Related papers: Affine solution sets of sparse polynomial systems
The note focuses on the differential geometric approach to the study of nonlinear systems that are affine in control. We first develop normal forms for nonlinear system affine in control. Based on these normal forms, we then address the…
We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active,…
Multiple scale homogenization problems are reduced to single scale problems in higher dimension. It is shown that sparse tensor product Finite Element Methods (FEM) allow the numerical solution in complexity independent of the dimension and…
We study the basic computational problem of detecting approximate stationary points for continuous piecewise affine (PA) functions. Our contributions span multiple aspects, including complexity, regularity, and algorithms. Specifically, we…
Sparse coding, which is the decomposition of a vector using only a few basis elements, is widely used in machine learning and image processing. The basis set, also called dictionary, is learned to adapt to specific data. This approach has…
We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. We propose a general algebraic framework to find the solutions and to…
We give the first exact determinantal formula for the resultant of an unmixed sparse system of four Laurent polynomials in three variables with arbitrary support. This follows earlier work by the author on exact formulas for bivariate…
In this article we study adaptive finite element methods (AFEM) with inexact solvers for a class of semilinear elliptic interface problems. We are particularly interested in nonlinear problems with discontinuous diffusion coefficients, such…
Equal-volume polygons are obtained from adequate discretizations of curves in 3-space, contained or not in surfaces. In this paper we explore the similarities of these polygons with the affine arc-length parameterized smooth curves to…
A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAP-EM algorithm. A dual mathematical interpretation of the…
Let $a_{i1}x_1+\cdots+a_{ik}x_k=0$, $i\in[m]$ be a balanced homogeneous system of linear equations with coefficients $a_{ij}$ from a finite field $\mathbb{F}_q$. We say that a solution $x=(x_1,\ldots, x_k)$ with $x_1,\ldots, x_k\in…
For several computational procedures such as finding radicals and Noether normalizations, it is important to choose as sparse as possible a system of parameters in a polynomial ideal or modulo a polynomial ideal. We describe new strategies…
Each point of a simplex is expressed as a unique convex combination of the vertices. The coefficients in the combination are the barycentric coordinates of the point. For each point in a general convex polytope, there may be multiple…
Finding sparse solutions of underdetermined systems of linear equations is a fundamental problem in signal processing and statistics which has become a subject of interest in recent years. In general, these systems have infinitely many…
An exact algorithm is presented for solving edge weighted graph partitioning problems. The algorithm is based on a branch and bound method applied to a continuous quadratic programming formulation of the problem. Lower bounds are obtained…
The paper is devoted to the local classification of generic control-affine systems on an n-dimensional manifold with scalar input for any n>3 or with two inputs for n=4 and n=5, up to state-feedback transformations, preserving the affine…
This is an expository paper which has two parts. In the first part, we study quiver varieties of affine $A$-type from a combinatorial point of view. We present a combinatorial method for obtaining a closed formula for the generating…
A solution for Smale's 17th problem, for the case of systems with bounded degree was recently given. This solution, an algorithm computing approximate zeros of complex polynomial systems in average polynomial time, assumed infinite…
We develop fixed-point algorithms for the approximation of structured matrices with rank penalties. In particular we use these fixed-point algorithms for making approximations by sums of exponentials, or frequency estimation. For the basic…
An adaptive algorithm for computing eigenmodes and propagation constants of optical fibers is proposed. The algorithm is built using a dual-weighted residual error estimator. The residuals are based on the eigensystem for leaky hybrid modes…