Finding points on real solution components and applications to differential polynomial systems

Wu, Wenyuan; Reid, Greg

doi:10.1145/2465506.2465954

Cited by 27 publications

(27 citation statements)

References 23 publications

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“…The reader is referred to [10,11,77,95] for the latest developments. Mature and robust software implementations for solving polynomial systems are, for example, Bertini [11,12], and PHCpack [86,87].…”

Section: General-purpose Methods For Finding Multiple Solutionsmentioning

confidence: 99%

A robust approach for finding all well-separated solutions of sparse systems of nonlinear equations

2016

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Tearing is a long-established decomposition technique, widely adapted across many engineering fields. It reduces the task of solving a large and sparse nonlinear system of equations to that of solving a sequence of low-dimensional ones. The most serious weakness of this approach is well-known: It may suffer from severe numerical instability. The present paper resolves this flaw for the first time. The new approach requires reasonable bound constraints on the variables. The worst-case time complexity of the algorithm is exponential in the size of the largest subproblem of the decomposed system. Although there is no theoretical guarantee that all solutions will be found in the general case, increasing the so-called sample size parameter of the method improves robustness. This is demonstrated on two particularly challenging problems. Our first example is the steady-state simulation a challenging distillation column, belonging to an infamous class of problems where tearing often fails due to numerical instability. This column has three solutions, one of which is missed using tearing, but even with problem-specific methods that are not based on tearing. The other example is the Stewart-Gough platform with 40 real solutions, an extensively studied benchmark in the field of numerical algebraic geometry. For both examples, all solutions are found with a fairly small amount of sampling.

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Section: General-purpose Methods For Finding Multiple Solutionsmentioning

confidence: 99%

A robust approach for finding all well-separated solutions of sparse systems of nonlinear equations

2016

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“…Such methods involving Gröbner bases include [10,24,31]. From the numerical algebraic geometry perspective, those in [32,2] are can be used and involve SVD decomposition and regeneration (respectively) to find critical points for the computation of witness sets. For computing the real critical points, one method is to compute all complex solutions and determine the real solutions among them.…”

Section: Introductionmentioning

confidence: 99%

Critical points via monodromy and local methods

Campo¹,

Rodriguez²

2017

Journal of Symbolic Computation

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Abstract. In many areas of applied mathematics and statistics, it is a fundamental problem to find the best representative of a model by optimizing an objective function. This can be done by determining critical points of the objective function restricted to the model.We compile ideas arising from numerical algebraic geometry to compute the critical points of an objective function. Our method consists of using numerical homotopy continuation and a monodromy action on the total critical space to compute all of the complex critical points of an objective function. To illustrate the relevance of our method, we apply it to the Euclidean distance function to compute ED-degrees and the likelihood function to compute maximum likelihood degrees.

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“…Previously in [20], we introduced a perturbation method aiming to obtain a real witness point on each real connected solution component with high accuracy of systems…”

Section: Introductionmentioning

confidence: 99%

“…As a sequel of our previous work [20] we consider numerical aspects of the methods in this paper with the assumption that a real witness point on a positive dimensional component is obtained already. The main contributions of this paper are two strategies to ensure the success of this method which are not provided in [20].…”

Section: Introductionmentioning

confidence: 99%

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Numerical and geometric properties of a method for finding points on real solution components

Reid

Feng

2014

Proceedings of the 2014 Symposium on Symbolic-Numeric Computation

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We consider a critical point method developed in our earlier work for finding certain solution (witness) points on real solution components of real polynomial systems of equations. The method finds points that are critical points of the distance from a plane to the component with the requirement that certain regularity conditions are satisfied. In this paper we analyze the numerical stability of the method. We aim to find at least one well conditioned witness point on each connected component by using perturbation, path tracking and projection techniques. An optimal-direction strategy and an adaptive step size control strategy for path following on high dimensional components are given.

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Finding points on real solution components and applications to differential polynomial systems

Cited by 27 publications

References 23 publications

A robust approach for finding all well-separated solutions of sparse systems of nonlinear equations

A robust approach for finding all well-separated solutions of sparse systems of nonlinear equations

Critical points via monodromy and local methods

Numerical and geometric properties of a method for finding points on real solution components

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