Multidimensional spline integration of scattered data

TL;DR

This paper presents a novel numerical method for reconstructing smooth, multidimensional surfaces from scattered gradient data, accommodating non-uniform inputs and enabling easy computation of derivatives and integrals.

Contribution

The method introduces a multidimensional spline approach that minimizes deviation from measured gradients, providing a continuous surface with error estimation, unlike traditional path-based integration.

Findings

Produces smooth, continuous surfaces from scattered gradient data

Handles non-equidistant, irregular input data

Includes error estimation for statistical and systematic uncertainties

Abstract

We introduce a numerical method for reconstructing a multidimensional surface using the gradient of the surface measured at some values of the coordinates. The method consists of defining a multidimensional spline function and minimizing the deviation between its derivatives and the measured gradient. Unlike a multidimensional integration along some path, the present method results in a continuous, smooth surface, furthermore, it also applies to input data that are non-equidistant and not aligned on a rectangular grid. Function values, first and second derivatives and integrals are easy to calculate. The proper estimation of the statistical and systematical errors is also incorporated in the method.