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reproducing kernel hilbert spaces and variable metric algorithms in PDE-constrained shape optimization

OPTIMIZATION METHODS & SOFTWARE 2018年第2期33卷 268-296页

作者： Eigel, M. Sturm, K. Weierstrass Inst Berlin Germany Univ Duisburg Essen Fak Math Essen Germany

In this paper we investigate and compare different gradient algorithms designed for the domain expression of the shape derivative. Our main focus is to examine the usefulness of kernel reproducing hilbert spaces for PDE-constrained shape optimization problems. We show that radial kernels provide convenient formulas for the shape gradient that can be efficiently used in numerical simulations. The shape gradients associated with radial kernels depend on a so-called smoothing parameter that allows a smoothness adjustment of the shape during the optimization process. Besides, this smoothing parameter can be used to modify the movement of the shape. The theoretical findings are verified in a number of numerical experiments.

关键词： Shape optimization reproducing kernel hilbert spaces gradient method variable metric radial kernels

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Adaptive estimation for nonlinear systems using reproducing kernel hilbert spaces

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2019年第2期45卷 869-896页

作者： Bobade, Parag Majumdar, Suprotim Pereira, Savio Kurdila, Andrew J. Ferris, John B. Virginia Tech Dept Engn Sci & Mech Blacksburg VA 24061 USA Virginia Tech Dept Elect & Comp Engn Blacksburg VA 24061 USA Virginia Tech Dept Mech Engn Blacksburg VA 24061 USA

This paper extends a conventional, general framework for online adaptive estimation problems for systems governed by unknown or uncertain nonlinear ordinary differential equations. The central feature of the theory introduced in this paper represents the unknown function as a member of a reproducing kernel hilbert space (RKHS) and defines a distributed parameter system (DPS) that governs state estimates and estimates of the unknown function. Under the assumption that full state measurements are available, this paper (1) derives sufficient conditions for the existence and stability of the infinite dimensional online estimation problem, (2) derives existence and stability of finite dimensional approximations of the infinite dimensional approximations, and (3) determines sufficient conditions for the convergence of finite dimensional approximations to the infinite dimensional online estimates. A new condition for persistency of excitation in a RKHS in terms of its evaluation functionals is introduced in the paper that enables proof of convergence of the finite dimensional approximations of the unknown function in the RKHS. This paper studies two particular choices of the RKHS, those that are generated by exponential functions and those that are generated by multiscale kernels defined from a multiresolution analysis.

关键词： Adaptive estimation reproducing kernel hilbert spaces Distributed parameter systems

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Frames, their relatives and reproducing kernel hilbert spaces

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JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 2020年第1期53卷

作者： Speckbacher, Michael Balazs, Peter Univ Bordeaux Inst Math Bordeaux UMR 5251 351 Cours Liberat F-33405 Talence France Austrian Acad Sci Acoust Res Inst Wohllebengasse 12-14 A-1040 Vienna Austria

This paper considers different facets of the interplay between reproducing kernel hilbert spaces (RKHS) and stable analysis/synthesis processes: first, we analyze the structure of the reproducing kernel of a RKHS using frames and reproducing pairs. Second, we present a new approach to prove the result that finite redundancy of a continuous frame implies atomic structure of the underlying measure space. Our proof uses the RKHS structure of the range of the analysis operator. This in turn implies that all the attempts to extend the notion of Riesz basis to general measure spaces are fruitless since every such family can be identified with a discrete Riesz basis. Finally, we show how the range of the analysis operators of a reproducing pair can be equipped with a RKHS structure.

