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Hao Ling - One of the best experts on this subject based on the ideXlab platform.

  • An efficient wavelet preconditioner for iterative solution of three-dimensional electromagnetic integral equations
    IEEE Transactions on Antennas and Propagation, 2003
    Co-Authors: Hai Deng, Hao Ling
    Abstract:

    A wavelet-based preconditioning method is proposed to facilitate the iterative solution of three-dimensional (3-D) electromagnetic integral equations. The preconditioner is derived from the wavelet transform of the Moment Matrix. It is based on the observation that both the Moment Matrix and its inverse exhibit a sparse, multilevel finger structure. A method based on the Forbenius-norm minimization is used to solve the inverse of the Matrix under the multilevel finger structure. Numerical results on a 3-D cavity show that the iteration numbers are significantly reduced with the wavelet-preconditioned system. The computational cost of the preconditioner is kept under O(NlogN).

  • An efficient preconditioner for electromagnetic integral equations using predefined wavelet packet basis
    IEEE Transactions on Antennas and Propagation, 2002
    Co-Authors: Hai Deng, Hao Ling
    Abstract:

    An approximate-inverse preconditioner based on the predefined wavelet packet (PWP) basis is proposed for the fast iterative solution of electromagnetic integral equations. The PWP basis is designed to achieve a sparse representation of the Moment Matrix and the preconditioner is constructed by inverting the block-diagonal approximation of the PWP-based Moment Matrix and transforming the results into the space domain. Numerical results show that the PWP preconditioner is effective in accelerating the convergence rate of iterative solution to Moment equations. It is also demonstrated that by properly designing the block-diagonal Matrix and computing the Matrix elements, the total computational complexity and memory costs for the preconditioner can be kept to O(NlogN).

  • fast solution of electromagnetic integral equations using adaptive wavelet packet transform
    IEEE Transactions on Antennas and Propagation, 1999
    Co-Authors: Hai Deng, Hao Ling
    Abstract:

    The adaptive wavelet packet transform is applied to sparsify the Moment matrices for the fast solution of electromagnetic integral equations. In the algorithm, a cost function is employed to adaptively select the optimal wavelet packet expansion/testing functions to achieve the maximum sparsity possible in the resulting transformed system. The search for the best wavelet packet basis and the Moment Matrix transformation are implemented by repeated two-channel filtering of the original Moment Matrix with a pair of quadrature filters. It is found that the sparsified Matrix has above-threshold elements that grow only as O(N/sup 1.4/) for typical scatterers. Consequently the operations to solve the transformed Moment equation using the conjugate gradient method scales as O(N/sup 1.4/). The additional computational cost for carrying out the adaptive wavelet packet transform is evaluated and discussed.

  • On a class of predefined wavelet packet bases for efficient representation of electromagnetic integral equations
    IEEE Transactions on Antennas and Propagation, 1999
    Co-Authors: Hai Deng, Hao Ling
    Abstract:

    A general wavelet packet tree is proposed to design predefined wavelet packet (PWP) bases for the efficient representation of electrodynamic integral equations. The wavelet packet decomposition tree is constructed by zooming in along the spectral oscillatory frequency of the free-space Green's function. Numerical results show that for typical two dimensional (2-D) scatterers the number of above-threshold elements in the PWP-based Moment Matrix is on the order of O(N/sup 1.3/) and tends to grow at a rate of O(N/spl middot/log N) for large-scale problems. Therefore, the complexity of solving the Moment equations can be reduced accordingly. Furthermore, it is shown that the elements of the Moment Matrix based on the PWP bases can be computed directly at approximately the same complexity as the fast wavelet transform approach. Consequently, with on-the-fly thresholding of the Matrix elements, the O(N/sup 2/) memory bottleneck in the formation of the PWP-based Moment Matrix can be circumvented.

  • Representation of Moment Matrix with predefined wavelet packet basis
    Electronics Letters, 1998
    Co-Authors: H. Deng, Hao Ling
    Abstract:

    A general wavelet packet tree is proposed to generate a predefined wavelet packet basis for the efficient representation of electrodynamic integral equations. The wavelet packet decomposition tree is constructed by zooming in along the spectral oscillation frequency of the free-space Green's function. This is in contrast to the conventional wavelet transform which zooms in along the lowest spectral frequency. Numerical results show the transformed Moment Matrix based on the pre-defined wavelet packet basis has about O(N/sup 1.3/) above-threshold elements for typical scatterers.

