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In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic…
To enhance solution accuracy and training efficiency in neural network approximation to partial differential equations, partitioned neural networks can be used as a solution surrogate instead of a single large and deep neural network…
In this paper, we are concerned with the problem of determining the existence of multiple equilibria in economic models. We propose a general and complete approach for identifying multiplicities of equilibria in semi-algebraic economies,…
We propose a presentation of classical propositional tableaux elaborated by application of methods that are noteworthy in program design, namely program derivation with separation of concerns. We start by deriving from a straightforward…
Cylindrical algebraic decomposition (CAD) is an important tool for the investigation of semi-algebraic sets, with applications within algebraic geometry and beyond. We recently reported on a new implementation of CAD in Maple which…
The reduction of computational costs in the numerical solution of nonstationary problems is achieved through splitting schemes. In this case, solving a set of less computationally complex problems provides the transition to a new level in…
We propose a decomposition framework for the parallel optimization of the sum of a differentiable (possibly nonconvex) function and a (block) separable nonsmooth, convex one. The latter term is usually employed to enforce structure in the…
As an attempt to bridge between numerical analysis and algebraic geometry, this paper formulates the multiplicity for the general nonlinear system at an isolated zero, presents an algorithm for computing the multiplicity structure, proposes…
Inspired by rational canonical forms, we introduce and analyze two decompositions of dynamic programming (DP) problems for systems with linear dynamics. Specifically, we consider both finite and infinite horizon DP problems in which the…
In this paper we propose a new inexact dual decomposition algorithm for solving separable convex optimization problems. This algorithm is a combination of three techniques: dual Lagrangian decomposition, smoothing and excessive gap. The…
We consider a composite open quantum system consisting of a fast subsystem coupled to a slow one. Using the time-scale separation, we develop an adiabatic elimination technique to derive at any order the reduced model describing the slow…
Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which…
In this work, we explore physical systems which support not only multipartite interparticle entanglement, but also intraparticle entanglement between different degrees of freedom of the constituent particles and entanglement between…
Joint diagonalization, the process of finding a shared set of approximate eigenvectors for a collection of matrices, arises in diverse applications such as multidimensional harmonic analysis or quantum information theory. This task is…
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 introduce semiframes (an algebraic structure) and investigate their duality with semitopologies (a topological one). Both semitopologies and semiframes are relatively recent developments, arising from a novel application of topological…
While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention…
Given an approximation to a multiple isolated solution of a polynomial system of equations, we have provided a symbolic-numeric deflation algorithm to restore the quadratic convergence of Newton's method. Using first-order derivatives of…
We derive a combined analytical and numerical scheme to solve the (1+1)-dimensional differential Kirchhoff system. Here the object is to obtain an accurate as well as an efficient solution process. Purely numerical algorithms typically have…
The approximate joint diagonalization of a set of matrices consists in finding a basis in which these matrices are as diagonal as possible. This problem naturally appears in several statistical learning tasks such as blind signal…