variational Methods,  Variational Inequality, and Complementarity Problems

Nonconservative Hamilton Systems and Chaotic Dynamics

Information theory and network communication

Euclidean distance geometry

Scientific Computation

Primal-Dual Algorithms for solving nonconvex/nonsmooth problems

Canonical dual finite element methods

Large Scale Scientific Computations and Simulations

Duality, Complementarity and Triality in  Sciences  and Philosophy

Postdoctoral Research Fellows

Start Date: flexible

Description: Post-doctoral positions are available for a multidisciplinary research project (five-years) on Canonical duality theory with applications in global optimization and decision science. Candidates should have a Ph.D. in computational mathematics or related fields (operations research, applied math, computational mechanics, computer science, etc).

研究成果

Main Research Contributions

1.    Mathematical Modeling

·         Beam Models:

1)      elastic model (with D. Russell, click here for details),

2)      elasto-plastic model,

3)      large deformed beam models (Gao, 1996)

wtt  + k wxxxx + (ν p – a wx) wxx  = f(x,t)  

This equation has at most three solutions w(x,t) at each given (x, t), represents the two possible buckling states (upper and down positions), and one unbuckled (unstable) position. These three solutions could lead to chaos in dynamical vibration for certain given data k, a, ν, the axial load p and the distributed load f(x,t). This equation is also equivalent to one-dimensional Landau-Ginzburg equation in phase transitions. Applications of this beam model in literature, please check  here (see also here)

4)       general large deformation  elastic beam models (Gao, 2000)

 

·         Multi-scale model of phase transitions in solids

·         Eigenvalue problem on optimal surface (with S.T. Yau)

·         Optimal shape design of engineering structures

·         Bi-Complementarity model and framework in general geometrical linear systems.

2.    Dynamical Systems

·         New Phenomena discovered in Chaotic Systems

a. Meta-chaos, b.  tri-chaos, and c. reason for chaos (3-D vision of chaos)

·         Understand and control chaos

·         Duality, triality, and polarity in general nonlinear dynamics

·         Canonical dual feedback control against chaos

3.    Canonical Duality Theory

This theory is composed mainly of

1)      The canonical dual transformation (nonlinear transformation), which can be used for (correctly) modeling complex systems and to formulate canonical dual problems (with zero duality gap). 

2)      The complementary-dual principle, which leads to a unified analytical solution to general problems. For detailed discussion on this principle in finite elasticity, please check here.

3)      The triality theory, which revels an intrinsic duality pattern in complex systems, and can be used to identify both global and local extremality conditions in nonconvex variational problems and global optimization, and to predict and against chaos in nonconvex dynamical systems.

This theory is based on the original joined work with Gil Strang at MIT in 1989. It is now understood that the popular Semi-Definite Programming (SDP) method is actually a special application of this joined work. The canonical duality theory has been successfully applied to the following fields.

a.     Mathematical Analysis and Continuum Mechanics

A unified analytic solution to a class of nonconvex/boundary value problems, including

·         Large deformed elasto-plastic mechanics

·         General nonconvex/nonsmooth variational/boundary value problems (3-D)

·         Nonlinear ODEs (including Einstein’s special relativity theory)

·         Phase transitions in solids with Ray Ogden

·         Pure azimuthal shear problem with Ray Ogden

b.    Nonlinear Algebraic Systems

·         How to find all eigenvalues of a symmetric matrix 

·         Analytical solution to m-quadratic equations in n-dimensional space

·         Analytic solution to third-order nonlinear algebraic equations in n-D

c.      Global Optimization

A unified analytic solution form to a class of NP-hard problems, including

·         quadratic minimization with box/integer constraints

·         polynomial minimization 

·         0-1 programming (with S-C Fang et al)

·         fractional programming (with S.-C. Fang et al)

·         mixed integer programming (with Hanif Sherali and N. Ruan),

·         semi-infinite programming problems

·         sensor network localization

·         general nonconvex minimization problems (with H. Sherali, N. Ruan)

4.      Numerical Methods and Algorithms

1.      Pan-penalty mixed finite element method and re-scaling algorithm

2.      Canonical dual finite element method and algorithms

3.      Primal-dual algorithms for nonconvex/nonsmooth optimization problems

4.      Multi-scale and large-scale computation and algorithms.

 

 Recent Results and publications:

Advances in canonical duality theory with  applications to  global optimization, invited lecture presented at Foundations of Computer-Aided Process Operations (FOCAPO) 2008, June 29-July 2, 2009, Cambridge, MA     This paper presents a brief review and  recent developments of this theory with applications to some well-know problems, including polynomial minimization,  mixed integer and fractional  programming,  nonconvex minimization with nonconvex quadratic constraints, etc. Results shown that under certain conditions, these difficult problems can be solved by deterministic methods within polynomial times, and   NP-hard discrete optimization problems can be transformed to certain minimal stationary problems in continuous space. Concluding remarks and  open problems are presented in the end.

Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation, with Ray Ogden

This paper presents a complete set of analytical solutions to  general variational/boundary value problems with either mixed or Dirichlet boundary conditions. It shows that the global minimizer may not be a smooth function and can’t be obtained by traditional methods. Criteria for the existence, uniqueness, smoothness and multiplicity of solutions are presented and discussed. The iterative finite-difference method (FDM) is used to illustrate the difficulty of capturing non-smooth solutions with traditional FDMs.

