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An ?1 Regularized Method for Numerical Differentiation Using empirical eigenfunctions

Journal of Mathematical Research with Applications 2017年第4期37卷 496-504页

作者： Junbin LI Renhong WANG Min XU School of Mathematical Sciences Dalian University of Technology

We propose an ?1regularized method for numerical differentiation using empirical eigenfunctions. Compared with traditional methods for numerical differentiation, the output of our method can be considered directly as the derivative of the underlying function. Moreover,our method could produce sparse representations with respect to empirical *** results show that our method is quite effective.

关键词： numerical differentiation empirical eigenfunctions ?~1 regularization mercer kernel

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Model Reduction for the Spatially Distributed Systems Using the Combined eigenfunctions and empirical eigenfunctions

Model Reduction for the Spatially Distributed Systems Using ...

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作者： Mian JIANG Yong GUO Jigang WU Wenyun WANG Hunan Provincial Key Laboratory of Health Maintenance for Mechanical Equipment Hunan University of Science and Technology

The selection of spatial basis functions will significantly affect the accuracy and efficiency of modeling for nonlinear spatially distributed processes(SDP). The performance using the general spatial basis functions is not good enough which restricts the applications of the approximated models. The current study compares the model reduction performance of empirical eigenfunctions(EEFs) and a kind of new basis functions for the spatially distributed processes, which are obtained from general spatial basis functions by linear transformation, and the transformation matrix is derived using empirical balanced truncation. The EEFs are assumed the optimal in the sense of the least squares errors for the model reduction of spatially distributed processes, however, the results of the simulations show that the accuracy of the modeling based on the present new basis functions is better than that based on the EEFs derived from the measured spatio-temporal data.

关键词： Spatially Distributed Process Basis Functions empirical eigenfunctions Linear Transformation empirical Balanced Truncation

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Low-Order Optimal Regulation of Parabolic PDEs with Time-Dependent Domain

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AICHE JOURNAL 2015年第2期61卷 494-502页

作者： Izadi, Mojtaba Dubljevic, Stevan Univ Alberta Dept Chem & Mat Engn Edmonton AB T6G 2V4 Canada

Observer and optimal boundary control design for the objective of output tracking of a linear distributed parameter system given by a two-dimensional (2-D) parabolic partial differential equation with time-varying domain is realized in this work. The transformation of boundary actuation to distributed control setting allows to represent the system's model in a standard evolutionary form. By exploring dynamical model evolution and generating data, a set of time-varying empirical eigenfunctions that capture the dominant dynamics of the distributed system is found. This basis is used in Galerkin's method to accurately represent the distributed system as a finite-dimensional plant in terms of a linear time-varying system. This reduced-order model enables synthesis of a linear optimal output tracking controller, as well as design of a state observer. Finally, numerical results are prepared for the optimal output tracking of a 2-D model of the temperature distribution in Czochralski crystal growth process which has nontrivial geometry. (c) 2014 American Institute of Chemical Engineers AIChE J, 61: 494-502, 2015

关键词： parabolic PDE with time-dependent domain empirical eigenfunctions optimal boundary control order-reduction

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On empirical eigenfunction-based ranking with l¹ norm regularization

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JOURNAL OF APPROXIMATION THEORY 2015年 192卷 273-290页

作者： Xu, Min Fang, Qin Wang, Shaofan Li, Junbin Dalian Univ Technol Sch Math Sci Dalian 116024 Peoples R China Dalian Univ Coll Informat Dalian 116622 Peoples R China Beijing Univ Technol Coll Metropolitan Transportat Beijing Key Lab Multimedia Intelligent Software T Beijing 100124 Peoples R China

The problem of ranking, in which the goal is to learn a real-valued ranking function that induces a ranking over an instance space, has recently gained increasing attention in machine learning. We study a learning algorithm for ranking generated by a regularized scheme with an l(1) regularizer. The algorithm is formulated in a data dependent hypothesis space. Such a space is spanned by empirical eigenfunctions which are constructed by a Mercer kernel and the learning data. We establish the computations of empirical eigenfunctions and the representer theorem for the algorithm. Particularly, we provide an analysis of the sparsity and convergence rates for the algorithm. The results show that our algorithm produces both satisfactory convergence rates and sparse representations under a mild condition, especially without assuming sparsity in terms of any basis. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Ranking Mercer kernel empirical eigenfunctions Sparsity l(1) regularization

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Optimal spatial sampling scheme for parameter estimation of nonlinear distributed parameter systems

