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Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton's method takes into…
We provide an algorithm that takes as an input a given parametric family of homogeneous polynomials, which is invariant under the action of the general linear group, and an integer $d$. It outputs the ideal of that family intersected with…
We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. We show that they essentially can be reduced to two computer…
We propose a new hybrid topology optimization algorithm based on multigrid approach that combines the parallelization strategy of CPU using OpenMP and heavily multithreading capabilities of modern Graphics Processing Units (GPU). In…
Synthesis problems for linkages in kinematics often yield large structured parameterized polynomial systems which generically have far fewer solutions than traditional upper bounds would suggest. This paper describes statistical models for…
Modeling of microlensing events poses computational challenges for the resolution of the lens equation and the high dimensionality of the parameter space. In particular, numerical noise represents a severe limitation to fast and efficient…
Given an undirected graph G, the edge orientation problem asks for assigning a direction to each edge to convert G into a directed graph. The aim is to minimize the maximum out degree of a vertex in the resulting directed graph. This…
Multiobjective discrete programming is a well-known family of optimization problems with a large spectrum of applications. The linear case has been tackled by many authors during the last years. However, the polynomial case has not been…
This paper presents new six solutions for sixth degree polynomial equation in general forms basing on new theorems, where the possibility to calculate the six roots of any sixth degree equation nearly simultaneously. The proposed roots for…
The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, yields insight into the hierarchy of complexity classes. In particular, the theory produces algebraic tests for a…
Polynomial preconditioning can improve the convergence of the Arnoldi method for computing eigenvalues. Such preconditioning significantly reduces the cost of orthogonalization; for difficult problems, it can also reduce the number of…
In this work, we introduce new families of nonconforming approximation methods for reconstructing functions on general polygonal meshes. These methods are defined using degrees of freedom based on weighted moments of orthogonal polynomials…
An iterative method is derived for image reconstruction. Among other attributes, this method allows constraints unrelated to the radiation measurements to be incorporated into the reconstructed image. A comparison is made with the widely…
We present a new algorithm for reconstructing an exact algebraic number from its approximate value using an improved parameterized integer relation construction method. Our result is consistent with the existence of error controlling on…
In this paper I present a kind of proof for classical Euclidean geometric problems which relies on both synthetic and analytic geometry. Using the elementary tools of polynomial algebra and multivariate calculus we manage to reduce the…
Given a zero-dimensional polynomial system consisting of n integer polynomials in n variables, we propose a certified and complete method to compute all complex solutions of the system as well as a corresponding separating linear form l…
The representation of polynomials by arithmetic circuits evaluating them is an alternative data structure which allowed considerable progress in polynomial equation solving in the last fifteen years. We present a circuit based computation…
Progressive Hedging is a popular decomposition algorithm for solving multi-stage stochastic optimization problems. A computational bottleneck of this algorithm is that all scenario subproblems have to be solved at each iteration. In this…
This work studies the problem of maximizing a higher degree real homogeneous multivariate polynomial over the unit sphere. This problem is equivalent to finding the leading eigenvalue of the associated symmetric tensor of higher order,…
In this paper, we introduce the new optimal perturbation iteration method based on the perturbation iteration algorithms for the approximate solutions of nonlinear differential equations of many types. The proposed method is illustrated by…