277 research outputs found

    Homotopies for solving polynomial systems within a bounded domain

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    AbstractThe problem considered in this paper is the computation of all solutions of a given polynomial system in a bounded domain. Proving Rouche's theorem by homotopy continuation concepts yields a new class of homotopy methods, the so-called regional homotopy methods. These methods rely on isolating a part of the system to be solved, which dominates the rest of the system on the border of the domain. As the dominant part has a sparser structure, it is easier to solve. It will be used as start system in the regional homotopy. The paper further describes practical homotopy construction methods by presenting estimators to obtain bounds for polynomials over a bounded domain. Applications illustrate the usefulness of the approach

    Computing Dynamic Output Feedback Laws

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    The pole placement problem asks to find laws to feed the output of a plant governed by a linear system of differential equations back to the input of the plant so that the resulting closed-loop system has a desired set of eigenvalues. Converting this problem into a question of enumerative geometry, efficient numerical homotopy algorithms to solve this problem for general Multi-Input-Multi-Output (MIMO) systems have been proposed recently. While dynamic feedback laws offer a wider range of use, the realization of the output of the numerical homotopies as a machine to control the plant in the time domain has not been addressed before. In this paper we present symbolic-numeric algorithms to turn the solution to the question of enumerative geometry into a useful control feedback machine. We report on numerical experiments with our publicly available software and illustrate its application on various control problems from the literature.Comment: 20 pages, 3 figures; the software described in this paper is publicly available via http://www.math.uic.edu/~jan/download.htm

    A Collection of Challenging Optimization Problems in Science, Engineering and Economics

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    Function optimization and finding simultaneous solutions of a system of nonlinear equations (SNE) are two closely related and important optimization problems. However, unlike in the case of function optimization in which one is required to find the global minimum and sometimes local minima, a database of challenging SNEs where one is required to find stationary points (extrama and saddle points) is not readily available. In this article, we initiate building such a database of important SNE (which also includes related function optimization problems), arising from Science, Engineering and Economics. After providing a short review of the most commonly used mathematical and computational approaches to find solutions of such systems, we provide a preliminary list of challenging problems by writing the Mathematical formulation down, briefly explaning the origin and importance of the problem and giving a short account on the currently known results, for each of the problems. We anticipate that this database will not only help benchmarking novel numerical methods for solving SNEs and function optimization problems but also will help advancing the corresponding research areas.Comment: Accepted as an invited contribution to the special session on Evolutionary Computation for Nonlinear Equation Systems at the 2015 IEEE Congress on Evolutionary Computation (at Sendai International Center, Sendai, Japan, from 25th to 28th May, 2015.

    Probability-one Homotopies in Computational Science

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    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 suc 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

    Homotopy Based Reconstruction from Acoustic Images

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    Gale duality, decoupling, parameter homotopies, and monodromy

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    2014 Spring.Numerical Algebraic Geometry (NAG) has recently seen significantly increased application among scientists and mathematicians as a tool that can be used to solve nonlinear systems of equations, particularly polynomial systems. With the many recent advances in the field, we can now routinely solve problems that could not have been solved even 10 years ago. We will give an introduction and overview of numerical algebraic geometry and homotopy continuation methods; discuss heuristics for preconditioning fewnomial systems, as well as provide a hybrid symbolic-numerical algorithm for computing the solutions of these types of polynomials and associated software called galeDuality; describe a software module of bertini named paramotopy that is scientific software specifically designed for large-scale parameter homotopy runs; give two examples that are parametric polynomial systems on which the aforementioned software is used; and finally describe two novel algorithms, decoupling and a heuristic that makes use of monodromy
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