A novel class of stabilized greedy kernel approximation algorithms: Convergence, stability and uniform point distribution

Wenzel, Tizian; Santin, Gabriele; Haasdonk, Bernard

doi:10.1016/j.jat.2020.105508

Cited by 32 publications

(53 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The sequence of Theorems 15, 19 and 20 of [33] further established that the resulting sequence of points are asymptotically uniformly distributed under some mild conditions. These results implied (optimal) stability statements in [33,Corollary 22].…”

Section: Greedy Kernel Interpolationmentioning

confidence: 85%

“…(b) The paper [33] showed further results for the case of kernels of finite smoothness τ > d/2: Theorem 12 in [33] showed that the decay rate on P n L ∞ (Ω) is sharp. The sequence of Theorems 15, 19 and 20 of [33] further established that the resulting sequence of points are asymptotically uniformly distributed under some mild conditions. These results implied (optimal) stability statements in [33,Corollary 22].…”

Section: Greedy Kernel Interpolationmentioning

confidence: 96%

“…For the P -greedy algorithm it holds P n (x n+1 ) = P n L ∞ (Ω) . Several results on the P -greedy algorithm have been derived in [21,33]:…”

Section: Greedy Kernel Interpolationmentioning

confidence: 99%

“…Namely, for the P -greedy algorithm it is possible to prove quasi-optimality statements (see [21]), namely that whatever is the best known decay rate of the power function for arbitrarily optimized points, this transfers to the same decay of the power function associated to the points selected by P -greedy. Especially in the case of Sobolev spaces, these results can be proven to be optimal [33]. On the other hand, the convergence theory for the target data-dependent algorithms is much weaker: The known results (see Section 2 for a detailed account) provide convergence of order at most n −1/2 , which is generally not only largely missing the practical observations, but also slower than the rates proven for P -greedy.…”

Section: Introductionmentioning

confidence: 99%

“…These algorithms have been used in a series of applications (see e.g. [6, 10-12, 18, 22, 27-29]), and overwhelming numerical evidence points to the fact that criteria which incorporate a residual-dependent term provide faster convergence, even if sometimes at the price of stability (see [33] for a discussion of this fact for f /P -greedy, and [6] for f • P -greedy).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Analysis of target data-dependent greedy kernel algorithms: Convergence rates for $f$-, $f \cdot P$- and $f/P$-greedy

Wenzel¹,

Santin²,

Haasdonk³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Data-dependent greedy algorithms in kernel spaces are known to provide fast converging interpolants, while being extremely easy to implement and efficient to run. Despite this experimental evidence, no detailed theory has yet been presented. This situation is unsatisfactory especially when compared to the case of the data-independent P -greedy algorithm, for which optimal convergence rates are available, despite its performances being usually inferior to the ones of target data-dependent algorithms.In this work we fill this gap by first defining a new scale of greedy algorithms for interpolation that comprises all the existing ones in a unique analysis, where the degree of dependency of the selection criterion on the functional data is quantified by a real parameter. We then prove new convergence rates where this degree is taken into account and we show that, possibly up to a logarithmic factor, target data-dependent selection strategies provide faster convergence.In particular, for the first time we obtain convergence rates for target data adaptive interpolation that are faster than the ones given by uniform points, without the need of any special assumption on the target function. The rates are confirmed by a number of examples.These results are made possible by a new analysis of greedy algorithms in general Hilbert spaces.

show abstract

Section: Greedy Kernel Interpolationmentioning

confidence: 85%

Section: Greedy Kernel Interpolationmentioning

confidence: 96%

“…For the P -greedy algorithm it holds P n (x n+1 ) = P n L ∞ (Ω) . Several results on the P -greedy algorithm have been derived in [21,33]:…”

Section: Greedy Kernel Interpolationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Analysis of target data-dependent greedy kernel algorithms: Convergence rates for $f$-, $f \cdot P$- and $f/P$-greedy

Wenzel¹,

Santin²,

Haasdonk³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Greedy algorithms for image approximation from scattered Radon data

Albrecht

Iske

2021

Proc Appl Math & Mech

View full text Add to dashboard Cite

Positive definite kernels are powerful tools for multivariate approximation from scattered data. This contribution discusses kernel-based image approximation from scattered Radon data. To this end, we use weighted kernels for the reconstruction. Moreover, we propose greedy algorithms, which are used to adaptively select suitable approximation spaces. This reduces the complexity of the resulting image reconstruction method and, moreover, it improves the numerical stability quite significantly.

show abstract

Analysis of Target Data-Dependent Greedy Kernel Algorithms: Convergence Rates for f-, $$f \cdot P$$- and f/P-Greedy

2022

View full text Add to dashboard Cite

Data-dependent greedy algorithms in kernel spaces are known to provide fast converging interpolants, while being extremely easy to implement and efficient to run. Despite this experimental evidence, no detailed theory has yet been presented. This situation is unsatisfactory, especially when compared to the case of the data-independent P-greedy algorithm, for which optimal convergence rates are available, despite its performances being usually inferior to the ones of target data-dependent algorithms. In this work, we fill this gap by first defining a new scale of greedy algorithms for interpolation that comprises all the existing ones in a unique analysis, where the degree of dependency of the selection criterion on the functional data is quantified by a real parameter. We then prove new convergence rates where this degree is taken into account, and we show that, possibly up to a logarithmic factor, target data-dependent selection strategies provide faster convergence. In particular, for the first time we obtain convergence rates for target data adaptive interpolation that are faster than the ones given by uniform points, without the need of any special assumption on the target function. These results are made possible by refining an earlier analysis of greedy algorithms in general Hilbert spaces. The rates are confirmed by a number of numerical examples.

show abstract

A novel class of stabilized greedy kernel approximation algorithms: Convergence, stability and uniform point distribution

Cited by 32 publications

References 24 publications

Analysis of target data-dependent greedy kernel algorithms: Convergence rates for $f$-, $f \cdot P$- and $f/P$-greedy

Analysis of target data-dependent greedy kernel algorithms: Convergence rates for $f$-, $f \cdot P$- and $f/P$-greedy

Greedy algorithms for image approximation from scattered Radon data

Analysis of Target Data-Dependent Greedy Kernel Algorithms: Convergence Rates for f-, $$f \cdot P$$- and f/P-Greedy

Contact Info

Product

Resources

About