关键词： continuous frames reproducing pairs reproducing kernel hilbert spaces redundancy atomic measures

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On the stability of reproducing kernel hilbert spaces of discrete-time impulse responses

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AUTOMATICA 2018年 95卷 529-533页

作者： Chen, Tianshi Pillonetto, Gianluigi Chinese Univ Hong Kong Sch Sci & Engn Shenzhen Peoples R China Chinese Univ Hong Kong Shenzhen Res Inst Big Data Shenzhen Peoples R China Univ Padua Dept Informat Engn Padua Italy

reproducing kernel hilbert spaces (RKHSs) have proved themselves to be key tools for the development of powerful machine learning algorithms, the so-called regularized kernel-based approaches. Recently, they have also inspired the design of new linear system identification techniques able to challenge classical parametric prediction error methods. These facts motivate the study of the RKHS theory within the control community. In this note, we focus on the characterization of stable RKHSs, i.e. RKHSs of functions representing stable impulse responses. Related to this, working in an abstract functional analysis framework, Carmeli et al. (2006) has provided conditions for an RKHS to be contained in the classical Lebesgue spaces L-p. In particular, we specialize this analysis to the discrete-time case with p=1. The necessary and sufficient conditions for the stability of an RKHS are worked out by a quite simple proof, more easily accessible to the control community. (C) 2018 Elsevier Ltd. All rights reserved.

关键词： Linear system identification reproducing kernel hilbert spaces BIBO stability kernel-based regularization Inverse problems

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reproducing kernel hilbert spaces of CR functions for the Euclidean motion group

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ANALYSIS AND APPLICATIONS 2015年第3期13卷 331-346页

作者： Barbieri, D. Citti, G. Univ Autonoma Madrid Dept Math E-28049 Madrid Spain Univ Bologna Dept Math I-40126 Bologna Italy

We study the geometric structure of the reproducing kernel hilbert space associated to the continuous wavelet transform generated by the irreducible representations of the group of Euclidean motions of the plane SE(2). A natural hilbert norm for functions on the group is constructed that makes the wavelet transform an isometry, but since the considered representations are not square integrable, the resulting hilbert space will not coincide with L-2(SE(2)). The reproducing kernel hilbert subspace generated by the wavelet transform, for the case of a minimal uncertainty mother wavelet, can be characterized in terms of the complex regularity defined by the natural CR structure of the group. Relations with the Bargmann transform are presented.

关键词： Euclidean motion group reproducing kernel hilbert spaces uncertainty principle CR functions

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Utilization of reproducing kernel hilbert spaces in Dynamic Discrete Choice Models: An Application to the High-speed Railway Timetabling Problem

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Transportation Research Procedia 2015年 10卷 544-553页

作者： Ricardo Garćıa-Ródenas Maŕıa Luz López-Garćıa Escuela Superior de Informtica. Universidad de Castilla-La Mancha Paseo de la Universidad Ciudad Real 13004 Spain

This paper introduces a generalized nested logit model that results from combining discrete and continuous response variables. reproducing kernel hilbert spaces are used to define the (dynamic) systematic utilities, allowing correlations between alternatives close together on the continuous spectrum, and reconciliation mechanisms between both types of response variables are established. The seminal motivation of this model is the passenger-centric train timetabling problem. For this reason, the discussion in this paper focuses on a high-speed railway (HSR) demand-forecasting model. The model proposes a maximum likelihood approach to estimating the parameters, and a Monte Carlo simulation study is conducted to test the proposed methodology.

关键词： Generalised nested logit model reproducing kernel hilbert spaces High-speed railway timetabling problem

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Gram matrices of reproducing kernel hilbert spaces over graphs

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LINEAR ALGEBRA AND ITS APPLICATIONS 2014年第0期445卷 56-68页

作者： Seto, Michio Suda, Sho Taniguchi, Tetsuji Shimane Univ Dept Math Matsue Shimane 6908504 Japan Aichi Univ Educ Dept Math Educ Kariya Aichi 4488542 Japan Matsue Coll Technol Matsue Shimane 6908518 Japan

In this paper, we introduce the notion of reproducing kernel hilbert spaces for graphs and the Gram matrices associated with them. Our aim is to investigate the Gram matrices of reproducing kernel hilbert spaces. We provide several bounds on the entries of the Gram matrices of reproducing kernel hilbert spaces and characterize the graphs which attain our bounds. (C) 2013 Elsevier Inc. All rights reserved.

