The area-based algorithm transforms several GMM with varying number of components into sets of equivalent polynomial regression splines.
This surprising application is backed by a recent success on Picard group computation.Īpplications include algebraic geometry, with the development of computational tools to experiment on concrete examples and build databases that document the largest possible range of behavior. In this study, we propose two algorithms rooted in numerical algebraic geometry (NAG), namely, an area-based algorithm and a local maxima algorithm, to identify the optimal number of components. Building upon transcendental continuation, I propose to design algorithms to compute certain algebraic invariants of complex varieties related to algebraic cycles and Hodge classes, far beyond the current reach of symbolic methods. In this article, we will give an overview of the new approach and some of the systems that have been successfully investigated and solved by these new methods.The new approach has. volume of semialgebraic sets, or periods of complex varieties) with rigorous error bounds and high precision, more than thousands of digits. In the last few years, methods of numerical algebraic geometry have begun to be used to investigate and solve systems of discretized nonlinear differential equations. A curve C on P 1 × E is said has projection degree ( d, e), if the projections C P 1 and C E are of degree d and e respectively. and the example 3 in lecture 1 in Mumfords book Lectures on curves on an algebraic surface. This would enable the computation of many integrals (e.g. Algebraic equivalence VS Numerical Equivalence - An Example.
#NUMERICAL ALGEBRAIC GEOMETRY SOFTWARE#
The joint development of theoretical aspects, algorithms and software implementations will turn these proof-of-concepts into breakthroughs.Ĭoncretely, I will develop a theory of transcendental continuation for the numerical computation of a wide range of multiple integrals, based on a striking combination of algebraic geometry, symbolic algorithms and numerical ODE solvers. The numerical difficulties are overcome if we work in the true synthetic spirit of the Schubert calculus, selecting the numerically most favorable equations to. What is the probability that a given spacecraft collides with a debris? How many smooth rational curves of degree 4 lie on a given quartic surface? These are questions with an underlying algebraic model and longing for computational answers.īased on recent proof-of-concept works, I propose new foundational methods in numerical nonlinear algebra, motivated by the need for reliability and applicability. Numerical algebraic geometry was founded 50 on the recognition that often one may be interested in positive-dimensional solution sets (curves, surfaces, etc.). As of applications, nonlinearity is also a formidable computational challenge.
Modeling nonlinear constraints by polynomial equations and inequalities raises fundamental theoretical issues, many of which have been answered by algebraic geometry. Volume 496, 2009 Polyhedral Methods in Numerical Algebraic Geometry Jan Verschelde To Andrew Sommese, on his 60th birthday Abstract.
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