Related papers: Robust Numerical Algebraic Geometry
Real algebraic geometry adapts the methods and ideas from (complex) algebraic geometry to study the real solutions to systems of polynomial equations and polynomial inequalities. As it is the real solutions to such systems modeling…
Enumerative Geometry is concerned with the number of solutions to a structured system of polynomial equations, when the structure comes from geometry. Enumerative real algebraic geometry studies real solutions to such systems, particularly…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation, and model selection. These are all optimization problems, well-known to be challenging due to…
We introduce the notion of a robust parameterized arithmetic circuit for the evaluation of algebraic families of multivariate polynomials. Based on this notion, we present a computation model, adapted to Scientific Computing, which captures…
Many combinatorial problems can be formulated as a polynomial optimization problem that can be solved by state-of-the-art methods in real algebraic geometry. In this paper we explain many important methods from real algebraic geometry, we…
Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semi-algebraic subsets of the…
Existing structural analysis methods may fail to find all hidden constraints for a system of differential-algebraic equations with parameters if the system is structurally unamenable for certain values of the parameters. In this paper, for…
In 1970s, a method was developed for integration of nonlinear equations by means of algebraic geometry. Starting from a Lax representation with spectral parameter, the algebro-geometric method allows to solve the system explicitly in terms…
The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gr\"{o}bner) methods are severely limited by algorithmic issues…
We produce algorithms to detect whether a complex affine variety computed and presented numerically by the machinery of numerical algebraic geometry corresponds to an associated component of a polynomial ideal.
Geometric number systems, obtained by extending the real number system to include new anticommuting square roots of +1 and -1, provide a royal road to higher mathematics by largely sidestepping the tedious languages of tensor analysis and…
The Numerical Assembly Technique is extended to investigate arbitrary planar frame structures with the focus on the computation of natural frequencies. This allows us to obtain highly accurate results without resorting to spatial…
An approximate formulation of a robust geometric program (RGP) as a convex program is proposed. Interest in using geometric programs (GPs) to model complex engineering systems has been growing, and this has motivated explicitly modeling the…
The goal of this paper is to provide computational tools able to find a solution of a system of polynomial inequalities. The set of inequalities is reformulated as a system of polynomial equations. Three different methods, two of which…
Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equations based on the classical theorem of B\'ezout. This was…
Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry…
Deep learning models are often considered black boxes due to their complex hierarchical transformations. Identifying suitable architectures is crucial for maximizing predictive performance with limited data. Understanding the geometric…
The concept of viewing graph solvability has gained significant interest in the context of structure-from-motion. A viewing graph is a mathematical structure where nodes are associated to cameras and edges represent the epipolar geometry…
The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI),…