Hai Deng - One of the best experts on this subject based on the ideXlab platform.

  • An efficient wavelet preconditioner for iterative solution of three-dimensional electromagnetic integral equations
    IEEE Transactions on Antennas and Propagation, 2003
    Co-Authors: Hai Deng, Hao Ling
    Abstract:

    A wavelet-based preconditioning method is proposed to facilitate the iterative solution of three-dimensional (3-D) electromagnetic integral equations. The preconditioner is derived from the wavelet transform of the Moment Matrix. It is based on the observation that both the Moment Matrix and its inverse exhibit a sparse, multilevel finger structure. A method based on the Forbenius-norm minimization is used to solve the inverse of the Matrix under the multilevel finger structure. Numerical results on a 3-D cavity show that the iteration numbers are significantly reduced with the wavelet-preconditioned system. The computational cost of the preconditioner is kept under O(NlogN).

  • An efficient preconditioner for electromagnetic integral equations using predefined wavelet packet basis
    IEEE Transactions on Antennas and Propagation, 2002
    Co-Authors: Hai Deng, Hao Ling
    Abstract:

    An approximate-inverse preconditioner based on the predefined wavelet packet (PWP) basis is proposed for the fast iterative solution of electromagnetic integral equations. The PWP basis is designed to achieve a sparse representation of the Moment Matrix and the preconditioner is constructed by inverting the block-diagonal approximation of the PWP-based Moment Matrix and transforming the results into the space domain. Numerical results show that the PWP preconditioner is effective in accelerating the convergence rate of iterative solution to Moment equations. It is also demonstrated that by properly designing the block-diagonal Matrix and computing the Matrix elements, the total computational complexity and memory costs for the preconditioner can be kept to O(NlogN).

  • fast solution of electromagnetic integral equations using adaptive wavelet packet transform
    IEEE Transactions on Antennas and Propagation, 1999
    Co-Authors: Hai Deng, Hao Ling
    Abstract:

    The adaptive wavelet packet transform is applied to sparsify the Moment matrices for the fast solution of electromagnetic integral equations. In the algorithm, a cost function is employed to adaptively select the optimal wavelet packet expansion/testing functions to achieve the maximum sparsity possible in the resulting transformed system. The search for the best wavelet packet basis and the Moment Matrix transformation are implemented by repeated two-channel filtering of the original Moment Matrix with a pair of quadrature filters. It is found that the sparsified Matrix has above-threshold elements that grow only as O(N/sup 1.4/) for typical scatterers. Consequently the operations to solve the transformed Moment equation using the conjugate gradient method scales as O(N/sup 1.4/). The additional computational cost for carrying out the adaptive wavelet packet transform is evaluated and discussed.

  • On a class of predefined wavelet packet bases for efficient representation of electromagnetic integral equations
    IEEE Transactions on Antennas and Propagation, 1999
    Co-Authors: Hai Deng, Hao Ling
    Abstract:

    A general wavelet packet tree is proposed to design predefined wavelet packet (PWP) bases for the efficient representation of electrodynamic integral equations. The wavelet packet decomposition tree is constructed by zooming in along the spectral oscillatory frequency of the free-space Green's function. Numerical results show that for typical two dimensional (2-D) scatterers the number of above-threshold elements in the PWP-based Moment Matrix is on the order of O(N/sup 1.3/) and tends to grow at a rate of O(N/spl middot/log N) for large-scale problems. Therefore, the complexity of solving the Moment equations can be reduced accordingly. Furthermore, it is shown that the elements of the Moment Matrix based on the PWP bases can be computed directly at approximately the same complexity as the fast wavelet transform approach. Consequently, with on-the-fly thresholding of the Matrix elements, the O(N/sup 2/) memory bottleneck in the formation of the PWP-based Moment Matrix can be circumvented.

Raj Mittra - One of the best experts on this subject based on the ideXlab platform.

