Conference Agenda
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Session Overview |
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MS160, part 4: Numerical methods for structured polynomial system solving
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3:00pm - 5:00pm
Numerical methods for structured polynomial system solving Improvements in the understanding of numerical methods for dense polynomial system solving led to the complete solution of Smale's 17th problem. At this point, it remains an open challenge to achieve the same success in the solution of structured polynomial systems: explain the typical behavior of current algorithms and devise polynomial-time algorithms for computing roots of polynomial systems. In this minisymposium, researchers will present the current progress on applying numerical methods to structured polynomial systems. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Numerical Schubert Calculus via the Littlewood-Richardson Homotopy Algorithm We describe the Littlewood-Richardson homotopy algorithm, which uses numerical continuation to compute solutions to Schubert problems on Grassmannians and is based on the geometric Littlewood-Richardson rule. We provide algorithmic details and discuss its mathematical aspects. Our implementation of this algorithm can solve problem instances with tens of thousands of solutions. We also give a new and optimal formulation of Schubert problems in local Stiefel coordinates as systems of equations. Computing Verified Real Solutions of Polynomials Systems via Low-rank Moment Matrix Completion We propose a new algorithm for computing verified real solutions of polynomial systems with equalities and inequalities. We recast Lasserre's hierarchy of moment relaxations for computing real solutions of polynomial systems into finding symmetric positive semidefinite matrices of minimum nuclear norm subject to linear equality constraints, and then apply fixed point iterations together with Barzilai-Borwein technique for solving a sequence of moment matrix completion problems. Although the method based on function values and gradient evaluations cannot yield as high accuracy as interior point methods, much larger problems can be solved since no second-order information needs to be computed and stored. Finally, we apply interval arithmetic to verify the existence of real solutions of polynomial systems near to the computed approximate real solutions. The algorithm has been implemented in Matlab and is available at http://159.226.47.210:8080/verifyrealroots/tryOnline.jsp Computing the Canonical Polyadic Decomposition of Tensors with Damped Gauss-Newton Method Low rank approximation of tensors can be formulated as a structured nonlinear minimization problem. Exploiting this structure allows to improve the speed and accuracy of a damped Gauss-Newton method. A preliminary implementation of this method performed better than availble published software. A most outrageous action The cost of homotopy algorithms for sparse polynomial systems can be bounded above by an integral of a condition length (Found Comput Math (2019) 19: 1. https://doi.org/10.1007/s10208-018-9375-2). This integral depends on a toric condition number and on a distortion invariant nu. In this talk, I will show how a certain renormalization operator induces a group action on the solution variety. This action will be used to produce a renormalized algorithm, where the distortion nu is constant. Then it becomes possible to integrate the square of the condition number for normal systems. This method provides upper bounds for the expected cost of sparse homotopy. | ||
