Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Location: Unitobler, F012
30 seats, 57m^2
Date: Tuesday, 09/Jul/2019
10:00am - 12:00pmRoom free
Unitobler, F012 
3:00pm - 5:00pmMS160, part 1: Numerical methods for structured polynomial system solving
Unitobler, F012 
 
3:00pm - 5:00pm

Numerical methods for structured polynomial system solving

Chair(s): Alperen Ergur (TU Berlin), Pierre Lairez (INRIA), Gregorio Malajovich (Universidade Federal do Rio de Janeiro, Brazil), Josue Tonelli-Cueto (TU Berlin)

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)

 

Introductory Talk

Alperen Ergur
TU Berlin

This talk will provide a brief overview of the state of the art in numerical methods for polynomial system solving, and introduce the goals of the mini-symposium.

 

On the condition number of some algebraic problems.

Diego Armentano1, Carlos Beltrán2
1Universidad de la Republica, 2Universidad de Cantábria

In this talk we will introduce a geometric framework to analyze the condition number of some classic algebraic problems. We will compute the average value of the squared condition number, when the input is taken at random, for a large family of these problems. In particular, we will show that the polynomial eigenvalue problem is quite well conditioned.

 

Numerical irreducible decomposition with one homotopy

Dan Bates1, David Eklund2, Jonathan Hauenstein3, Chris Peterson4
1US Naval Academy, 2KTH, 3University of Notre Dame, 4Colorado State University

A numerical irreducible decomposition (NID) of an algebraic set V includes a set of witness points (approximations of generic points) on each irreducible component of V, along with various auxiliary data. The computation of an NID typically involves a sequence of homotopies. Pairing together the machinery of excess intersection and isosingular sets, we show how to compute an NID with only one homotopy. This is joint work with David Eklund, Jonathan Hauenstein, and Chris Peterson.

 

Computing the Homology of arbitrary Semialgebraic Sets

Felipe Cucker1, Peter Bürgisser2, Josue Tonelli-Cueto2
1City University of Hong Kong, 2TU Berlin

We describe recent advances regarding the computation of the homology groups of arbitrary semialgebraic sets. These advances follow the line of results obtained for the particular cases of smooth projective varieties and basic semialgebraic sets.

Coauthors are Peter Buergisser and Josue Tonelli-Cueto.

 

Date: Wednesday, 10/Jul/2019
10:00am - 12:00pmRoom free
Unitobler, F012 
3:00pm - 5:00pmMS160, part 2: Numerical methods for structured polynomial system solving
Unitobler, F012 
 
3:00pm - 5:00pm

Numerical methods for structured polynomial system solving

Chair(s): Alperen Ergur (TU Berlin), Pierre Lairez (INRIA), Gregorio Malajovich (Universidade Federal do Rio de Janeiro, Brazil), Josue Tonelli-Cueto (TU Berlin)

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)

 

Polyhedral Real Homotopy Continuation

Timo de Wolff, Alperen Ergür
TU Berlin

We design a homotopy continuation algoritm to find real roots of sparse polynomial systems based on numerically tracking a well-known geometric deformation process called Viro’s patchworking. The main advantage of our algorithm is that it entirely operates over the field of real numbers and tracks the optimal number of solution paths. The price for this property is that the algorithm is not guaranteed to work universally, but requires to solve and track polynomial systems located in an unbounded component of the complement of the A-discriminant. We provide a relative entropy programming based relaxation to certify this requirement on the input data.

