Efficient Reduced Basis Algorithm (ERBA) for kernel-based approximation

TL;DR

This paper introduces ERBA, an efficient algorithm for constructing reduced kernel-based interpolation models by removing data points using residuals or error bounds, improving computational efficiency over traditional knot insertion methods.

Contribution

The paper presents ERBA, a novel reduced basis algorithm for kernel interpolation that efficiently removes data points using residuals and error bounds, with a fast implementation inspired by Rippa's algorithm.

Findings

ERBA reduces computational cost compared to traditional methods.

ERBA employs two iterative data removal rules: ERBA-r and ERBA-p.

The algorithm is inspired by extended Rippa's algorithm for efficiency.

Abstract

The main purpose of this work is the one of providing an efficient scheme for constructing reduced interpolation models for kernel bases. In literature such problem is mainly addressed via the well-established knot insertion or knot removal schemes. Such iterative strategies are usually quite demanding from a computational point of view and our goal is to study an efficient implementation for data removal approaches, namely Efficient Reduced Basis Algorithm (ERBA). Focusing on kernel-based interpolation, the algorithm makes use of two iterative rules for removing data. The former, called ERBA-r, is based on classical residual evaluations. The latter, namely ERBA-p, is independent of the function values and relies on error bounds defined by the power function. In both cases, inspired by the so-called extended Rippa's algorithm, our ERBA takes advantage of a fast implementation.