Pruning Algorithms for Pretropisms of Newton Polytopes

Sommars, Jeff; Verschelde, Jan

doi:10.1007/978-3-319-45641-6_31

Cited by 4 publications

(6 citation statements)

References 31 publications

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“…Our contributions. This paper extends the results of the last two authors, presented in [28,29], extending the pruning algorithms with dynamic enumeration and a work stealing strategy. Our parallel implementation gives the first computation of the tropical prevariety for the cyclic 16-roots problem.…”

Section: Introductionsupporting

confidence: 76%

Computing Tropical Prevarieties in Parallel

Jensen

Sommars

Verschelde

2017

Proceedings of the International Workshop on Parallel Symbolic Computation

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The computation of the tropical prevariety is the first step in the application of polyhedral methods to compute positive dimensional solution sets of polynomial systems. In particular, pretropisms are candidate leading exponents for the power series developments of the solutions. The computation of the power series may start as soon as one pretropism is available, so our parallel computation of the tropical prevariety has an application in a pipelined solver.We present a parallel implementation of dynamic enumeration. Our first distributed memory implementation with forked processes achieved good speedups, but quite often resulted in large variations in the execution times of the processes. The shared memory multithreaded version applies work stealing to reduce the variability of the run time. Our implementation applies the thread safe Parma Polyhedral Library (PPL), in exact arithmetic with the GNU Multiprecision Arithmetic Library (GMP), aided by the fast memory allocations of TCMalloc.Our parallel implementation is capable of computing the tropical prevariety of the cyclic 16-roots problem. We also report on computational experiments on the n-body and n-vortex problems; our computational results compare favorably with Gfan.

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Section: Introductionsupporting

confidence: 76%

Computing Tropical Prevarieties in Parallel

Jensen

Sommars

Verschelde

2017

Proceedings of the International Workshop on Parallel Symbolic Computation

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“…In the existed structure-adjustable algorithms based on growth, there is only one node in the hidden layer at the initial time generally; however, in the algorithms based on pruning, the number of hidden nodes is set to a maximum value in a reasonable range as possible at first [4,24]. In the methods based on both growing and pruning, the random number is taken as the initial number of hidden nodes.…”

Section: Data Description Improved Negative Correlation Learning Algomentioning

confidence: 99%

Negative Correlation Incremental Integration Classification Method for Underwater Target Recognition

He¹

2018

IJPE

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In this paper, an incremental learning algorithm based on negative correlation learning (NCL) is used as an identification classifier for underwater targets. Based on Selective negative incremental learning SNCL (Selective NCL) algorithm in the process of training, there are numbers of hidden layer nodes that are difficult to determine training time. Problems such as over fitting analysis arise. The algorithm combined with Bagging makes the difference between individual network further increase, and ensures the generalization performance of the whole. On the basis of this method, the use of the selective integration method based on clustering and a new proposed algorithm called SANCLBag, combined with the convolution of underwater target recognition neural network shows that the proposed integration approach can make the difference between individual network in the classification process further increase, and ensure the whole generalization performance. The model has higher identification accuracy, and can effectively solve the problem of incremental learning. . Growing NCL (GNCL). In the former, the number of learner is fixed, and when new data training sets are added, the neural network integration trained before would be used for training. Different from the former, in GNCL, the number of learners in integration would increase as the new data training sets are added. When there are new training sets, a new ANN is constructed and the NCL algorithm is used to train in new training sets, which would be added into the previous integration after training. Selective negative incremental learning based on clusteringThis paper has referred to the selective negative learning framework of Minlong Lin and modified the model training and model selection of it. The selective negative incremental learning framework is shown in Figure 1.

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“…Algorithms to compute the tropical prevariety are presented in [21]. For preprocessing purposes, the software of [21] is useful.…”

Section: Introductionmentioning

confidence: 99%

“…Algorithms to compute the tropical prevariety are presented in [21]. For preprocessing purposes, the software of [21] is useful. However, the focus on this paper concerns the tropical variety for which Gfan [14] provides a tropical basis.…”

Section: Introductionmentioning

confidence: 99%

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Computing All Space Curve Solutions of Polynomial Systems by Polyhedral Methods

Bliss

Verschelde

2016

Computer Algebra in Scientific Computing

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A polyhedral method to solve a system of polynomial equations exploits its sparse structure via the Newton polytopes of the polynomials. We propose a hybrid symbolic-numeric method to compute a Puiseux series expansion for every space curve that is a solution of a polynomial system. The focus of this paper concerns the difficult case when the leading powers of the Puiseux series of the space curve are contained in the relative interior of a higher dimensional cone of the tropical prevariety. We show that this difficult case does not occur for polynomials with generic coefficients. To resolve this case, we propose to apply polyhedral end games to recover tropisms hidden in the tropical prevariety.

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Pruning Algorithms for Pretropisms of Newton Polytopes

Cited by 4 publications

References 31 publications

Computing Tropical Prevarieties in Parallel

Computing Tropical Prevarieties in Parallel

Negative Correlation Incremental Integration Classification Method for Underwater Target Recognition

Computing All Space Curve Solutions of Polynomial Systems by Polyhedral Methods

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