 

Root counts of structured algebraic systems

Ioannis Z. Emiris1, Raimundas Vidunas2, Evangelos Bartzos3, Josef Schicho4
1National and Kapodistrian University of Athens, 2U.Vilnius, Lithuania, 3NKU Athens, 4JKU Linz, Austria

We consider bounds on the number of complex roots of well-constrained algebraic systems, namely mixed volume and multi-homogeneous Bezout bound. We relate these bounds to permanent expressions, generalizing this relationship to the case of systems whose Newton polytopes are products of arbitrary polytopes in complementary subspaces. This improves the computational complexity of determining the bounds. We apply the bounds to obtaining new counts on the number of complex embeddings of minimally rigid graphs in the plane, in space, and on the sphere. We relate the bounds to certain combinatorial properties of the graphs such as the number of ways to orient the edges according to certain rules. We focus on bounds for semi-mixed algebraic systems where equations are partitioned to subsets with common Newton polytopes. This is applied to counting the number of totally mixed Nash equilibria in games of several players.

 

A local complexity theory

Teresa Krick1, Felipe Cucker2
1Universidad de Buenos Aires, 2City University of Hong Kong

I will describe advances on an on-going project of establishing a local complexity theory. This is inspired on the concept of smoothed analysis introduced some years ago by Spielman and Teng, which takes as cost the supremum of the average on small balls around points. Here we are interested in styuding directly the average on small balls around points, without considering the supremum on the whole space. We believe that this would be a more accurate notion of cost for practical problems than the smoothed complexity. Our first analysis studies local complexity for real conic condition numbers, under uniform and Gaussian distributions.
 

Low-degree approximation of real singularities

Antonio Lerario1, Paul Breiding2, Daouda Niang Diatta3, Hanieh Keneshlou2
1SISSA, 2MPI-MSI Leipzig, 3Université Assane SECK de Ziguinchor

In this talk I will discuss some recent results that allow to approximate a real singularity given by polynomial equations of degree d (e.g. the zero set of a polynomial, or the number of its critical points of a given Morse index) with a singularity which is diffeomorphic to the original one, but it is given by polynomials of degree O(d^{1/2} log d). The approximation procedure is constructive (in the sense that one can read the approximating polynomial from a linear projection of the given one) and quantitative (in the sense that the approximating procedure will hold for a subset of the space of polynomials with measure increasing very quickly to full measure as the degree goes to infinity).

I will also discuss the potential of this procedure for improving the average complexity of some algorithms.

This is based on a combination of joint works with P. Breiding, D. N. Diatta and H. Keneshlou.

 

Date: Thursday, 11/Jul/2019
10:00am - 12:00pmMS173, part 1: Numerical methods in algebraic geometry
Unitobler, F012 
 
10:00am - 12:00pm

Numerical methods in algebraic geometry

Chair(s): Jose Israel Rodriguez (UW Madison, United States of America), Paul Breiding (MPI MiS)

This minisymposium is meant to report on recent advances in using numerical methods in algebraic geometry: the foundation of algebraic geometry is the solving of systems of polynomial equations. When the equations to be considered are defined over a subfield of the complex numbers, numerical methods can be used to perform algebraic geometric computations forming the area of numerical algebraic geometry (NAG). Applications which have driven the development of this field include chemical and biological reaction networks, robotics and kinematics, algebraic statistics, and tropical geometry. The minisymposium will feature a diverse set of talks, ranging from the application of NAG to problems in either theory and practice, to discussions on how to implement new insights from numerical mathematics to improve existing methods.

 

(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)

 

Minimal problems in multiview 3D reconstruction via homotopy continuation

Anton Leykin
Georgia Tech

This will be a short survey of a class of problems in computer vision, for which it is plausible to construct efficient solvers based on polynomial homotopy continuation. For some of these problems alternative solvers do not exist at the moment.
The problems we consider concern relative camera pose recovery from points and lines in more than 2 views. In addition to classical point correspondences and line correspondences, we use incidence correspondences, which result from points lying on lines. We show how to build a solver based on a parameter homotopy coming from this framework.
(This survey is based on collaboration and discussions with many people involved in the Nonlinear Algebra semester at ICERM in Fall 2018.)

