Probability-one homotopies in computational science

Authors:
Layne T. Watson
Affiliations:
Departments of Computer Science and Mathematics, Virginia Polytechnic Institute & State University, 630 McBryde Hall, Blacksburg, VA
Venue:
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the 9th International Congress on computational and applied mathematics
Year:
2002

Citing 7
Cited 2

Numerical linear algebra aspects of globally convergent homotopy methods

SIAM Review
Modern homotopy methods in optimization

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Numerical continuation methods: an introduction

Numerical continuation methods: an introduction
Algorithm 652: HOMPACK: a suite of codes for globally convergent homotopy algorithms

ACM Transactions on Mathematical Software (TOMS)
Algorithm 777: HOMPACK90: a suite of Fortran 90 codes for globally convergent homotopy algorithms

ACM Transactions on Mathematical Software (TOMS)
Algorithm 801: POLSYS_PLP: a partitioned linear product homotopy code for solving polynomial systems of equations

ACM Transactions on Mathematical Software (TOMS)
Theory of Globally Convergent Probability-One Homotopies for Nonlinear Programming

SIAM Journal on Optimization

Probability-one homotopy maps for mixed complementarity problems

Computational Optimization and Applications
Solving the inverse problem of identifying an unknown source term in a parabolic equation

Computers & Mathematics with Applications

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Abstract

Probability-one homotopy algorithms are a class of methods for solving nonlinear systems of equations that, under mild assumptions, are globally convergent for a wide range of problems in science and engineering. Convergence theory, robust numerical algorithms, and production quality mathematical software exist for general nonlinear systems of equations, and special cases such as Brouwer fixed point problems, polynomial systems, and nonlinear constrained optimization. Using a sample of challenging scientific problems as motivation, some pertinent homotopy theory and algorithms are presented. The problems considered are analog circuit simulation (for nonlinear systems), reconfigurable space trusses (for polynomial systems), and fuel-optimal orbital rendezvous (for nonlinear constrained optimization). The mathematical software packages HOMPACK90 and POLSYS_PLP are also briefly described.