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We present a new algorithm to decide isomorphism between finite graded algebras. For a broad class of nilpotent Lie algebras, we demonstrate that it runs in time polynomial in the order of the input algebras. We introduce heuristics that…
Galois/monodromy groups attached to parametric systems of polynomial equations provide a method for detecting the existence of symmetries in solution sets. Beyond the question of existence, one would like to compute formulas for these…
In this paper, a novel multigrid method based on Newton iteration is proposed to solve nonlinear eigenvalue problems. Instead of handling the eigenvalue $\lambda$ and eigenfunction $u$ separately, we treat the eigenpair $(\lambda, u)$ as…
A new method for solving systems of linear algebraic equations of a special type arising in solving problems of image reconstruction has been proposed. This method, due to a certain symmetry of the matrix and the choice of the voxel…
In this paper we present a unified method for solving general polynomial equations of degree less than five.
We present an algorithm based on continuation techniques that can be applied to solve numerically minimization problems with equality constraints. We focus on problems with a great number of local minima which are hard to obtain by local…
A new version of the Graeffe algorithm for finding all the roots of univariate complex polynomials is proposed. It is obtained from the classical algorithm by a process analogous to renormalization of dynamical systems. This iteration is…
We present a numerical algorithm for finding real non-negative solutions to polynomial equations. Our methods are based on the expectation maximization and iterative proportional fitting algorithms, which are used in statistics to find…
The graphical realization of a given degree sequence and given partition adjacency matrix simultaneously is a relevant problem in data driven modeling of networks. Here we formulate common generalizations of this problem and the Exact…
We consider the numerical irreducible decomposition of a positive dimensional solution set of a polynomial system into irreducible factors. Path tracking techniques computing loops around singularities connect points on the same irreducible…
We propose a new type of multilevel method for solving eigenvalue problems based on Newton iteration. With the proposed iteration method, solving eigenvalue problem on the finest finite element space is replaced by solving a small scale…
An efficient gradient-based method to solve the volume constrained topology optimization problems is presented. Each iterate of this algorithm is obtained by the projection of a Barzilai-Borwein step onto the feasible set consisting of box…
We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…
We present a new algorithm for solving the real roots of a bivariate polynomial system $\Sigma=\{f(x,y),g(x,y)\}$ with a finite number of solutions by using a zero-matching method. The method is based on a lower bound for bivariate…
We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the…
An efficient algorithm to enumerate the vertices of a two-dimensional (2D) projection of a polytope, is presented in this paper. The proposed algorithm uses the support function of the polytope to be projected and enumerated for vertices.…
We study algorithms for solving quadratic systems of equations based on optimization methods over polytopes. Our work is inspired by a recently proposed convex formulation of the phase retrieval problem, which estimates the unknown signal…
We define a new grading, that we call the "level grading", on the algebra of polynomials generated by the derivatives $u_{k+i}=\partial^{k+i}u/\partial x^{k+i}$ over the ring $K^{(k)}$ of $C^{\infty}$ functions of $u,u_1,...,u_k$. This…
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…
Gradient descent method, as one of the major methods in numerical optimization, is the key ingredient in many machine learning algorithms. As one of the most fundamental way to solve the optimization problems, it promises the function value…