Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonsingularity of unsymmetric Kansa matrices: random collocation by MultiQuadrics and Inverse MultiQuadrics

R. Cavoretto, F. Dell'Accio, A. De Rossi, A. Sommariva, M. Vianello

Abstract

Unisolvence of unsymmetric Kansa collocation is still a substantially open problem. We prove that Kansa matrices with MultiQuadrics and Inverse MultiQuadrics for the Dirichlet problem of the Poisson equation are almost surely nonsingular, when the collocation points are chosen by any continuous random distribution in the domain interior and arbitrarily on its boundary.

Nonsingularity of unsymmetric Kansa matrices: random collocation by MultiQuadrics and Inverse MultiQuadrics

Abstract

Unisolvence of unsymmetric Kansa collocation is still a substantially open problem. We prove that Kansa matrices with MultiQuadrics and Inverse MultiQuadrics for the Dirichlet problem of the Poisson equation are almost surely nonsingular, when the collocation points are chosen by any continuous random distribution in the domain interior and arbitrarily on its boundary.
Paper Structure (2 sections, 1 theorem, 32 equations)

This paper contains 2 sections, 1 theorem, 32 equations.

Table of Contents

  1. Introduction
  2. Unisolvence of random MQ and IMQ Kansa collocation

Key Result

Theorem 1

Let $K_n$ be the MQ or IMQ Kansa collocation matrix defined above, where $\{Q_h\}$ is any fixed set of $m$ distinct points on $\partial\Omega$, and $\{P_i\}$ is a sequence of i.i.d. (independent and identically distributed) random points in $\Omega$ with respect to any probability density $\sigma \i

Theorems & Definitions (1)

  • Theorem 1