Hom4PS-3: A Parallel Numerical Solver for Systems of Polynomial Equations Based on Polyhedral Homotopy Continuation Methods

Chen, Tianran; Lee, Tsung-Lin; Li, Tien Yien

doi:10.1007/978-3-662-44199-2_30

Cited by 54 publications

(47 citation statements)

References 17 publications

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“…This is well within the reach of modern polyhedral homotopy implementations. In particular, with a random choice of the Y -matrix, Hom4PS-3 [11,12] was able to find all solutions in less than 5 minutes (around 297 seconds) on a single workstation (with two Intel Xeon processor). 6.…”

Section: Clustersmentioning

confidence: 99%

On the Network Topology Dependent Solution Count of the Algebraic Load Flow Equations

Chen

Mehta

2018

IEEE Trans. Power Syst.

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A large amount of research activity in power systems areas has focused on developing computational methods to solve load flow equations where a key question is the maximum number of isolated solutions. Though several concrete upper bounds exist, recent studies have hinted that much sharper upper bounds that depend the topology of underlying power networks may exist. This paper establishes such a topology dependent solution bound which is actually the best possible bound in the sense that it is always attainable. We also develop a geometric construction called adjacency polytope which accurately captures the topology of the underlying power network and is immensely useful in the computation of the solution bound. Finally we highlight the significant implications of the development of such solution bound in solving load flow equations.2. Algebraic formulation. In this paper, we focus on the mathematical abstraction of a power network which is captured by a graph G = (B, E) together with a complex matrix Y = (Y ij ). Here B is the finite set of nodes representing the "buses", E is the set of edges (also called lines or branches) representing the connections between buses, and the matrix Y is the nodal admittance matrix 1 which assigns a nonzero †

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Section: Clustersmentioning

confidence: 99%

On the Network Topology Dependent Solution Count of the Algebraic Load Flow Equations

Chen

Mehta

2018

IEEE Trans. Power Syst.

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“…Intermediate steps in the determination of the BKK bound are reused to create an appropriate start system. Using the fully parallel implementation Hom4PS-3 [45] of the polyhedral homotopy method, we compute the BKK bound, that is tight in all our experiments, and obtain all isolated positive solutions.…”

Section: Polyhedral Homotopymentioning

confidence: 99%

Fixed Points of Belief Propagation—An Analysis via Polynomial Homotopy Continuation

Knoll

Mehta

Chen

et al. 2018

IEEE Trans. Pattern Anal. Mach. Intell.

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Belief propagation (BP) is an iterative method to perform approximate inference on arbitrary graphical models. Whether BP converges and if the solution is a unique fixed point depends on both the structure and the parametrization of the model. To understand this dependence it is interesting to find all fixed points. In this work, we formulate a set of polynomial equations, the solutions of which correspond to BP fixed points. To solve such a nonlinear system we present the numerical polynomial-homotopy-continuation (NPHC) method. Experiments on binary Ising models and on error-correcting codes show how our method is capable of obtaining all BP fixed points. On Ising models with fixed parameters we show how the structure influences both the number of fixed points and the convergence properties. We further asses the accuracy of the marginals and weighted combinations thereof. Weighting marginals with their respective partition function increases the accuracy in all experiments. Contrary to the conjecture that uniqueness of BP fixed points implies convergence, we find graphs for which BP fails to converge, even though a unique fixed point exists. Moreover, we show that this fixed point gives a good approximation, and the NPHC method is able to obtain this fixed point.

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“…A big advantage of the polynomial continuation methods is their parallel nature [42]. Each path can be traced independently, which means we have significant reduction in solution time if we perform parallel computations using multi-core processors or GPUs.…”

Section: Polynomial Continuationmentioning

confidence: 99%

A Matlab tool for regions of attraction estimation via numerical algebraic geometry

Demenkov¹

2015

2015 International Conference on Mechanics - Seventh Polyakhov's Reading

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For a locally stable polynomial dynamical system its region of attraction can be estimated by a polynomial Lyapunov function level set. We formulate this problem in terms of global minimization of a polynomial function over single polynomial constraint. We describe a simple Matlab tool, which employs numerical algebraic geometry methods for computing all local solutions of the optimization problem and therefore its global solution.

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Hom4PS-3: A Parallel Numerical Solver for Systems of Polynomial Equations Based on Polyhedral Homotopy Continuation Methods

Cited by 54 publications

References 17 publications

On the Network Topology Dependent Solution Count of the Algebraic Load Flow Equations

On the Network Topology Dependent Solution Count of the Algebraic Load Flow Equations

Fixed Points of Belief Propagation—An Analysis via Polynomial Homotopy Continuation

A Matlab tool for regions of attraction estimation via numerical algebraic geometry

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