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The Besov Subspace Consisting of Most non-smooth functions

JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS-ARMENIAN ACADEMY OF SCIENCES 2009年第3期44卷 163-171页

作者： Berezhnoi, E. I. Yaroslavl State Univ Yaroslavl Russia

Under the assumptions that Delta(f, h)(t) = vertical bar f(t + h) - f(t)vertical bar, X is a symmetric space of functions in [0, 1], alpha is an element of (0, 1) and p is an element of [1,8) are any fixed number, by the triple (X, alpha, p) a Besov type space Lambda(alpha)(X,p) is constructed, where the norm is given by the equality parallel to f vertical bar Lambda(alpha)(X,p)parallel to = ((i=1)Sigma(infinity)(2(alpha i)parallel to Delta(f;2(-1))(.)vertical bar X parallel to)(p))(1/p). For any alpha(0) is an element of (0,1), it is shown that there exists an infinite-dimensional, closed subspace of Lambda(alpha)(X,p), such that any non-identically zero function does not belong to the subspace Lambda(alpha)(X,p) with alpha > alpha(0).

关键词： Symmetric space subspace of the Besov space non-smooth functions

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Critical points of non-smooth functions with a weak compactness condition

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2009年第1期358卷 189-201页

作者： Marano, Salvatore A. Motreanu, Dumitru Univ Catania Dipartimento Matemat & Informat I-95125 Catania Italy Univ Perpignan Dept Math F-66860 Perpignan France

In the framework of non-differentiable functionals expressed as a locally Lipschitz continuous term plus a convex, proper, lower semi-continuous function, a critical point result is established under a new weak Palais-Smale hypothesis, which contains the so-called Cerami condition. Some meaningful special cases are then pointed out. (C) 2009 Elsevier Inc. All rights reserved.

关键词： Critical points non-smooth functions Weak Palais-Smale condition

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High order approximation to non-smooth multivariate functions

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COMPUTER AIDED GEOMETRIC DESIGN 2018年 63卷 31-65页

作者： Amir, Anat Levin, David School of Mathematical Sciences Tel Aviv University Ramat Aviv Tel Aviv 6997801 Israel

Common approximation tools return low-order approximations in the vicinities of singularities. Most prior works solve this problem for univariate functions. In this work we introduce a method for approximating non-smooth multivariate functions of the form f = g + r(+) where g, r is an element of CM+1 (R-n) and the function r(+) is defined by r(+)(Y) = {r(y), r(y) >= 0 , for all y is an element of R-n. 0, r(y) < 0 Given scattered (or uniform) data points X subset of R-n, we investigate approximation by quasi- interpolation. We design a correction term, such that the corrected approximation achieves full approximation order on the entire domain. We also show that the correction term is the solution to a Moving Least Squares (MLS) problem, and as such can both be easily computed and is smooth. Last, we prove that the suggested method includes a high-order approximation to the locations of the singularities. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Quasi-interpolation Multivariate functions non-smooth functions

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Determination of SRMC with non-smooth fuel cost functions using evolutionary programming

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ELECTRIC POWER COMPONENTS AND SYSTEMS 2005年第1期33卷 85-102页

作者： Gnanadass, R Palanivelu, TG Indian Inst Technol Dept Elect Engn Roorkee 247667 Uttar Pradesh India Pondicherry Engn Coll Dept Elect Engn Pondicherry India

In this article, a new algorithm for determination of short run marginal cost (SRMC) for feasible bilateral transactions using optimal power flow (OPF) solution has been presented. Determination of SRMC using conventional methods suffer due to the presence of non-smooth fuel cost generators in the modern, deregulated utilities. Hence in this article, evolutionary programming (EP) based OPF solution has been developed for obtaining optimal generator settings with four non-smooth fuel functions. System transmission loss and penalty factor at each and every bus are computed using the OPF solution. Further bus incremental cost at all the buses is computed using penalty factors. Generalized loss coefficients are also obtained from OPF solution and they are the functions of the system operating state with non-smooth fuel functions. The performance of the proposed algorithm has been validated with IEEE 30 bus and Indian utility 62-bus practical test systems.

关键词： bus incremental cost evolutionary programming non-smooth functions SRMC

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Bounded Palais-Smale sequences for non-differentiable functions

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nonLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2011年第16期74卷 5446-5454页

作者： Candito, P. Livrea, R. Motreanu, D. Univ Reggio Calabria Dipartimento DIMET Fac Ingn I-89100 Reggio Di Calabria Italy Univ Reggio Calabria Dipartimento Patrimonio Architetton & Urbanist Fac Architettura I-89100 Reggio Di Calabria Italy Univ Perpignan Dept Math F-66860 Perpignan France

The existence of bounded Palais-Smale sequences (briefly BPS) for functionals depending on a parameter belonging to a real interval and which are the sum of a locally Lipschitz continuous term and of a convex, proper, lower semicontinuous function, is obtained when the parameter runs in a full measure subset of the given interval. Specifically, for this class of non-smooth functions, we obtain BPS related to mountain pass and to global infima levels. This is done by developing a unifying approach, which applies to both cases and relies on a suitable deformation lemma. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Critical points non-smooth functions Bounded Palais-Smale sequences Mountain pass geometry Deformation

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Learning non-stationary and discontinuous functions using clustering, classification and Gaussian process modelling

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COMPUTERS & STRUCTURES 2023年第1期281卷

作者： Moustapha, Maliki Sudret, Bruno Swiss Fed Inst Technol Chair Risk Safety & Uncertainty Quantificat Stefano Franscini Pl 5 CH-8093 Zurich Switzerland

