Multivariate spline and algebraic geometry

Multivariate Splines and Algebraic Geometry

Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-Béz...

Using Algebraic Geometry for Multivariate Hermite Interpolation

This paper uses some well known theorems of algebraic geometry to characterize polynomial Hermite interpolation in any dimension. Efficient numerical algorithms are presented for interpolatory curve fits through points in the plane, surface fits through points and curves in space, and in general, hypersuface fits through. points, curves, surfaces, and sub-varieties in n dimensional space. These...

Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry

We study multivariate normal models that are described by linear constraints on the inverse of the covariance matrix. Maximum likelihood estimation for such models leads to the problem of maximizing the determinant function over a spectrahedron, and to the problem of characterizing the image of the positive definite cone under an arbitrary linear projection. These problems at the interface of s...

Introductory Geometry: Algebraic Geometry

What is a multivariate spline?

The various concepts and ideas that have contributed to univariate spline theory are considered with a view to finding a suitable definition of a multivariate spline. In this way, an overview of the existing more or less complete univariate spline theory is given along with a survey of some of the high points of the current research in multivariate splines. My very first paper dealt with multiv...