arXiv subjects

Xiangcheng Yu

Publications and source records attributed to Xiangcheng Yu.

4 recordsLinked to original sources

arXiv2015

Solving Polynomial Systems in the Cloud with Polynomial Homotopy Continuation

Polynomial systems occur in many fields of science and engineering. Polynomial homotopy continuation methods apply symbolic-numeric algorithms to solve polynomial systems. We describe the design and implementation of our web interface and reflect on the application of polynomial homotopy continuation methods to solve polynomial systems in the cloud. Via the graph isomorphism problem we organize and classify the polynomial systems we solved. The classification with the canonical form of a graph identifies newly submitted systems with systems that have already been solved.

cs.MS↗

arXiv2015

Tracking Many Solution Paths of a Polynomial Homotopy on a Graphics Processing Unit

Polynomial systems occur in many areas of science and engineering. Unlike general nonlinear systems, the algebraic structure enables to compute all solutions of a polynomial system. We describe our massive parallel predictor-corrector algorithms to track many solution paths of a polynomial homotopy. The data parallelism that provides the speedups stems from the evaluation and differentiation of the monomials in the same polynomial system at different data points, which are the points on the solution paths. Polynomial homotopies that have tens of thousands of solution paths can keep a sufficiently large amount of threads occupied. Our accelerated code combines the reverse mode of algorithmic differentiation with double double and quad double precision to compute more accurate results faster.

cs.MS↗

arXiv2015

Accelerating Polynomial Homotopy Continuation on a Graphics Processing Unit with Double Double and Quad Double Arithmetic

Numerical continuation methods track a solution path defined by a homotopy. The systems we consider are defined by polynomials in several variables with complex coefficients. For larger dimensions and degrees, the numerical conditioning worsens and hardware double precision becomes often insufficient to reach the end of the solution path. With double double and quad double arithmetic, we can solve larger problems that we could not solve with hardware double arithmetic, but at a higher computational cost. This cost overhead can be compensated by acceleration on a Graphics Processing Unit (GPU). We describe our implementation and report on computational results on benchmark polynomial systems.

cs.MS↗

arXiv2014

GPU acceleration of Newton's method for large systems of polynomial equations in double double and quad double arithmetic

In order to compensate for the higher cost of double double and quad double arithmetic when solving large polynomial systems, we investigate the application of NVIDIA Tesla K20C general purpose graphics processing unit. The focus on this paper is on Newton's method, which requires the evaluation of the polynomials, their derivatives, and the solution of a linear system to compute the update to the current approximation for the solution. The reverse mode of algorithmic differentiation for a product of variables is rewritten in a binary tree fashion so all threads in a block can collaborate in the computation. For double arithmetic, the evaluation and differentiation problem is memory bound, whereas for complex quad double arithmetic the problem is compute bound. With acceleration we can double the dimension and get results that are twice as accurate in about the same time.

cs.DC↗