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

TL;DR

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

Contribution

The paper presents a new algorithm based on gradually varied functions that simplifies data reconstruction without domain decomposition, applicable to various data types and higher dimensions.

Findings

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

Demonstrated flexibility across six different algorithms

Potential for extension to higher dimensions

Abstract

This paper presents some applications 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 the Spline method (for polynomial) or finite elements method (for PDE) is used to fit the data. Our method is based on the gradually varied function that does not assume the property of being linearly separable among guiding points, i.e. no domain decomposition methods are needed. We also demonstrate the flexibility of the new method and its potential to solve a variety of problems. The examples include some real data from water well logs and harmonic functions on closed 2D manifolds. This paper presents the results from six different algorithms. This method can be easily extended to higher multi-dimensions. We also include an advanced consideration related to the use of gradually varied mapping.