 

Canonical dual least squares method for solving general nonlinear systems of quadratic equationsThis paper presents a canonical dual approach for solving general nonlinear algebraic systems. By using least square method, the nonlinear system of  m-quadratic equations in n-dimensional space is first formulated as a nonconvex optimization problem. We then proved that, by the canonical duality theory developed by the second author, this nonconvex problem is equivalent to a concave maximization  problem in Rm, which can be solved easily by well-developed convex optimization techniques. Both existence and uniqueness of global optimal solutions are discussed, and several illustrative examples are presented.

 

 

 Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem, with Ray Ogden,

This paper solved nonlinear variational/boundary value problems in nonlinear elasticity with both convex and nonconvex strain energy densities. The results for the non-convex strain energy function provide an illustration of a situation in which smooth analytic solutions of a nonlinear boundary-value problem are not global minimizers of the energy in the variational statement of the problem. Both the global minimizer and the local extrema are identified and the results are illustrated for particular values of the material parameters.

 

CANONICAL DUAL APPROACH TO SOLVING 0-1 QUADRATIC PROGRAMMING PROBLEMS This paper presents an application of the canonical duality theory for 0-1 programming problems. It shows that by the canonical dual transformation, discrete integer minimization problems can be converted into canonical dual problems in continuous space, which can be solved easily under certain conditions. Both global and local minimizers can be identified by Triality theory. Multi-integer programming problems can be solved too, results will be posed soon.

 

Multi-scale modelling and canonical dual finite element  method in phase transitions of solids Int. J. Solids and Structure. This paper presents a multi-scale model in phase transitions of solid materials with both macro and micro effects. This model is governed by a semi-linear nonconvex partial differential equation which can be converted into a coupled quadratic mixed variational problem by the canonical dual transformation method. The extremality conditions of this variational problem are controlled by a triality theory, which reveals the multi-scale effects in phase transitions. Therefore, a potentially useful canonical dual finite element method is proposed for the first time to solve the nonconvex variational problems in multi-scale phase transitions of solids

 

Solutions and Optimality Criteria  to Box Constrained Nonconvex Minimization ProblemsJ. Industrial and Management Optimization, 3(2), 293-304, 2007. This paper solved a class of box constrained nonconvex minimization problems, including quadratic minimization, integer programming, and Boolean least squares problems. This paper shows that some “NP-hard problems” can be solved by polynomial algorithms.

Complete solutions and extremality criteria to polynomial optimization problems, J. Global Optimization,  35 : 131-143, 2006

Canonical duality theory and solutions to constrained nonconvex quadratic programmingJ. Global Optimization. This paper presents a set of complete solutions to quadratic minimization over a sphere, as well as a canonical dual form for quadratic minimization with linear inequality constraints.

Perfect duality theory and complete solutions to a class of global optimization problemsOptimization. This paper presents a potentially useful methodology for solving a class of nonconvex variational/optimization problems. A set of complete solutions is obtained for Landau-Ginzburg equation, nonlinear Schrödinger, equation and Cahn-Hilliard theory in finite dimensional space.

Sufficient conditions and perfect duality in nonconvexs minimization with inequalityJ. Ind. Management Optimization. This paper solved quadratic minimization problem with a quadratic inequality constraint.

Complete solutions to a class of polynomial minimization.  J. Global Optimization

Analytic solutions and triality theory for nonconvex and nonsmooth variational problems with applicationsNonlinear Analysis.  This paper present a complete set of solutions to a class of nonconvex/nonsmooth variational/boundary value problems. The global minimizer and local extrema can be identified by the triality theory.

General analytic solutions and complementary variational principles for large deformation nonsmooth mechanicsMeccanica. This paper solved an open problem in finite deformation theory and proposed a pure complementary energy principle (click here for details) which can be used to solve a large class of nonconvex variational/boundary value problems.

 Pure complementary energy principle and triality theory in finite elasticity. Mechanics Research Communication.

Nonconvex Semi-Linear Problems and Canonical Duality Solutions, Advances in Mechanics and Mathematics, Vol II, 2003, Springer, 261-311. 

Editorial:

  1. Co-Editor-in-Chief for two book series:

o    Springer Book Series: 

  Advances in  Mechanics and Mathematics (AMMA)

  1. Associate Editor for the following journals

Organizations and Activity Research Groups:

Some movies about finite element simulations for dynamical post-buckling analysis of a rubber diaphragm created by my Ph.D. student Axinte Ionita. This is a 3-D nonconvex, nonsmooth, nonconservative dynamical system subjected to follower force. 

Publications

 Complete publications list

Publications listed by research areas

Encyclopedia Articles

  1. Gao, David Y., Duality-Mathematics,   Wiley Encyclopedia of Electronical and Eletronical Engineering 6, 1999, 68-77 ps file
  2. Gao, David Y., Mono-duality in Convex Systems, ENCYCLOPEDIA OF  OPTIMIZATION, Kluwer Academic Publishers
  3. Gao, David Y., Bi-duality in convex dynamically systems and D.C. programming, ENCYCLOPEDIA OF  OPTIMIZATION, Kluwer Academic Publishers
  4. Gao, David Y., Triality in Global Optimization,  ENCYCLOPEDIA OF  OPTIMIZATION, Kluwer Academic Publishers

Scientific philosophy

  1. Gao, David Y.,  Dao of the complementarity-duality: I. Complementarity and dual principles in natural sciences,   in Selected Philosophical Papers of Tsinghua's Ph.D. Thesis, Tsinghua Univ. Press, 1996
  2. Gao, D.Y.,  Dao of the complementarity-duality: II.  Complementarity and dual principles in general systems.  Excellent paper award in The First National Congress on Natural PhilosophyAnhui, July 1986.   J. of Hefei University of Technology (Sociology Ed.), 2 (1986).
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