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COMPUTERS & CHEMICAL ENGINEERING 2012年 45卷 38-49页

作者： Alana, Jorge E. Theodoropoulos, Constantinos Univ Manchester Sch Chem Engn & Analyt Sci Manchester M13 9PL Lancs England Univ Zulia Fac Ingn Escuela Ingn Quim Maracaibo 4011 Zulia Venezuela

In this paper a methodology for the estimation of parameters of distributed parameter systems based on optimal spatial measurements is discussed. The concept of the covariance matrix for sampling design is exploited and D-optimality criteria concerning relevant metrics are linked with the computation of optimal measurement locations. These are obtained at the points where sensitivity functions reach their extrema values and coincide with the locations where system eigenmodes obtained by proper orthogonal decomposition (POD) are maximised or minimised as has been shown in a recent work (Alana & Theodoropoulos, 2011). A tubular reactor with recycle is used as illustrative example to demonstrate the sampling methodology, including cases where the system behaviour is unstable exhibiting sustained oscillations. The estimates with the highest deviation from the nominal parameter values are the ones with the lowest (absolute) sensitivity coefficients. Using a POD-based reduced model can significantly reduce computational costs and in addition improve the estimation procedure by using measurements at the locations where the POD modes extrema occur. The results are strongly influenced by experimental noise, and filtering techniques are needed to mitigate the related uncertainties. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Covariance matrix Sensitivity coefficients Measurement points Proper orthogonal decomposition empirical eigenfunctions Nonlinear distributed parameter system

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empirical eigenfunctions and medial surface dynamics of a human vocal fold

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METHODS OF INFORMATION IN MEDICINE 2005年第3期44卷 384-391页

作者： Döllinger, M Tayama, N Berry, DA Univ Calif Los Angeles Sch Med Laryngeal Dynam Lab Div Head & Neck Surg Los Angeles CA 90095 USA Int Med Ctr Japan Div Otolaryngol Tracheo Esophagol Tokyo Japan

Objectives: The purpose of this investigation was to use an excised human larynx to substantiate physical mechanisms of sustained vocal fold oscillation over a variety of phonatory conditions. During sustained, flow-induced oscillation, dynamical data was collected from the medial surface of the vocal fold. The method of empirical eigenfunctions was used to analyze the data and to probe physical mechanisms of sustained oscillation. Methods. Thirty microsutures were mounted on the medial margin of a human vocal fold. Across five distinct phonatory conditions, the vocal fold was set into oscillation and imaged with a high-speed digital imaging system. The position coordinates of the sutures were extracted from the images and converted into physical coordinates. empirical eigenfunctions were computed from the time-varying physical coordinates, and mechanisms of sustained oscillation were explored. Results: Using the method of empirical eigenfunctions, physical mechanisms of sustained vocal fold oscillation were substantiated. In particular, the essential dynamics of vocal fold vibration were captured by two dominant empirical eigenfunctions. The largest Eigenfunction primarily captured the alternating convergent/divergent shape of the medial surface of the vocal fold, while the second largest Eigenfunction primarily captured the lateral vibrations of the voal fold. Conclusions: The hemi-larynx setup yielded a view of the medial surface of the vocal folds, revealing the tissue vibrations which produced sound. Through the use of empirical eigenfunctions, the underlying modes of vibration were computed, disclosing physical mechanisms of sustained vocal fold oscillation. The investigation substantiated previous theoretical analyses and yielded significant data to help evaluate and refine computational models of vocal fold vibration.

关键词： human vocal folds medial surface dynamics empirical eigenfunctions hemi-larynx sustained phonation nonlinear dynamics

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Spatiotemporal classification of vocal fold dynamics by a multimass model comprising time-dependent parameters

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JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 2008年第4期123卷 2324-2334页

作者： Wurzbacher, Tobias Doellinger, Michael Schwarz, Raphael Hoppe, Ulrich Eysholdt, Ulrich Lohscheller, Joerg Univ Hosp Erlangen Sch Med Dept Phoniatr & Pediat Audiol Erlangen Germany Univ Hosp Erlangen Dept Audiol Erlangen Germany

A model-based approach is proposed to objectively measure and classify vocal fold vibrations by left-right asymmetries along the anterior-posterior direction, especially in the case of nonstationary phonation. For this purpose, vocal fold dynamics are recorded in real time with a digital high-speed camera during phonation of sustained vowels as well as pitch raises. The dynamics of a multimass model with time-dependent parameters are matched to vocal fold vibrations extracted at dorsal, medial, and ventral positions by an automatic optimization procedure. The block-based optimization accounts for nonstationary vibrations and compares the vocal fold and model dynamics by wavelet coefficients. The optimization is verified with synthetically generated data sets and is applied to 40 clinical high-speed recordings comprising normal and pathological voice subjects. The resulting model parameters allow an intuitive visual assessment of vocal fold instabilities within an asymmetry diagram and are applicable to an objective quantification of asymmetries. (C) 2008 Acoustical Society of America.

