Jan Verschelde

Danko Adrovic, Jan Verschelde

In this paper we outline an algorithmic approach to compute Puiseux series expansions for algebraic surfaces. The series expansions originate at the intersection of the surface with as many coordinate planes as the dimension of the surface. Our approach starts with a polyhedral method to compute cones of normal vectors to the Newton polytopes of the given polynomial system that defines the surface. If as many vectors in the cone as the dimension of the surface define an initial form system that has isolated solutions, then those vectors are potential tropisms for the initial term of the Puiseux series expansion. Our preliminary methods produce exact representations for solution sets of the cyclic $n$-roots problem, for $n = m^2$, corresponding to a result of Backelin.

Jan Verschelde

In numerical algebraic geometry witness sets are numerical representations of positive dimensional solution sets of polynomial systems. Considering the asymptotics of witness sets we propose certificates for algebraic curves. These certificates are the leading terms of a Puiseux series expansion of the curve starting at infinity. The vector of powers of the first term in the series is a tropism. For proper algebraic curves, we relate the computation of tropisms to the calculation of mixed volumes. With this relationship, the computation of tropisms and Puiseux series expansions could be used as a preprocessing stage prior to a more expensive witness set computation. Systems with few monomials have fewer isolated solutions and fewer data are needed to represent their positive dimensional solution sets.

Danko Adrovic, Jan Verschelde

We present a polyhedral algorithm to manipulate positive dimensional solution sets. Using facet normals to Newton polytopes as pretropisms, we focus on the first two terms of a Puiseux series expansion. The leading powers of the series are computed via the tropical prevariety. This polyhedral algorithm is well suited for exploitation of symmetry, when it arises in systems of polynomials. Initial form systems with pretropisms in the same group orbit are solved only once, allowing for a systematic filtration of redundant data. Computations with cddlib, Gfan, PHCpack, and Sage are illustrated on cyclic $n$-roots polynomial systems.

Jan Verschelde, Genady Yoffe

In order to obtain more accurate solutions of polynomial systems with numerical continuation methods we use multiprecision arithmetic. Our goal is to offset the overhead of double double arithmetic accelerating the path trackers and in particular Newton's method with a general purpose graphics processing unit. In this paper we describe algorithms for the massively parallel evaluation and differentiation of sparse polynomials in several variables. We report on our implementation of the algorithmic differentiation of products of variables on the NVIDIA Tesla C2050 Computing Processor using the NVIDIA CUDA compiler tools.

Kathy Piret, Jan Verschelde

Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the solution paths. A point along a solution path is critical when the Jacobian matrix is rank deficient. The simplest case of quadratic turning points is well understood, but these methods no longer work for general types of singularities. In order not to miss any singular solutions along a path we propose to monitor the determinant of the Jacobian matrix. We examine the operation range of deflation and relate the effectiveness of deflation to the winding number. Computational experiments on systems coming from different application fields are presented.

Nathan Bliss, Jan Verschelde

We consider the extension of the method of Gauss-Newton from complex floating-point arithmetic to the field of truncated power series with complex floating-point coefficients. With linearization we formulate a linear system where the coefficient matrix is a series with matrix coefficients, and provide a characterization for when the matrix series is regular based on the algebraic variety of an augmented system. The structure of the linear system leads to a block triangular system. In the regular case, solving the linear system is equivalent to solving a Hermite interpolation problem. We show that this solution has cost cubic in the problem size. In general, at singular points, we rely on methods of tropical algebraic geometry to compute Puiseux series. With a few illustrative examples, we demonstrate the application to polynomial homotopy continuation.

Jan Verschelde, Genady Yoffe

Our problem is to accurately solve linear systems on a general purpose graphics processing unit with double double and quad double arithmetic. The linear systems originate from the application of Newton's method on polynomial systems. Newton's method is applied as a corrector in a path following method, so the linear systems are solved in sequence and not simultaneously. One solution path may require the solution of thousands of linear systems. In previous work we reported good speedups with our implementation to evaluate and differentiate polynomial systems on the NVIDIA Tesla C2050. Although the cost of evaluation and differentiation often dominates the cost of linear system solving in Newton's method, because of the limited bandwidth of the communication between CPU and GPU, we cannot afford to send the linear system to the CPU for solving during path tracking. Because of large degrees, the Jacobian matrix may contain extreme values, requiring extended precision, leading to a significant overhead. This overhead of multiprecision arithmetic is our main motivation to develop a massively parallel algorithm. To allow overdetermined linear systems we solve linear systems in the least squares sense, computing the QR decomposition of the matrix by the modified Gram-Schmidt algorithm. We describe our implementation of the modified Gram-Schmidt orthogonalization method for the NVIDIA Tesla C2050, using double double and quad double arithmetic. Our experimental results show that the achieved speedups are sufficiently high to compensate for the overhead of one extra level of precision.

