[Paper Review] A non grid-based interpolation scheme for the eigenvalue problem

Kai Lin, Wei‐Liang Qian|arXiv (Cornell University)|Sep 8, 2016

Advanced Numerical Analysis Techniques2 references12 citations

TL;DR

This paper proposes a non-grid-based interpolation scheme using local scattered data and Taylor series expansions to approximate function values and derivatives at a query point. By solving a linear system derived from Taylor expansions around nearby points, the method enables adaptive precision control and achieves high accuracy in solving eigenvalue problems for quantum mechanical potentials, as validated by comparisons with exact solutions.

ABSTRACT

We propose a non grid-based interpolation scheme based on the information from the data collected from the vicinity of the query point. As a non-grid-based interpolation, the data points can be distributed randomly in a small region, and the interpolation is constructed so that it naturally makes use of the information not only on the function value but also on its higher order derivatives. The main advantage of the present approach is that the precision of the interpolation can be adjusted in accordance to the quantity of the data, in other words, a balance between the precision and the computational cost can be achieved by properly choosing the size of the neighborhood where the data points are collected. The method is applicable to univariate as well as multivariate functions. We show that the proposed scheme is efficient and precise. The present approach is then employed to study the eigenvalue problem.

Motivation & Objective

• To develop a flexible interpolation method that does not require regular grid data distributions.
• To enable adaptive precision control in interpolation by adjusting the number of nearby data points.
• To apply the interpolation scheme to solve differential equations and eigenvalue problems in quantum mechanics.
• To demonstrate the method's efficiency and accuracy through numerical comparisons with standard software.
• To explore applications in physics where scattered data or irregular sampling is common, such as quasinormal modes in general relativity.

Proposed method

• The method uses Taylor series expansions of a function about a query point to relate function values and derivatives at nearby scattered data points.
• It formulates a linear system F = MD, where F contains function values at N scattered points, D contains the function and its derivatives at the query point, and M contains coefficients derived from relative coordinate differences.
• The unknown vector D is solved via matrix inversion, enabling interpolation at the query point using only local data.
• The scheme is applicable to both univariate and multivariate functions, with the order of approximation controlled by the number of terms in the Taylor expansion.
• The method naturally incorporates higher-order derivative information, improving accuracy without requiring structured grids.
• The neighborhood size and number of data points can be tuned to balance computational cost and precision.

Experimental results

Research questions

• RQ1 Can a non-grid-based interpolation method achieve high accuracy using only scattered data points near a query location?
• RQ2 How does the precision of the interpolation scale with the number of nearby data points?
• RQ3 Can this interpolation scheme be effectively applied to solve differential equations and eigenvalue problems in quantum mechanics?
• RQ4 How does the method compare in accuracy and efficiency to standard numerical solvers like MATLAB or Mathematica?
• RQ5 What are the practical advantages of this method in scenarios with irregular or sparse data sampling?

Key findings

• For the Pöschl-Teller potential, the method achieved relative errors below 1% for the first three eigenvalues, with the largest deviation being 0.5% for (k,λ)=(10,10).
• For the trigonometric potential, the method produced results within 0.05% error for V_a=1000, with eigenvalues E_1 and E_2 matching exact values to four significant digits.
• The method showed consistent convergence, with eigenvalue estimates approaching exact values as the number of data points increased.
• In the case of V_a=1000, the computed E_1=104.402 was within 0.001 of the exact E_1=104.403, demonstrating high precision.
• For the trigonometric potential with V_a=10, the method slightly overestimated E_1 (15.9083 vs. 16.0275), but the error was less than 0.8%.
• The method maintained good performance even for high potential values, with E_4 for V_a=1000 deviating by only 0.1% from the exact value.