 

Computing the real CANDECOMP/PARAFAC decomposition of real tensors

Tsung-Lin Lee
National Sun Yat-sen University

The real Candecomp/Parafac decomposition (CPD) has many applications in psychometrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Several methods have been provided for computing the CPD such as alternating least squares (ALS), nonlinear least squares (NLS) and unconstrained nonlinear optimization. Those methods may take many iterations to converge and are not guar-anteed to converge to the solution. Recently, homotopy continuation techniques have been applied in computing tensor decomposition. In this talk, the real CPD of a real unbalanced tensor will be considered.

 

Computing transcendental invariants of hypersurfaces via homotopy

Emre Sertoz
Max-Planck-Institute MiS, Leipzig

Deep geometric properties of each projective variety is encoded in a matrix of complex numbers, called its periods. Knowing the periods of a variety, one can often say quite a lot about the type of subvarieties it contains using LLL methods, without resorting to symbolic elimination. However, numerical computation of periods have been previously confined to curves in the plane and to varieties enjoying many symmetries. We will demonstrate how periods of hypersurfaces can be computed using a form of homotopy and how they can be studied to reveal the geometry of the hypersurface.

 

On the nonlinearity interval in parametric semidefinite optimization

Tingting Tang
University of Notre Dame

We consider the parametric analysis of semidefinite optimization problems with respect to the perturbation of the objective vector along a fixed direction. We characterize the so-called transition point of the optimal partition where the ranks of a maximally complementary optimal solution suddenly change, and the nonlinearity interval of the optimal partition where the ranks of maximally complementary optimal solutions stay constant. The continuity of the optimal set mapping on the basis of Painleve-Kuratowski set convergence in a nonlinearity interval is investigated. We show that not only the continuity might fail, even the sequence of maximally complementary optimal solutions might jump in the interior of a nonlinearity interval. Finally, we present a procedure stemming from numerical algebraic geometry to efficiently compute nonlinearity intervals.

 
3:00pm - 5:00pmMS160, part 3: Numerical methods for structured polynomial system solving
Unitobler, F012 
 
3:00pm - 5:00pm

Numerical methods for structured polynomial system solving

Chair(s): Alperen Ergur (TU Berlin), Pierre Lairez (INRIA), Gregorio Malajovich (Universidade Federal do Rio de Janeiro, Brazil), Josue Tonelli-Cueto (TU Berlin)

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)

 

Certifying solutions to a square system involving analytic functions

Michael Burr1, Kisun Lee2, Anton Leykin2
1Clemson University, 2Georgia Institute of Technology

In this talk, we introduce two different approaches for the certification of solutions to a square system built from univariate analytic functions. The first approach is based on alpha-theory, and the second is based on interval Newton's Method. The certification methods are based on the existence of oracles from input analytic functions. We show that these two oracles exist for D-finite functions. Finally, we compare these two approaches using our SageMath implementation for D-finite function cases.

 

Toric witness sets for sampling positive dimensional solution sets of polynomial systems

Tianran Chen
Auburn University at Montgomery

The linear slicing technique and the construction of "witness sets" proposed by Sommese and Wampler in 1996 form the foundation for numerical algebraic geometry. They are the indispensable tools for numerically finding and studying positive dimensional (non-isolated) solution sets defined by systems of polynomial equations. Combining the strength of the polyhedral homotopy method and the toric approach for studying algebraic sets, we propose a general framework for efficiently sampling positive dimensional solution sets that can potentially take advantage of the sparsity in the system. The practical usefulness of this approach is demonstrated through an application to the "power-flow equation" from electric engineering.

 

Farewell to Weyl: Condition-based analysis with a Banach norm in numerical algebraic geometry

Josue Tonelli-Cueto1, Felipe Cucker2, Alperen Ergür1
1TU Berlin, 2City University of Hong Kong

Condition-based complexity analyses of numerical algorithms in algebraic geometry seem to rely heavily on inner product norms, such as the Weyl norm. This contrasts with the situation in numerical linear algebra where it is common to use plenty of Banach norm that do not come from an inner product. We show that similar advantages can be obtained in numerical algebraic geometry by showing how such an analysis can be carried out with respect a Banach norm in various settings, obtaining substancial improvements over the known complexity estimates for linear homotopy continuation and grid based methods. This is on going work with Felipe Cucker and Alperen A. Ergür.

