Implementations

Buhmann, Martin D.

doi:10.1017/cbo9780511543241.008

Cited by 51 publications

(63 citation statements)

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“…Rather than dwelling much on explaining conditionally positive definite kernel functions and the structure of their native reproducing kernel Hilbert spaces, we refer to the text books [5,7,12,24]. For the following of our discussion, it is sufficient to say that scattered data interpolation by positive definite kernels (where m = 0) leads to a unique reconstruction of the form (6).…”

Section: Kernel-based Reconstruction In Particle Flow Simulationsmentioning

confidence: 99%

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On the Construction of Kernel-Based Adaptive Particle Methods in Numerical Flow Simulation

Iske

2013

Notes on Numerical Fluid Mechanics and Multidisciplinary Design

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This contribution discusses the construction of kernel-based adaptive particle methods for numerical flow simulation, where the finite volume particle method (FVPM) is used as a prototype. In the FVPM, scattered data approximation algorithms are required in the recovery step of the WENO reconstruction. We first show how kernel-based approximation schemes can be used in the recovery step of particle methods, where we give preference to the radial polyharmonic spline kernel. Then we discuss important aspects concerning the numerical stability and approximation behaviour of polyharmonic splines. Moreover, we propose customized coarsening and refinement rules for the adaptive resampling of the particles. Supporting numerical examples and comparisons with other radial kernels are provided.

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Section: Kernel-based Reconstruction In Particle Flow Simulationsmentioning

confidence: 99%

“…Now the scale invariance of the polyharmonic spline reconstructions' absolute condition number allows us to construct a simple preconditioner to obtain a stable evaluation of the reconstruction s in (5). To this end, we regard for any h > 0 the scaled reconstruction problem s h hΞ = u hΞ , i.e.,…”

Section: Stable Evaluation Of the Reconstructionmentioning

confidence: 99%

On the Construction of Kernel-Based Adaptive Particle Methods in Numerical Flow Simulation

Iske

2013

Notes on Numerical Fluid Mechanics and Multidisciplinary Design

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“…Atualmente, o emprego de funções escalares dependentes da distância euclidiana entre dois pontos, denominadas de Funções de Base Radial (FBR) substitui com vantagem os procedimentos de interpolação polinomiais em várias aplicações, particularmente quando se trata de aproximar dados esparsos em várias dimensões [3], mas também se mostram efetivas no ajuste de curvas e na solução de equações diferenciais. Grande parte do avanço no desenvolvimento de tais funções foi devidoà aplicação delas na solução de problemas de geração e reestruturação de malhas em técnicas adaptativas, particularmente no contexto do Método dos Elementos Finitos (MEF).…”

Section: Introductionunclassified

Desempenho do Método dos Elementos de Contorno com Integração Direta em Problemas de Autovalor com Domínio Não Regular

Loeffler¹,

Barcelos²

2018

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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Resumo. A pesquisa como um todo tem como objetivo testar os principais tipos de funções de base radial juntoà nova técnica de solução do Método dos Elementos de Contorno, denominada MECID, na solução de problemas de autovalor em que o domínio bidimensional nãoé regular. Diversas funções radiais, tanto as funções radiais clássicas quanto as funções radiais de Wendland e Wu, com suporte pleno, já foram testadas anteriormente e os resultados colhidos foram comparados com soluções analíticas disponíveis, indicando aquelas com melhor desempenho. Os domínios, contudo, foram relativamente simples, apresentando formatos retangulares e circulares. Neste trabalho, mostra-se um exemplo de simulação da MECID com duas das melhores funções na solução de problemas geometricamente mais elaborados. A comparação dos resultados foi feita com o Método dos Elementos Finitos.Palavras-chave. Problemas de Autovalores, Problemas de Helmholtz, Funcões de Base Radial no Método dos Elementos de Contorno.

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“…Applications come from such different fields as physics, biology, geology, meteorology and finance. The books [4,7,20,21] show how to use (conditionally) positive definite kernels to construct interpolants for observation data sampled from some unknown functions in the native spaces induced by the kernel functions. In the books [2,18], the optimal support vector machine solutions are obtained in reproducing kernel Hilbert spaces (RKHSs), and these solutions are formulated in terms of the related reproducing kernels and given data values.…”

Section: Introductionmentioning

confidence: 99%

Solving support vector machines in reproducing kernel Banach spaces with positive definite functions

Fasshauer

Hickernell

2015

Applied and Computational Harmonic Analysis

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In this paper we solve support vector machines in reproducing kernel Banach spaces with reproducing kernels defined on nonsymmetric domains instead of the traditional methods in reproducing kernel Hilbert spaces. Using the orthogonality of semi-innerproducts, we can obtain the explicit representations of the dual (normalized-duality-mapping) elements of support vector machine solutions. In addition, we can introduce the reproduction property in a generalized native space by Fourier transform techniques such that it becomes a reproducing kernel Banach space, which can be even embedded into Sobolev spaces, and its reproducing kernel is set up by the related positive definite function. The representations of the optimal solutions of support vector machines (regularized empirical risks) in these reproducing kernel Banach spaces are formulated explicitly in terms of positive definite functions, and their finite numbers of coefficients can be computed by fixed point iteration. We also give some typical examples of reproducing kernel Banach spaces induced by Matérn functions (Sobolev splines) so that their support vector machine solutions are well computable as the classical algorithms. Moreover, each of their reproducing bases includes information from multiple training data points. The concept of reproducing kernel Banach spaces offers us a new numerical tool for solving support vector machines.

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Implementations

Cited by 51 publications

References 0 publications

On the Construction of Kernel-Based Adaptive Particle Methods in Numerical Flow Simulation

On the Construction of Kernel-Based Adaptive Particle Methods in Numerical Flow Simulation

Desempenho do Método dos Elementos de Contorno com Integração Direta em Problemas de Autovalor com Domínio Não Regular

Solving support vector machines in reproducing kernel Banach spaces with positive definite functions

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