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A boundary formula for reproducing kernel hilbert spaces of real harmonic functions in Lipschitz domains

COMPLEX VARIABLES AND ELLIPTIC EQUATIONS 2021年第1期66卷 94-117页

作者： Chaira, Abdellatif Touhami, Soumia Univ Moulay Ismail Fac Sci Lab Math & Leures Applicat Equipe EDP & Calcul Sc Zitoune Meknes Morocco

This paper develops a new hilbert space method to characterize a family of reproducing kernel hilbert spaces of real harmonic functions in a bounded Lipschitz domain . Such method involves some families of positive self-adjoint operators and makes use of characterizations of their trace data and of a special inner product on . We also establish boundary representation results for this family in terms of the -Bergman kernel. In particular, a boundary integral representation for the very weak solution of the Dirichlet problem for Laplace's equation with -boundary data is provided. reproducing kernels and orthonormal bases for the harmonic spaces are also found.

关键词： reproducing kernel hilbert spaces Lipschitz domains harmonic spaces trace spaces and Moore-Penrose pseudo-inverse

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Learning partial differential equations in reproducing kernel hilbert spaces

The Journal of Machine Learning Research

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The Journal of Machine Learning Research 2023年第1期24卷 3887-3958页

作者： George Stepaniants Department of Mathematics Massachusetts Institute of Technology Cambridge MA

We propose a new data-driven approach for learning the fundamental solutions (Green's functions) of various linear partial differential equations (PDEs) given sample pairs of input-output functions. Building off the theory of functional linear regression (FLR), we estimate the best-fit Green's function and bias term of the fundamental solution in a reproducing kernel hilbert space (RKHS) which allows us to regularize their smoothness and impose various structural constraints. We derive a general representer theorem for operator RKHSs to approximate the original infinite-dimensional regression problem by a finite-dimensional one, reducing the search space to a parametric class of Green's functions. In order to study the prediction error of our Green's function estimator, we extend prior results on FLR with scalar outputs to the case with functional outputs. Finally, we demonstrate our method on several linear PDEs including the Poisson, Helmholtz, Schrödinger, Fokker-Planck, and heat equation. We highlight its robustness to noise as well as its ability to generalize to new data with varying degrees of smoothness and mesh discretization without any additional training.

关键词： Green's functions partial differential equations reproducing kernel hilbert spaces functional linear regression simultaneous diagonalization

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New reproducing kernel hilbert spaces on semi-infinite domains with existence and uniqueness results for the nonhomogeneous telegraph equation

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2020年第17期43卷 9615-9636页

作者： Hassan, Jabar S. Grow, David Salahaddin Univ Erbil Dept Math Erbil Iraq Missouri Univ Sci & Technol Dept Math Rolla MO 65409 USA

We introduce new reproducing kernel hilbert spaces on a semi-infinite domain and demonstrate existence and uniqueness of solutions to the nonhomogeneous telegraph equation in these spaces if the driver is square-integ... 详细信息

关键词： existence and uniqueness reproducing kernel hilbert spaces telegraph equation

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Koopman spectra in reproducing kernel hilbert spaces

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APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 2020年第2期49卷 573-607页

作者： Das, Suddhasattwa Giannakis, Dimitrios NYU Courant Inst Math Sci 251 Mercer St New York NY 10012 USA

Every invertible, measure-preserving dynamical system induces a Koopman operator, which is a linear, unitary evolution operator acting on the L-2 space of observables associated with the invariant measure. Koopman eigenfunctions represent the quasiperiodic, or non-mixing, component of the dynamics. The extraction of these eigenfunctions and their associated eigenfrequencies from a given time series is a non-trivial problem when the underlying system has a dense point spectrum, or a continuous spectrum behaving similarly to noise. This paper describes methods for identifying Koopman eigenfrequencies and eigenfunctions from a discretely sampled time series generated by such a system with unknown dynamics. Our main result gives necessary and sufficient conditions for a Fourier function, defined on Nstates sampled along an orbit of the dynamics, to be extensible to a Koopman eigenfunction on the whole state space, lying in a reproducing kernel hilbert space (RKHS). In particular, we show that such an extension exists if and only if the RKHS norm of the Fourier function does not diverge as N -> infinity. In that case, the corresponding Fourier frequency is also a Koopman eigenfrequency, modulo a unique translate by a Nyquist frequency interval, and the RKHS extensions of the N-sample Fourier functions converge to a Koopman eigenfunction in RKHS norm. For Koopman eigenfunctions in L-2 that do not have RKHS representatives, the RKHS extensions of Fourier functions at the corresponding eigenfrequencies are shown to converge in L-2 norm. Numerical experiments on mixed-spectrum systems with weak periodic components demonstrate that this approach has significantly higher skill in identifying Koopman eigenfrequencies compared to conventional spectral estimation techniques based on the discrete Fourier transform. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Koopman operators Spectral estimation reproducing kernel hilbert spaces Ergodic dynamical systems

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Eigendecompositions of Transfer Operators in reproducing kernel hilbert spaces

