Applications of the Digital-Discrete Method in Smooth-Continuous Data Reconstruction

TL;DR

This paper explores applications of a novel digital-discrete method for smooth-continuous data reconstruction that avoids domain decomposition, demonstrating its flexibility and effectiveness on real-world and mathematical data.

Contribution

Introduces a digital-discrete approach using gradually varied functions for data reconstruction, eliminating the need for domain decomposition and extending applicability to higher dimensions.

Findings

Successfully applied to water well logs and harmonic functions on 2D manifolds

Demonstrated flexibility across six algorithms

Potential for extension to higher dimensions

Abstract

This paper presents some applications of using recently developed algorithms for smooth-continuous data reconstruction based on the digital-discrete method. The classical discrete method for data reconstruction is based on domain decomposition according to guiding (or sample) points. Then uses Splines (for polynomial) or finite elements method (for PDE) to fit the data. Our method is based on the gradually varied function that does not assume the property of the linearly separable among guiding points, i.e. no domain decomposition methods are needed. We also demonstrate the flexibility of the new method and the potential to solve variety of problems. The examples include some real data from water well logs and harmonic functions on closed 2D manifolds. This paper presented the results from six different algorithms. This method can be easily extended to higher multi-dimensions.