关键词： MEDIAL SURFACE DYNAMICS SOUNDING VOICE SYNTHESIS HIGH-SPEED RECORDINGS FINITE-ELEMENT MODEL 2-MASS MODEL empirical eigenfunctions PATHOLOGICAL VOICES VIBRATION CORDS VIDEOKYMOGRAPHY

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Optimal control of diffusion-convection-reaction processes using reduced-order models

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COMPUTERS & CHEMICAL ENGINEERING 2008年第9期32卷 2123-2135页

作者： Li, Mingheng Christofides, Panagiotis D. Univ Calif Los Angeles Dept Chem & Biomol Engn Los Angeles CA 90095 USA Calif State Polytech Univ Pomona Dept Chem & Mat Engn Pomona CA 91768 USA

Two approaches for optimal control of diffusion-convection-reaction processes based on reduced-order models are presented. The approaches differ in the way spatial discretization is carried out to compute a reduced-order model suitable for controller design. In the first approach, the partial differential equation (PDE) that describes the process is first discretized in space and time using the finite difference method to derive a large number of recursive algebraic equations, which are written in the form of a discrete-time state-space model with sparse state, input and output matrices. Snapshots based on this high-dimensional state-space model are generated to calculate empirical eigenfunctions using proper orthogonal decomposition. The Galerkin projection with the computed empirical eigenfunctions as basis functions is then directly applied to the high-dimensional state-space model to derive a reduced-order model. In the second approach, a continuous-time finite-dimensional state-space model is constructed directly from the PDE through application of orthogonal collocation on finite elements in the spatial domain. The dimension of the derived state-space model can be further reduced using standard model reduction techniques. In both cases, optimal controllers are designed based on the low-order state-space models using discrete-time and continuous-time linear quadratic regulator (LQR) techniques. The effectiveness of the proposed methods are illustrated through applications to a diffusion-convection process and a diffusion-convection-reaction process. (C) 2007 Elsevier Ltd. All rights reserved.

关键词： optimal control diffusion-convection-reaction processes model reduction empirical eigenfunctions orthogonal collocation

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Reduced order distributed boundary control of thermal transients in microsystems

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 2005年第6期13卷 853-867页

作者： Bleris, LG Kothare, MV Lehigh Univ Dept Elect & Comp Engn Microchem Syst Lab Bethlehem PA 18015 USA

We study the problem of regulation of thermal transients in a microsystem using empirical eigenfunctions. Proper orthogonal decomposition (POD) is applied to an ensemble of data to obtain the dominant structures, called empirical eigenfunctions, that characterize the dynamics of the process. These eigenfunctions are the most efficient basis for capturing the dynamics of an infinite dimensional process with a finite number of modes. In contrast to published approaches, we propose a new receding horizon boundary control scheme using the empirical eigenfunctions in a constrained optimization procedure to track a desired spatiotemporal profile. Finite element method (FEM) simulations of heat transfer are provided and used in order to implement and test the performance of the controller.

关键词： boundary reduced order control empirical eigenfunctions microsystems receding horizon control (RHC)

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System identification and model-based control for distributed parameter systems

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COMPUTERS & CHEMICAL ENGINEERING 2004年第8期28卷 1361-1375页

作者： Zheng, D Hoo, KA Texas Tech Univ Dept Chem Engn Lubbock TX 79409 USA

A linear, low-order input/output model is identified for a nonlinear distributed parameter system using a combination of singular value decomposition and the Karhunen Loeve expansion. The model captures the dominant behavior of the system around a nominal operating point. A quadratic dynamic model-based controller (QDMC) is designed based on this low-order model. Sufficient conditions for closed-loop stability are presented and proven. The system identification method and the resulting QDMC controller are demonstrated on a nonlinear multiple-input multiple-output (MIMO) chemical reactor that produces benzene from the hydro-dealkylation of toluene. (C) 2003 Elsevier Ltd. All rights reserved.

关键词： empirical eigenfunctions Karhunen Loeve expansion singular value decomposition QDMC multiple-input multiple-output

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