Related papers: Homotopy methods for multiplication modulo triangu…
The multiplication of matrices is an important arithmetic operation in computational mathematics. In the context of hierarchical matrices, this operation can be realized by the multiplication of structured block-wise low-rank matrices,…
Homotopy optimization is a traditional method to deal with a complicated optimization problem by solving a sequence of easy-to-hard surrogate subproblems. However, this method can be very sensitive to the continuation schedule design and…
In this paper we introduce a generic model for multiplicative algorithms which is suitable for the MapReduce parallel programming paradigm. We implement three typical machine learning algorithms to demonstrate how similarity comparison,…
Numerical algebraic geometry provides a number of efficient tools for approximating the solutions of polynomial systems. One such tool is the parameter homotopy, which can be an extremely efficient method to solve numerous polynomial…
We study the algebraic dynamical systems generated by triangular systems of rational functions and estimate the height growth of iterations generated by such systems. Further, using a result on the reduction modulo primes of systems of…
We study the graded polynomial identities with a homogeneous involution on the algebra of upper triangular matrices endowed with a fine group grading. We compute their polynomial identities and a basis of the relatively free algebra,…
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…
Using three different approaches, we analyze the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach consists in the study of the images of lines,…
Already for bivariate tropical polynomials, factorization is an NP-Complete problem. In this paper, we give an efficient algorithm for factorization and rational factorization of a rich class of tropical polynomials in $n$ variables.…
This is a survey on the computational complexity of nonlinear mixed-integer optimization. It highlights a selection of important topics, ranging from incomputability results that arise from number theory and logic, to recently obtained…
The matrix formation associated to high-order discretizations is known to be numerically demanding. Based on the existing procedure of interpolation and lookup, we design a multiscale assembly procedure to reduce the exorbitant assembly…
Multilinear systems of equations arise in various applications, such as numerical partial differential equations, data mining, and tensor complementarity problems. In this paper, we propose a homotopy method for finding the unique positive…
It is well-known how to compute the structure of the second homotopy group of a space, $X$, as a module over the fundamental group, $\pi_1X$, using the homology of the universal cover and the Hurewicz isomorphism. We describe a new method…
First order shape optimization methods, in general, require a large number of iterations until they reach a locally optimal design. While higher order methods can significantly reduce the number of iterations, they exhibit only local…
We introduce the notion of a strongly homotopy-comultiplicative resolution of a module coalgebra over a chain Hopf algebra, which we apply to proving a comultiplicative enrichment of a well-known theorem of Moore concerning the homology of…
We present fully polynomial approximation schemes for a broad class of Holant problems with complex edge weights, which we call Holant polynomials. We transform these problems into partition functions of abstract combinatorial structures…
We provide elementary identities relating the three known types of non-symmetric interpolation Macdonald polynomials. In addition we derive a duality for non-symmetric interpolation Macdonald polynomials. We consider some applications of…
We provide formulas and algorithms for computing the excess numbers of certain ideals. The solution for monomial ideals is given by the mixed volumes of certain polytopes. These results enable us to design specific homotopies for numerical…
In this work, Lienard equations are considered. The limit cycles of these systems are studied by applying the homotopy analysis method. The amplitude and frequency obtained with this methodology are in good agreement with those calculated…
In this paper, we develop fast multipole methods for 3D Helmholtz kernel in layered media. Two algorithms based on different forms of Taylor expansion of layered media Green's function are developed. A key component of the first algorithm…