 

Singular polynomial eigenvalue problems are not ill-conditioned

Martin Lotz1, Vanni Noferini2
1Warwick University, 2Aalto University

Numerical methods are not supposed to work on ill-posed inputs, and even less so if the function to be computed is discontinuous. Yet, there are examples where arbitrary small perturbations in the input can lead to literally any function value, but where standard numerical algorithms that are oblivious to the special structure of the problem work perfectly fine. Neither the classical nor the stochastic theories of conditioning are adequate to predict the typical forward accuracy in such cases. Motivated by this limitation, and using singular polynomial eigenvalue problems as running example, we define and study weak worst-case and a weak stochastic condition numbers. This new theory can be a more powerful predictor of the accuracy of computations than existing tools, especially when the worst-case and the expected sensitivity of a problem to perturbations of the input are not finite. We illustrate how such condition number can be estimated and used in practice.

 

Date: Friday, 12/Jul/2019
10:00am - 12:00pmMS173, part 2: Numerical methods in algebraic geometry
Unitobler, F012 
 
10:00am - 12:00pm

Numerical methods in algebraic geometry

Chair(s): Jose Israel Rodriguez (UW Madison, United States of America), Paul Breiding (MPI MiS)

This minisymposium is meant to report on recent advances in using numerical methods in algebraic geometry: the foundation of algebraic geometry is the solving of systems of polynomial equations. When the equations to be considered are defined over a subfield of the complex numbers, numerical methods can be used to perform algebraic geometric computations forming the area of numerical algebraic geometry (NAG). Applications which have driven the development of this field include chemical and biological reaction networks, robotics and kinematics, algebraic statistics, and tropical geometry. The minisymposium will feature a diverse set of talks, ranging from the application of NAG to problems in either theory and practice, to discussions on how to implement new insights from numerical mathematics to improve existing methods.

 

(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 Root Finding via Cox Rings

Simon Telen
KU Leuven

In this talk, we consider the problem of solving a system of (sparse) Laurent polynomial equations defining finitely many nonsingular points on a compact toric variety. The Cox ring of this toric variety is a generalization of the homogeneous coordinate ring of projective space. We work with multiplication maps in graded pieces of this ring to generalize the eigenvalue, eigenvector theorem for root finding in affine space. We present a numerical linear algebra algorithm for computing the corresponding matrices, and from these matrices a set of homogeneous coordinates of the roots of the system. Several numerical experiments show the effectiveness of the resulting method, especially for solving (nearly) degenerate, high degree systems in small numbers of variables.

 

Numerical computation of monodromy action over R

Margaret Regan
University of Notre Dame

The monodromy group (over the complex numbers) is a geometric invariant that encodes the structure of the solutions for a parameterized family of polynomial systems and can be computed using numerical algebraic geometry. Since a naive extension to the real numbers is very restrictive, this talk will explore a new approach over the real numbers which is computed piece-wise to obtain tiered characteristics of the real solution set. This talk will conclude with an application in kinematics to help highlight the computational method and impact on calibration.

 

Adaptive step size control for homotopy continuation methods

Sascha Timme
TU Berlin

At the heart of homotopy continuation methods lies the numerical tracking of implicitly defined paths by a predictor-corrector scheme. For efficient path tracking the predictor step size must be chosen appropriately. We present a new adaptive step size control which changes the step size based on computational estimates of local geometric information as well as the order of the used predictor method. We also give an update on the Julia package HomotopyContinuation.jl.

 

Numerical homotopies from Khovanskii bases

Elise Walker
Texas A&M

Homotopies are useful numerical methods for solving systems of polynomial equations. I will present such a homotopy method using Khovanskii bases. Finite Khovanskii bases provide a flat degeneration to a toric variety, which consequentially gives a homotopy. The polyhedral homotopy, which is implemented in PHCPack, can be used to solve for points on a general linear slice of this toric variety. These points can then be traced via the Khovanskii homotopy to points on a general linear slice of the original variety. This is joint work with Michael Burr and Frank Sottile.