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JOURNAL OF NONLINEAR SCIENCE 2020年第1期30卷 283-315页

作者： Klus, Stefan Schuster, Ingmar Muandet, Krikamol Free Univ Berlin Dept Math & Comp Sci Berlin Germany Zalando SE Zalando Res Berlin Germany Max Planck Inst Intelligent Syst Tubingen Germany

Transfer operators such as the Perron-Frobenius or Koopman operator play an important role in the global analysis of complex dynamical systems. The eigenfunctions of these operators can be used to detect metastable sets, to project the dynamics onto the dominant slow processes, or to separate superimposed signals. We propose kernel transfer operators, which extend transfer operator theory to reproducing kernel hilbert spaces and show that these operators are related to hilbert space representations of conditional distributions, known as conditional mean embeddings. The proposed numerical methods to compute empirical estimates of these kernel transfer operators subsume existing data-driven methods for the approximation of transfer operators such as extended dynamic mode decomposition and its variants. One main benefit of the presented kernel-based approaches is that they can be applied to any domain where a similarity measure given by a kernel is available. Furthermore, we provide elementary results on eigendecompositions of finite-rank RKHS operators. We illustrate the results with the aid of guiding examples and highlight potential applications in molecular dynamics as well as video and text data analysis.

关键词： reproducing kernel hilbert spaces Koopman operator Perron-Frobenius operator Eigendecompositions kernel mean embeddings

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Weighted p-regular kernels for reproducing kernel hilbert spaces and Mercer Theorem

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ANALYSIS AND APPLICATIONS 2020年第3期18卷 359-383页

作者： Agud, L. Calabuig, J. M. Sanchez Perez, E. A. Univ Politecn Valencia Dept Matemat Aplicada Campus Alcoy Valencia Spain Univ Politecn Valencia Inst Univ Matemat Pura & Aplicada Camino Vera S-N E-46022 Valencia Spain

Let (X, Sigma, mu) be a finite measure space and consider a Banach function space Y(mu). Motivated by some previous papers and current applications, we provide a general framework for representing reproducing kernel hilbert spaces as subsets of Kothe Bochner (vectorvalued) function spaces. We analyze operator-valued kernels Gamma that define integration maps L-Gamma between Kothe-Bochner spaces of hilbert-valued functions Y(mu;kappa). We show a reduction procedure which allows to find a factorization of the corresponding kernel operator through weighted Bochner spaces L-P(gd mu;kappa) and L-P (hd mu;kappa) - where 1/p + 1/p' = 1 - under the assumption of p-concavity of Y(mu). Equivalently, a new kernel obtained by multiplying Gamma by scalar functions can be given in such a way that the kernel operator is defined from L-P (mu;kappa) to L-P (mu;kappa) in a natural way. As an application, we prove a new version of Mercer Theorem for matrix-valued weighted kernels.

关键词： reproducing kernel hilbert spaces integral operator kernel operator factorization representation Mercer Theorem

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Optimal Transport in reproducing kernel hilbert spaces: Theory and Applications

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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 2020年第7期42卷 1741-1754页

作者： Zhang, Zhen Wang, Mianzhi Nehorai, Arye Washington Univ Dept Elect & Syst Engn St Louis MO 63130 USA

In this paper, we present a mathematical and computational framework for comparing and matching distributions in reproducing kernel hilbert spaces (RKHS). This framework, called optimal transport in RKHS, is a generalization of the optimal transport problem in input spaces to (potentially) infinite-dimensional feature spaces. We provide a computable formulation of Kantorovich's optimal transport in RKHS. In particular, we explore the case in which data distributions in RKHS are Gaussian, obtaining closed-form expressions of both the estimated Wasserstein distance and optimal transport map via kernel matrices. Based on these expressions, we generalize the Bures metric on covariance matrices to infinite-dimensional settings, providing a new metric between covariance operators. Moreover, we extend the correlation alignment problem to hilbert spaces, giving a new strategy for matching distributions in RKHS. Empirically, we apply the derived formulas under the Gaussianity assumption to image classification and domain adaptation. In both tasks, our algorithms yield state-of-the-art performances, demonstrating the effectiveness and potential of our framework.

关键词： kernel Covariance matrices hilbert space Task analysis Geometry Modeling Optimal transport reproducing kernel hilbert spaces kernel methods optimal transport map Wasserstein distance Wasserstein geometry covariance operator image classification domain adaptation

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Linear dynamics in reproducing kernel hilbert spaces

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BULLETIN DES SCIENCES MATHEMATIQUES 2020年第0期159卷 102826-000页

作者： Mundayadan, Aneesh Sarkar, Jaydeb Indian Stat Inst Stat & Math Unit 8th MileMysore Rd Bangalore 560059 Karnataka India

Complementing earlier results on dynamics of unilateral weighted shifts, we obtain a sufficient (but not necessary, with supporting examples) condition for hypercyclicity, mixing and chaos for M-z*, the adjoint of M-z, on vector-valued analytic reproducing kernel hilbert spaces H in terms of the derivatives of kernel functions on the open unit disc D in C. Here M-z denotes the multiplication operator by the coordinate function z, that is (M(z)f)(w) = wf (w), for all f is an element of H and w is an element of D. We analyze the special case of quasiscalar reproducing kernel hilbert spaces. We also present a complete characterization of hypercyclicity of M-z* on tridiagonal reproducing kernel hilbert spaces and some special classes of vector-valued analytic reproducing kernel hilbert spaces. (C) 2019 Elsevier Masson SAS. All rights reserved.

