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Unisolvence of randomized MultiQuadric Kansa collocation for convection-diffusion with mixed boundary conditions

Maryam Mohammadi, Alvise Sommariva, Marco Vianello

TL;DR

The paper tackles unisolvence for unsymmetric Kansa collocation by proving that the MQ Kansa discretization for stationary convection-diffusion with mixed Dirichlet-Neumann boundaries is almost surely nonsingular when using fixed collocation points and randomly drawn fictitious centers with a density $\sigma$. The approach relies on an interpolation-in-analytic-spaces lemma and a complex-embedding argument to establish linear independence of the interpolation functions, ensuring almost-sure unisolvence for general domains. Key contributions include extending prior results from Polyharmonic Splines to MQ RBFs, accommodating general center distributions (not limited to i.i.d. centers), and providing a 2D numerical illustration with centers perturbed near collocation points. The work offers theoretical guarantees and practical insight for employing randomized Kansa collocation with MQs in convection-diffusion problems, including Neumann boundary terms, while noting conditioning and parameter considerations in practice.

Abstract

We make a further step in the open problem of unisolvence for unsymmetric Kansa collocation, proving that the MultiQuadric Kansa method with fixed collocation points and random fictitious centers is almost surely unisolvent, for stationary convection-diffusion equations with mixed boundary conditions on general domains. For the purpose of illustration, the method is applied in 2D with fictitious centers that are local random perturbations of predetermined collocation points.

Unisolvence of randomized MultiQuadric Kansa collocation for convection-diffusion with mixed boundary conditions

TL;DR

The paper tackles unisolvence for unsymmetric Kansa collocation by proving that the MQ Kansa discretization for stationary convection-diffusion with mixed Dirichlet-Neumann boundaries is almost surely nonsingular when using fixed collocation points and randomly drawn fictitious centers with a density . The approach relies on an interpolation-in-analytic-spaces lemma and a complex-embedding argument to establish linear independence of the interpolation functions, ensuring almost-sure unisolvence for general domains. Key contributions include extending prior results from Polyharmonic Splines to MQ RBFs, accommodating general center distributions (not limited to i.i.d. centers), and providing a 2D numerical illustration with centers perturbed near collocation points. The work offers theoretical guarantees and practical insight for employing randomized Kansa collocation with MQs in convection-diffusion problems, including Neumann boundary terms, while noting conditioning and parameter considerations in practice.

Abstract

We make a further step in the open problem of unisolvence for unsymmetric Kansa collocation, proving that the MultiQuadric Kansa method with fixed collocation points and random fictitious centers is almost surely unisolvent, for stationary convection-diffusion equations with mixed boundary conditions on general domains. For the purpose of illustration, the method is applied in 2D with fictitious centers that are local random perturbations of predetermined collocation points.

Paper Structure

This paper contains 3 sections, 2 theorems, 16 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Unisolvence of MQ Kansa collocation
  3. Numerical examples

Key Result

Lemma 1

Let $A\subseteq \mathbb{R}^d$, be an open connected set and $\{f_j\}_{1\leq j\leq N}$ be linearly independent real analytic functions in $A$. Then the set of non-unisolvent $N$-uples for interpolation in $\hbox{span}\{f_1,\dots,f_N\}$ has null Lebesgue measure in $A^N$.

Figures (1)

  • Figure 1: $441$ collocation grid points (left) and the random fictitious centers distribution (right) for $\delta=0.01$.

Theorems & Definitions (3)

  • Lemma 1
  • Theorem 1
  • Remark 1