Monte Carlo quasi-interpolation of spherical data

TL;DR

This work develops a robust framework for spherical function approximation that blends deterministic and stochastic discretizations via scaled zonal kernels. By combining convolution-based quasi-interpolation with both quasi-Monte Carlo and Monte Carlo quadrature, and by introducing a multilevel scheme, the authors achieve stable, high-order Sobolev-space approximations on the sphere without solving linear systems. They establish Sobolev error bounds, probabilistic concentration results for MCQI, and convergence guarantees for the multilevel construction, including noisy-data scenarios. Numerical experiments on diverse point sets and kernel orders demonstrate superior noise robustness and computational efficiency compared with hyperinterpolation-based methods, highlighting practical impact for spherical data analysis in geophysics, astronomy, and graphics.

Abstract

We establish a deterministic and stochastic spherical quasi-interpolation framework featuring scaled zonal kernels derived from radial basis functions on the ambient Euclidean space. The method incorporates both quasi-Monte Carlo and Monte Carlo quadrature rules to construct easily computable quasi-interpolants, which provide efficient approximation to Sobolev-space functions for both clean and noisy data. To enhance the approximation power and robustness of our quasi-interpolants, we develop a multilevel method in which quasi-interpolants constructed with graded resolutions join force to reduce the error of approximation. In addition, we derive probabilistic concentration inequalities for our quasi-interpolants in pertinent stochastic settings. The construction of our quasi-interpolants does not require solving any linear system of equations. Numerical experiments show that our quasi-interpolation algorithm is more stable and robust against noise than comparable ones in the literature.

Figures (6)

• Figure 1: Numerical errors and convergence rates of SKQI using Gaussian kernels with orders $m=2,4,6$ for approximating spherical harmonic $\mathcal{Y}_{6,4}$ on RD, MD, GS and TD point sets.
• Figure 2: Numerical errors and convergence rates of SKQI using compactly-supported kernels with $m=2,4,6$ for approximating spherical harmonic $\mathcal{Y}_{6,4}$ on RD, MD, GS and TD point sets.
• Figure 3: $L_\infty$ and $L_2$ errors for approximating $\mathcal{Y}_{6,4}$ using single-level and multilevel SKQI methods with compactly-supported kernels ($m=2,4$) on MD point sets. The left figure corresponds to $L_{\infty}$ error, and the right corresponds to $L_2$ error.
• Figure 4: $L_\infty$ and $L_2$ errors for approximating Franke function \ref{['eq:frank']} under various noise levels ($\sigma_{\bm{\varepsilon}}=0.001, 0.01, 0.1$) using single-level and multilevel SKQI methods with the compactly-supported kernel ($m=2$) on MD point sets.
• Figure 5: $L_2$ errors of FHI, QMCQI ($m=2,4$) and MCQI ($m=2,4$) for approximating Franke function \ref{['eq:frank']} under two noise levels, $\sigma_{\varepsilon}=0.01$ and $\sigma_{\varepsilon}=0.1$.

Theorems & Definitions (30)

• Lemma 2.3
• Lemma 2.4
• Proof 1
• Lemma 2.5
• Proof 2
• Lemma 3.1
• Theorem 3.2
• Proof 3
• Corollary 3.3
• Proof 4