关键词： Hypercyclicity Chaos Mixing reproducing kernel hilbert spaces Multiplication operator Linear dynamics

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Distance between reproducing kernel hilbert spaces and geometry of finite sets in the unit ball

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2021年第2期500卷 125140-125140页

作者： Ofek, Danny Pandey, Satish K. Shalit, Orr Moshe Tel Aviv Univ Sch Math Sci IL-6997801 Tel Aviv Israel Technion Israel Inst Technol Fac Math IL-3200003 Haifa Israel

In this paper we study the relationships between a reproducing kernel hilbert space, its multiplier algebra, and the geometry of the point set on which they live. We introduce a variant of the Banach-Mazur distance suited for measuring the distance between reproducing kernel hilbert spaces, that quantifies how far two spaces are from being isometrically isomorphic as reproducing kernel hilbert spaces. We introduce an analogous distance for multiplier algebras, that quantifies how far two algebras are from being completely isometrically isomorphic. We show that, in the setting of finite dimensional quotients of the Drury-Arveson space, two spaces are "close" to one another if and only if their multiplier algebras are "close", and that this happens if and only if one of the underlying point sets is close to an image of the other under a biholomorphic automorphism of the unit ball. These equivalences are obtained as corollaries of quantitative estimates that we prove. (c) 2021 Elsevier Inc. All rights reserved.

关键词： reproducing kernel hilbert spaces Multiplier algebras reproducing kernel Banach-Mazur distance Multiplier Banach-Mazur distance

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Qualité prédictive des méta-modèles construits sur des espaces de hilbert à noyau auto-reproduisant et analyse de sensibilité des modèles complexes.: Predictive quality of meta-models constructed on the reproducing kernel hilbert spaces and sensitivity analysis of complex models.

Qualité prédictive des méta-modèles construits sur des e...

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作者： Kamari, Halaleh Université Paris-Saclay

学位级别：博士

Ce travail porte sur le problème de l'estimation d'un méta-modèle d'un modèle complexe, noté m. Le modèle m dépend de d variables d'entrées X1,...,Xd qui sont in... 详细信息

Ce travail porte sur le problème de l'estimation d'un méta-modèle d'un modèle complexe, noté m. Le modèle m dépend de d variables d'entrées X1,...,Xd qui sont indépendantes et ont une loi connue. Le méta-modèle, noté f∗, approche la décomposition de Hoeffding de m et permet d'estimer ses indices de Sobol. Il appartient à un espace de hilbert à noyau auto-reproduisant (RKHS), noté H, qui est construit comme une somme directe d'espaces de hilbert (Durrande et al. (2013)). L'estimateur du méta-modèle, noté f^, est calculé en minimisant un critère des moindres carrés pénalisé par la somme de la norme de hilbert et de la norme empirique L2 (Huet and Taupin (2017)). Cette procédure, appelée RKHS ridge groupe sparse, permet à la fois de sélectionner et d'estimer les termes de la décomposition de Hoeffding, et donc de sélectionner les indices de Sobol non-nuls et de les estimer. Il permet d'estimer les indices de Sobol même d'ordre élevé, un point connu pour être difficile à mettre en *** travail se compose d'une partie théorique et d'une partie pratique. Dans la partie théorique, j'ai établi les majorations du risque empirique L2 et du risque quadratique de l'estimateur f^ d'un modèle de régression où l'erreur est non-gaussienne et non-bornée. Il s'agit des bornes supérieures par rapport à la norme empirique L2 et à la norme L2 pour la distance entre le modèle m et son estimation f^ dans le RKHS H. Dans la partie pratique, j'ai développé un package R appelé RKHSMetaMod, pour la mise en œuvre des méthodes d'estimation du méta-modèle f∗ de m. Ce package s'applique indifféremment dans le cas où le modèle m est calculable et le cas du modèle de régression. Afin d'optimiser le temps de calcul et la mémoire de stockage, toutes les fonctions de ce package ont été écrites en utilisant les bibliothèques GSL et Eigen de C++ à l'exception d'une fonction qui est écrite en R. Elles sont ensuite interfacées avec l'environnement R afin de proposer un package facilement exploitabl

关键词： Méta-Modèle Des espaces de hilbert à noyau auto-Reproduisant Indice de Sobol Modèle de régression Estimation non-Paramétrique Majoration du risque Meta-Model reproducing kernel hilbert spaces Sobol indices Regression model Non-Parametric estimation Risk upper bound

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