Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multivariate Splines and Their Applications

Ming-Jun Lai

TL;DR

The paper surveys multivariate splines, emphasizing Schumaker’s foundational contributions, and surveys definitions, dimensions, approximation orders, and computation. It demonstrates how high-order, smooth splines can solve linear and nonlinear PDEs via spline collocation and constrained minimization, and shows practical curve/surface fitting and medical data applications. A key focus is overcoming the curse of dimensionality through Kolmogorov–superposition-based KB/LKB splines, enabling denoising and high-dimensional function approximation with tractable data. Numerical results include Poisson and Monge–Ampère related computations, spherical splines for geopotential reconstruction, and 3D data fitting, underscoring the practical impact of multivariate splines in analysis and computation.

Abstract

This paper begins by reviewing numerous theoretical advancements in the field of multivariate splines, primarily contributed by Professor Larry L. Schumaker. These foundational results have paved the way for a wide range of applications and computational techniques. The paper then proceeds to highlight various practical applications of multivariate splines. These include scattered data fitting and interpolation, the construction of smooth curves and surfaces, and the numerical solutions of various partial differential equations, encompassing both linear and nonlinear PDEs. Beyond these conventional and well-established uses, the paper introduces a novel application of multivariate splines in function value denoising. This innovative approach facilitates the creation of LKB splines, which are instrumental in approximating high-dimensional functions and effectively circumventing the curse of dimensionality.

Multivariate Splines and Their Applications

TL;DR

The paper surveys multivariate splines, emphasizing Schumaker’s foundational contributions, and surveys definitions, dimensions, approximation orders, and computation. It demonstrates how high-order, smooth splines can solve linear and nonlinear PDEs via spline collocation and constrained minimization, and shows practical curve/surface fitting and medical data applications. A key focus is overcoming the curse of dimensionality through Kolmogorov–superposition-based KB/LKB splines, enabling denoising and high-dimensional function approximation with tractable data. Numerical results include Poisson and Monge–Ampère related computations, spherical splines for geopotential reconstruction, and 3D data fitting, underscoring the practical impact of multivariate splines in analysis and computation.

Abstract

This paper begins by reviewing numerous theoretical advancements in the field of multivariate splines, primarily contributed by Professor Larry L. Schumaker. These foundational results have paved the way for a wide range of applications and computational techniques. The paper then proceeds to highlight various practical applications of multivariate splines. These include scattered data fitting and interpolation, the construction of smooth curves and surfaces, and the numerical solutions of various partial differential equations, encompassing both linear and nonlinear PDEs. Beyond these conventional and well-established uses, the paper introduces a novel application of multivariate splines in function value denoising. This innovative approach facilitates the creation of LKB splines, which are instrumental in approximating high-dimensional functions and effectively circumventing the curse of dimensionality.
Paper Structure (14 sections, 12 theorems, 49 equations, 15 figures, 6 tables)

This paper contains 14 sections, 12 theorems, 49 equations, 15 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Definition of Multivariate Splines and Main Research Problems
  3. Dimension of Spline Spaces
  4. Spherical Spline Functions
  5. Approximation Order of Spline Spaces
  6. Computational Methods for Multivariate Splines
  7. Solutions of Linear Partial Differential Equations
  8. Construction of Smooth Curves and Surfaces
  9. Trivariate Splines for Medical Data Fitting and Surface Construction
  10. Numerical Solution of the Monge-Ampere Equation
  11. Existence, Uniqueness, and Regularity
  12. Numerical Solutions of the Monge-Ampére equation
  13. Overcome the Curse of Dimensionality
  14. Conclusions and Remarks

Key Result

Theorem 1

Suppose that $\triangle$ is a triangulation of a given domain $\Omega\subset \mathbb{R}^2$. For any $0\le r\le d$, where and $\sigma_v=\sum_{j=1}^{d-r} (r+j+1-jm_v)_+$ and $\tilde{\sigma}_v= \sum_{j=1}^{d-r} (r+j+1-j \tilde{m}_v)_+$.

Figures (15)

  • Figure 1: A poster (left) is deformed to the one on oval shaped domain (right) like a printed balloon
  • Figure 2: A set of data locations associated with the mercury pollution as explained in WWLG20
  • Figure 3: An associated triangulation of the given data set in Figure \ref{['data1']}
  • Figure 4: Geo-potential measurement locations and a triangulation of the earth
  • Figure 5: Geopotential data values (left) and spherical spline surface of the data (right)
  • ...and 10 more figures

Theorems & Definitions (20)

  • Theorem 1: Schumaker, 1979S79 and Schumaker, 1984S84
  • Theorem 2: Alfeld, Schumaker, and Sirvent, 1992ASS92
  • Example 1: Geopotential Reconstruction (cf. LSBW09
  • Theorem 3: Lai and Schumaker 1998LS98
  • Theorem 4: Neamtu and Schumaker, 2004NS04
  • Theorem 5: Lai and Lee, 2022LL22
  • Example 2: Multiple Curves
  • Example 3: Curves with Corners
  • Example 4: Construction of Smooth Surfaces
  • Theorem 6: The Brenier Theorem
  • ...and 10 more