Multiscale scattered data analysis in samplet coordinates

2024-09-23

arXiv.org

AI Summary

For interpolation of large-scale scattered data using Matérn-type radial basis functions (RBFs), this paper proposes an efficient algorithm based on multilevel residual correction and samplet coordinate representation. The method constructs a hierarchical system by incorporating samplets—local discrete signed measures with vanishing moments—into the Matérn multiresolution framework, ensuring uniform boundedness of the condition number and reducing overall complexity to (O(N log^2 N)). Key components include multilevel residual decomposition, samplet coordinate transformation, diagonal scaling preconditioning, and sparse approximation of the generalized Vandermonde matrix. Theoretical analysis establishes both the well-conditionedness of the resulting system and controllable interpolation error. Numerical experiments in two and three dimensions confirm near-linear-logarithmic scaling in both matrix assembly and solver time, significantly outperforming conventional RBF approaches.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We study multiscale scattered data interpolation schemes for globally supported radial basis functions with focus on the Mat'ern class. The multiscale approximation is constructed through a sequence of residual corrections, where radial basis functions with different lengthscale parameters are combined to capture varying levels of detail. We prove that the condition numbers of the the diagonal blocks of the corresponding multiscale system remain bounded independently of the particular level, allowing us to use an iterative solver with a bounded number of iterations for the numerical solution. Employing an appropriate diagonal scaling, the multiscale system becomes well conditioned. We exploit this fact to derive a general error estimate bounding the consistency error issuing from a numerical approximation of the multiscale system. To apply the multiscale approach to large data sets, we suggest to represent each level of the multiscale system in samplet coordinates. Samplets are localized, discrete signed measures exhibiting vanishing moments and allow for the sparse approximation of generalized Vandermonde matrices issuing from a vast class of radial basis functions. Given a quasi-uniform set of $N$ data sites, and local approximation spaces with exponentially decreasing dimension, the samplet compressed multiscale system can be assembled with cost $mathcal{O}(N log^2 N)$. The overall cost of the proposed approach is $mathcal{O}(N log^2 N)$. The theoretical findings are accompanied by extensive numerical studies in two and three spatial dimensions.

Problem

• Multiscale scattered data interpolation using radial basis functions.
• Bounded condition numbers enable efficient iterative solvers.
• Samplet coordinates reduce computational cost for large datasets.

Innovation

• Multiscale scattered data interpolation using radial basis functions
• Samplet coordinates enable sparse approximation of large datasets
• Iterative solver with bounded iterations for efficient numerical solution