Related papers: Polynomial meshes on algebraic sets
Polynomial inequalities lie at the heart of many mathematical disciplines. In this paper, we consider the fundamental computational task of automatically searching for proofs of polynomial inequalities. We adopt the framework of…
We enhance the approximation capabilities of algebraic polynomials by composing them with homeomorphisms. This composition yields families of functions that remain dense in the space of continuous functions, while enabling more accurate…
We show that minimizing a convex function over the integer points of a bounded convex set is polynomial in fixed dimension.
In this work, we introduce new families of nonconforming approximation methods for reconstructing functions on general polygonal meshes. These methods are defined using degrees of freedom based on weighted moments of orthogonal polynomials…
We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for…
A polynomial matrix inequality is a formula asserting that a polynomial matrix is positive semidefinite. Polynomial matrix optimization concerns minimizing the smallest eigenvalue of a symmetric polynomial matrix subject to a tuple of…
The problem of minimizing a polynomial over a set of polynomial inequalities is an NP-hard non-convex problem. Thanks to powerful results from real algebraic geometry, one can convert this problem into a nested sequence of…
We solve the following problem: given a polynomial of order $n$ and the corresponding $B\'ezier$ tensor product patches over an unstructured regular quadrilateral mesh of any valence, find a solution to the $G^{1}1$ or $C^{1}1$…
Motivated by applications in robotics and computer vision, we study problems related to spatial reasoning of a 3D environment using sublevel sets of polynomials. These include: tightly containing a cloud of points (e.g., representing an…
Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required…
We here specialize the standard matrix-valued polynomial interpolation to the case where on the imaginary axis the interpolating polynomials admit various symmetries: Positive semidefinite, Skew-Hermitian, $J$-Hermitian, Hamiltonian and…
Computational analysis with the finite element method requires geometrically accurate meshes. It is well known that high-order meshes can accurately capture curved surfaces with fewer degrees of freedom in comparison to low-order meshes.…
We focus on two central themes in this dissertation. The first one is on decomposing polytopes and polynomials in ways that allow us to perform nonlinear optimization. We start off by explaining important results on decomposing a polytope…
We investigate the uniform approximation provided by least squares polynomials on the unit Euclidean sphere $\mathbb{S}^q$ in $\mathbb{R}^{q+1}$, with $q\ge 2$. Like any other polynomial projection, the study concerns the growth, as the…
We introduce a novel method for bounding high-order multi-dimensional polynomials in finite element approximations. The method involves precomputing optimal piecewise-linear bounding boxes for polynomial basis functions, which can then be…
Using the notion of Dubiner distance, we give an elementary proof of the fact that good covering point configurations on the 2-sphere are optimal polynomial meshes. From these we extract Caratheodory-Tchakaloff (CATCH) submeshes for…
Only a few numerical methods can treat boundary value problems on polygonal and polyhedral meshes. The BEM-based Finite Element Method is one of the new discretization strategies, which make use of and benefits from the flexibility of these…
We expand the Chebyshev polynomials and some of its linear combination in linear combinations of the q-Hermite, the Rogers (q-utraspherical) and the Al-Salam--Chihara polynomials and vice versa. We use these expansions to obtain expansions…
Partially ordered sets (posets) play a universal role as an abstract structure in many areas of mathematics. For finite posets, an explicit enumeration of distinct partial orders on a set of unlabelled elements is known only up to a…
We give new positive and negative results (some conditional) on speeding up computational algebraic geometry over the reals: (1) A new and sharper upper bound on the number of connected components of a semialgebraic set. Our bound is novel…