 
3:00pm - 5:00pmMS160, part 4: Numerical methods for structured polynomial system solving
Unitobler, F012 
 
3:00pm - 5:00pm

Numerical methods for structured polynomial system solving

Chair(s): Alperen Ergur (TU Berlin), Pierre Lairez (INRIA), Gregorio Malajovich (Universidade Federal do Rio de Janeiro, Brazil), Josue Tonelli-Cueto (TU Berlin)

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

Jan Verschelde1, Anton Leykin2, Abraham Martín del Campo3, Frank Sottile4, Ravi Vakil5
1University of Illinois at Chicago, 2GeorgiaTech, 3CIMAT, Guanajuato, 4Texas A&M University, 5Stanford University

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

Lihong Zhi, Yue Ma, Zhengfeng Yang
Academia Sinica

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
It is a joint work with Zhengfeng Yang, Yue Ma, Yijun Zhu.

 

Computing the Canonical Polyadic Decomposition of Tensors with Damped Gauss-Newton Method

Felipe Diniz
Universidade Federal do Rio de Janeiro

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

Gregorio Malajovich
Universidade Federal do Rio de Janeiro

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.

 

Date: Saturday, 13/Jul/2019
10:00am - 12:00pmMS173, part 3: Numerical methods in algebraic geometry
Unitobler, F012 
 
10:00am - 12:00pm

Numerical methods in algebraic geometry

Chair(s): Jose Israel Rodriguez (UW Madison, United States of America), Paul Breiding (MPI MiS)

This minisymposium is meant to report on recent advances in using numerical methods in algebraic geometry: the foundation of algebraic geometry is the solving of systems of polynomial equations. When the equations to be considered are defined over a subfield of the complex numbers, numerical methods can be used to perform algebraic geometric computations forming the area of numerical algebraic geometry (NAG). Applications which have driven the development of this field include chemical and biological reaction networks, robotics and kinematics, algebraic statistics, and tropical geometry. The minisymposium will feature a diverse set of talks, ranging from the application of NAG to problems in either theory and practice, to discussions on how to implement new insights from numerical mathematics to improve existing methods.

 

(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)

 

Certification of approximate roots of exact ill-posed polynomial systems

Agnes Szanto
NCSU

In this talk, I will survey some of our recent work on certifying approximate roots of exact polynomial systems and will describe some applications. In particular, we will concentrate on systems with root multiplicity, and show ways to certify approximations to singular roots, as well as their multiplicity structure. The difficulty lies in the fact that having singular roots is not a continuous property, so traditional numerical certification techniques do not work. Our certification methods are based on hybrid symbolic-numeric techniques. This is joint work with Jonathan Hauenstein and Bernard Mourrain.

 

Numerical Implicitization

Justin Chen
Georgia Tech

It is increasingly important nowadays to perform explicit computations on varieties, even in the realm where symbolic (e.g. Grobner basis) methods are too slow. We give an overview of the Macaulay2 package NumericalImplicitization, which aims to provide numerical information about images of varieties, such as dimension, degree, and Hilbert function. We also discuss some changes and additions to the package, such as improvements to point sampling, completions of partial pseudo-witness sets, and parallelization. This is joint work with Joe Kileel.

 

The Distribution of Numbers of Operating Points of Power Networks

Julia Lindberg
Wisconsin Institute for Discovery

The operating points of an n-node power network are real solutions of the power flow equations, a system of 2n-2 quadratic polynomials in 2n-2 variables. Our work finds the distribution of the number of real solutions, which is important in determining the stability of the network. In general, the number of nontrivial operating points equals the number of real solutions of a single polynomial. We use this polynomial to visualize regions with a fixed number of solutions, finding that some cluster around hyperplanes.

 
3:00pm - 5:00pmRoom free
Unitobler, F012