Approximate polynomial expansion for joint density

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

We investigate the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs, and it is a subset of the family of graphs that have polynomial expansion. We present efficient (1 + ε)-approximation algorithms for polynomial expansion graphs, for Independent Set, Set Cover, and Dominating Set problems, among ot...

Approximate Optimal Designs for Multivariate Polynomial Regression

Abstract: We introduce a new approach aiming at computing approximate optimal designs for multivariate polynomial regressions on compact (semi-algebraic) design spaces. We use the moment-sum-of-squares hierarchy of semidefinite programming problems to solve numerically the approximate optimal design problem. The geometry of the design is recovered via semidefinite programming duality theory. Th...

Approximate Fekete Points for Weighted Polynomial Interpolation

We compute approximate Fekete points for weighted polynomial interpolation by a recent algorithm based on QR factorizations of Vandermonde matrices. We consider in particular the case of univariate and bivariate functions with prescribed poles or other singularities, which are absorbed in the basis by a weight function. Moreover, we apply the method to the construction of real and complex weigh...

Approximate Reachability Computation for Polynomial Systems

In this paper we propose an algorithm for approximating the reachable sets of systems defined by polynomial differential equations. Such systems can be used to model a variety of physical phenomena. We first derive an integration scheme that approximates the state reachable in one time step by applying some polynomial map to the current state. In order to use this scheme to compute all the stat...

Explicit Polynomial Expansion