Surrogate models have shown to be an extremely efficient aid in solving engineering problems that require repeated evaluations of an expensive computational model. They are built by sparsely evaluating the costly original model and have provided a way to solve otherwise intractable problems. A crucial aspect in surrogate modelling is the assumption of smoothness and regularity of the model to approxi-mate. This assumption is however not always met in reality. For instance in civil or mechanical engineer-ing, some models may present discontinuities or non-smoothness e.g., in case of instability patterns such as buckling or snap-through. Building a single surrogate model capable of accounting for these funda-mentally different behaviours or discontinuities is not an easy task. In this paper, we propose a three-stage approach for the approximation of non-smooth functions which combines clustering, classification and regression. The idea is to split the space following the localized behaviors or regimes of the system and build local surrogates that are eventually assembled. A sequence of well-known machine learning techniques are used: Dirichlet process mixtures models (DPMM), support vector machines and Gaussian process modelling. The approach is tested and validated on two analytical functions and a finite element model of a tensile membrane structure.(c) 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://***/licenses/by/4.0/).

关键词： Surrogate modelling non-smooth functions Discontinuities Dirichlet process mixture models Uncertainty quantification

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H¹-interpolation on quadrilateral and hexahedral meshes with hanging nodes

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COMPUTING 2007年第3期80卷 203-220页

作者： Heuveline, V. Schieweck, F. Univ Karlsruhe Rechenzentrum D-76128 Karlsruhe Germany Univ Karlsruhe Inst Angew Math D-76128 Karlsruhe Germany Otto Von Guericke Univ Inst Anal & Numer D-39016 Magdeburg Germany

We propose a Scott-Zhang type finite element interpolation operator of first order for the approximation of H-1-functions by means of continuous piecewise mapped bilinear or trilinear polynomials. The novelty of the proposed interpolation operator is that it is defined for general non-affine equivalent quadrilateral and hexahedral elements and so-called 1-irregular meshes with hanging nodes. We prove optimal local approximation properties of this interpolation operator for functions in H-1. As necessary ingredients we provide a definition of a hanging node and a rigorous analysis of the issue of constrained approximation which cover both the two- and three-dimensional case in a unified fashion.

关键词： finite-element interpolation hanging nodes constrained approximation non-smooth functions quadrilateral and hexahedral elements

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A Nash type solution for hemivariational inequality systems

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nonLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2011年第16期74卷 5585-5590页

作者： Repovs, Dusan Varga, Csaba Univ Babes Bolyai Fac Math & Comp Sci RO-400084 Cluj Napoca Romania Univ Ljubljana Fac Math & Phys Ljubljana 1001 Slovenia Univ Ljubljana Fac Educ Ljubljana 1001 Slovenia

In this paper, we prove an existence result for a general class of hemivariational inequality systems using the Ky Fan version of the KKM theorem Fan (1984) [10] or Tarafdar fixed points Tarafdar (1987) [11]. As application, we give an infinite-dimensional version for the existence result of Nash generalized derivative points introduced recently by Kristaly (2010) [5]. We also give an application to a general hemivariational inequality system. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Fixed point theory non-smooth functions Hemivariational inequality systems

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An Efficient Sampling Algorithm for non-smooth Composite Potentials

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JOURNAL OF MACHINE LEARNING RESEARCH 2022年第1期23卷 1-50页

作者： Mou, Wenlong Flammarion, Nicolas Wainwright, Martin J. Bartlett, Peter L. Univ Calif Berkeley Dept EECS Berkeley CA 94720 USA Ecole Polytech Fed Lausanne Sch Comp & Commun Sci CH-1015 Lausanne Switzerland Univ Calif Berkeley Dept Stat Berkeley CA 94720 USA

We consider the problem of sampling from a density of the form p(x) ? exp(-f (x) - g(x)), where f : Rd-+ R is a smooth function and g : R-d-+ R is a convex and Lipschitz function. We propose a new algorithm based on the Metropolis-Hastings framework. Under certain isoperimetric inequalities on the target density, we prove that the algorithm mixes to within total variation (TV) distance e of the target density in at most O(d log(d/e)) iterations. This guarantee extends previous results on sampling from distributions with smooth log densities (g = 0) to the more general composite non-smooth case, with the same mixing time up to a multiple of the condition number. Our method is based on a novel proximal-based proposal distribution that can be efficiently computed for a large class of non-smooth functions g. Simulation results on posterior sampling problems that arise from the Bayesian Lasso show empirical advantage over previous proposal distributions.

关键词： Markov Chain Monte Carlo mixing time Metropolis-Hastings algorithms Langevin diffusion non-smooth functions Bayesian inference

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Analysis of commutation errors for functions with low regularity

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2007年第2期206卷 1027-1045页

作者： Berselli, Luigi C. Grisanti, Carlo R. John, Volker Univ Pisa Dipartimento Mat Appl U Dini I-56100 Pisa Italy Univ Saarland FR Math 61 D-66041 Saarbrucken Germany

Commutation errors arise in the derivation of the space averaged Navier-Stokes equations, the basic equations for the large eddy simulation of turbulent flows, if the filter is non-uniform or asymmetric (skewed) with non-constant skewness. These errors need to be analyzed for turbulent flow fields, where one expects a limited regularity of the solution. This paper studies the order of convergence of commutation errors, as the filter width tends to zero, for functions with low regularity. Several convergence results are proved and it is also shown that convergence may fail (or its order decreases) if the functions become less smooth. The main results are those dealing with Holder-continuous functions and with functions having singularities. The sharpness of the analytic results is confirmed with numerical illustrations. (c) 2006 Elsevier B.V. All rights reserved.

关键词： commutation errors space averaged Navier-Stokes equations Gaussian and box filter non-smooth functions

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