关键词： reproducing kernel hilbert spaces Graph Laplacian matrix Generalized inverse Resistance distance

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Weighted Fourier-Laplace transforms in reproducing kernel hilbert spaces on the sphere

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2014年第2期411卷 732-741页

作者： Jordao, T. Menegatto, V. A. ICMC USP Sao Carlos BR-13560970 Sao Carlos SP Brazil

We study the action of a weighted Fourier-Laplace transform on the functions in the reproducing kernel hilbert space (RKHS) associated with a positive definite kernel on the sphere. After defining a notion of smoothness implied by the transform, we show that smoothness of the kernel implies the same smoothness for the generating elements (spherical harmonics) in the Mercer expansion of the kernel. We prove a reproducing property for the weighted Fourier-Laplace transform of the functions in the RKHS and embed the RKHS into spaces of smooth functions. Some relevant properties of the embedding are considered, including compactness and boundedness. The approach taken in the paper includes two important notions of differentiability characterized by weighted Fourier-Laplace transforms: fractional derivatives and Laplace-Beltrami derivatives. (C) 2013 Elsevier Inc. All rights reserved.

关键词： Sphere reproducing kernel hilbert spaces Fourier-Laplace transforms Fractional derivative Laplace-Beltrami derivative

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A Primer on reproducing kernel hilbert spaces

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FOUNDATIONS AND TRENDS IN SIGNAL PROCESSING 2014年第1-2期8卷 1-126页

作者： Manton, Jonathan H. Amblard, Pierre-Olivier Univ Melbourne Melbourne Vic 3010 Australia CNRS F-38402 Grenoble France

reproducing kernel hilbert spaces are elucidated without assuming prior familiarity with hilbert spaces. Compared with extant pedagogic material, greater care is placed on motivating the definition of reproducing kernel hilbert spaces and explaining when and why these spaces are efficacious. The novel viewpoint is that reproducing kernel hilbert space theory studies extrinsic geometry, associating with each geometric configuration a canonical overdetermined coordinate system. This coordinate system varies continuously with changing geometric configurations, making it well-suited for studying problems whose solutions also vary continuously with changing geometry. This primer can also serve as an introduction to infinite-dimensional linear algebra because reproducing kernel hilbert spaces have more properties in common with Euclidean spaces than do more general hilbert spaces.

关键词： reproducing kernel hilbert spaces hilbert space Coordinate system

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Model averaging for genome-enabled prediction with reproducing kernel hilbert spaces: a case study with pig litter size and wheat yield

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JOURNAL OF ANIMAL BREEDING AND GENETICS 2014年第2期131卷 105-115页

作者： Tusell, L. Perez-Rodriguez, P. Forni, S. Gianola, D. Univ Wisconsin Dept Anim Sci Madison WI 53706 USA Colegio Postgrad Montecillo Estado De Mexic Mexico Genus Plc Hendersonville TN USA Univ Wisconsin Dept Dairy Sci Madison WI 53706 USA Univ Wisconsin Dept Biostat & Med Informat Madison WI 53706 USA

Predictive ability of yet-to-be observed litter size (pig) grain yield (wheat) records of several reproducing kernel hilbert spaces (RKHS) regression models combining different number of Gaussian or t kernels was evaluated. Predictive performance was assessed as the average (over 50 replicates) predictive correlation in the testing set. Predictions from these models were combined using three different types of model averaging: (i) mean of predicted phenotypes obtained in each model, (ii) weighted average using mean squared error as weight or (iii) using the marginal likelihood as weight. (ii) and (iii) were obtained in a validation set with 5% of the data. Phenotypes consisted of 2598, 1604 and 1879 average litter size records from three commercial pig lines and wheat grain yield of 599 lines evaluated in four macro-environments. SNPs from the PorcineSNP60 BeadChip and 1447 DArT markers were used as predictors for the pig and wheat data analyses, respectively. Gaussian and univariate t kernels led to same predictive performance. Multikernel RKHS regression models overcame shortcomings of single kernel models (increasing the predictive correlation of RKHS models by 0.05 where 3 Gaussian or t kernels were fitted in the RKHS models simultaneously). None of the proposed averaging strategies improved the predictive correlations attained with single models using multiple kernel fitting.

关键词： Genomic prediction genomic selection model averaging pigs reproducing kernel hilbert spaces wheat

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