Jan Verschelde

A numerical irreducible decomposition for a polynomial system provides representations for the irreducible factors of all positive dimensional solution sets of the system, separated from its isolated solutions. Homotopy continuation methods are applied to compute a numerical irreducible decomposition. Load balancing and pipelining are techniques in a parallel implementation on a computer with multicore processors. The application of the parallel algorithms is illustrated on solving the cyclic $n$-roots problems, in particular for $n = 8, 9$, and~12.

Nathan Bliss, Jan Verschelde

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.

Johnny Joyce, Jan Verschelde

Convolutional neural networks (CNNs) are a popular choice of model for tasks in computer vision. When CNNs are made with many layers, resulting in a deep neural network, skip connections may be added to create an easier gradient optimization problem while retaining model expressiveness. In this paper, we show that arbitrarily complex, trained, linear CNNs with skip connections can be simplified into a single-layer model, resulting in greatly reduced computational requirements during prediction time. We also present a method for training nonlinear models with skip connections that are gradually removed throughout training, giving the benefits of skip connections without requiring computational overhead during during prediction time. These results are demonstrated with practical examples on Residual Networks (ResNet) architecture.

Johnny Joyce, Jan Verschelde

Neural networks are important tools in machine learning. Representing piecewise linear activation functions with tropical arithmetic enables the application of tropical geometry. Algorithms are presented to compute regions where the neural networks are linear maps. Through computational experiments, we provide insights on the difficulty to train neural networks, in particular on the problems of overfitting and on the benefits of skip connections.

Frank Sottile, Ravi Vakil, Jan Verschelde

We present a new numerical homotopy continuation algorithm for finding all solutions to Schubert problems on Grassmannians. This Littlewood-Richardson homotopy is based on Vakil's geometric proof of the Littlewood-Richardson rule. Its start solutions are given by linear equations and they are tracked through a sequence of homotopies encoded by certain checker configurations to find the solutions to a given Schubert problem. For generic Schubert problems the number of paths tracked is optimal. The Littlewood-Richardson homotopy algorithm is implemented using the path trackers of the software package PHCpack.

Yun Guan, Jan Verschelde

Numerical data structures for positive dimensional solution sets of polynomial systems are sets of generic points cut out by random planes of complimentary dimension. We may represent the linear spaces defined by those planes either by explicit linear equations or in parametric form. These descriptions are respectively called extrinsic and intrinsic representations. While intrinsic representations lower the cost of the linear algebra operations, we observe worse condition numbers. In this paper we describe the local adaptation of intrinsic coordinates to improve the numerical conditioning of sampling algebraic sets. Local intrinsic coordinates also lead to a better stepsize control. We illustrate our results with Maple experiments and computations with PHCpack on some benchmark polynomial systems.

Danko Adrovic, Jan Verschelde

Finding a common factor of two multivariate polynomials with approximate coefficients is a problem in symbolic-numeric computing. Taking a tropical view on this problem leads to efficient preprocessing techniques, applying polyhedral methods on the exact exponents with numerical techniques on the approximate coefficients. With Maple we will illustrate our use of tropical algebraic geometry.

Anton Leykin, Jan Verschelde, Ailing Zhao

Given an approximation to a multiple isolated solution of a polynomial system of equations, we have provided a symbolic-numeric deflation algorithm to restore the quadratic convergence of Newton's method. Using first-order derivatives of the polynomials in the system, our method creates an augmented system of equations which has the multiple isolated solution of the original system as a regular root. In this paper we consider two approaches to computing the ``multiplicity structure'' at a singular isolated solution. An idea coming from one of them gives rise to our new higher-order deflation method. Using higher-order partial derivatives of the original polynomials, the new algorithm reduces the multiplicity faster than our first method for systems which require several first-order deflation steps. We also present an algorithm to predict the order of the deflation.

Andrew J. Sommese, Jan Verschelde, Charles W. Wampler

By a numerical continuation method called a diagonal homotopy we can compute the intersection of two positive dimensional solution sets of polynomial systems. This paper proposes to use this diagonal homotopy as the key step in a procedure to intersect general solution sets. Of particular interest is the special case where one of the sets is defined by a single polynomial equation. This leads to an algorithm for finding a numerical representation of the solution set of a system of polynomial equations introducing the equations one-by-one. Preliminary computational experiments show this approach can exploit the special structure of a polynomial system, which improves the performance of the path following algorithms.

Anton Leykin, Jan Verschelde, Ailing Zhao

We present a modification of Newton's method to restore quadratic convergence for isolated singular solutions of polynomial systems. Our method is symbolic-numeric: we produce a new polynomial system which has the original multiple solution as a regular root. Using standard bases, a tool for the symbolic computation of multiplicities, we show that the number of deflation stages is bounded by the multiplicity of the isolated root. Our implementation performs well on a large class of applications.