  • Parametric interpolation of the Moment Matrix in surface integral equation formulation
    International Journal of Rf and Microwave Computer-aided Engineering, 1999
    Co-Authors: Krishna Naishadham, Todd W. Nuteson, Raj Mittra
    Abstract:

    In the analysis of electromagnetic scattering and radiation from objects of arbitrary shape using the method of Moments (MoM), it is desirable to fill the impedance (or Moment) Matrix efficiently so that larger size problems can be solved. This article describes a general MoM technique in which the Matrix is filled by spatial interpolation with respect to a parametrized electrical separation between source and test elements. The parametrization is accomplished such that the same algorithm also provides frequency interpolation, thus facilitating efficient computations over a wide frequency band. The spatial interpolation method is illustrated by application to the analysis of radiation from tunable microstrip patch antennas over multiple frequency bands. By specializing the interpolation scheme to a surface integral equation formulation that employs rooftop basis functions on a grid of rectangular cells, it is shown that the interpolation method results in considerable reduction of the storage and CPU time requirements. ©1999 John Wiley & Sons, Inc. Int J RF and Microwave CAE 9: 474–489, 1999.

  • Spatial interpolation of the Moment Matrix for efficient analysis of microstrip circuits
    1993 IEEE MTT-S International Microwave Symposium Digest, 1
    Co-Authors: Todd W. Nuteson, Krishna Naishadham, Raj Mittra
    Abstract:

    An efficient Moment method technique, based on spatial interpolation of the Moment Matrix, is developed for the analysis of microstrip circuit elements of arbitrary shape. Redundant calculations in the Moment Matrix are eliminated by utilizing various symmetries. The quasi-dynamic approximations of the Green's functions and closed-form analytical approximations of the Sommerfeld integrals are invoked to simplify the analysis. Sample computed results are presented on the current distribution obtained by interpolation of the Moment Matrix and agree very well with those evaluated without interpolation. >

  • Spatial interpolation of the Moment Matrix in electromagnetic scattering and radiation problems
    Proceedings of IEEE Antennas and Propagation Society International Symposium, 1
    Co-Authors: Todd W. Nuteson, Krishna Naishadham, Raj Mittra
    Abstract:

    In the analysis of antennas and scatterers of arbitrary shape by the method of Moments (MoM), it is desirable to fill the Matrix efficiently so that larger size problems can be solved. The authors describe an efficient MoM technique in which the Matrix is filled by spatial interpolation of the Matrix elements. Redundant calculations in the Moment Matrix are eliminated by utilizing various symmetries. The method is illustrated by application to planar problems including scattering from a square plate and radiation from a microstrip patch antenna. Computed results compare very well with those in the published literature. Using interpolation to fill the Moment Matrix has proven to be very efficient and produces negligible error in the computed current distribution. >

Edouard Pauwels - One of the best experts on this subject based on the ideXlab platform.

  • Data Analysis from Empirical Moments and the Christoffel Function
    Foundations of Computational Mathematics, 2020
    Co-Authors: Edouard Pauwels, Mihai Putinar, Jean-bernard Lasserre
    Abstract:

    Spectral features of the empirical Moment Matrix constitute a resourceful tool for unveiling properties of a cloud of points, among which, density, support and latent structures. This Matrix is readily computed from an input dataset, and its eigen decomposition can then be used to identify algebraic properties of the support or density/support estimates with the Christoffel function. It is already well known that the empirical Moment Matrix encodes a great deal of subtle attributes of the underlying measure. Starting from this object as base of observations, we combine ideas from statistics, real algebraic geometry, orthogonal polynomials and approximation theory for opening new insights relevant for machine learning problems with data supported on algebraic sets. Refined concepts and results from real algebraic geometry and approximation theory are empowering a simple tool (the empirical Moment Matrix) for the task of solving non-trivial questions in data analysis. We provide (1) theoretical support, (2) numerical experiments and (3) connections to real-world data as a validation of the stamina of the empirical Moment Matrix approach.

  • Data analysis from empirical Moments and the Christoffel function
    arXiv: Machine Learning, 2018
    Co-Authors: Edouard Pauwels, Mihai Putinar, Jean B. Lasserre
    Abstract:

    Spectral features of the empirical Moment Matrix constitute a resourceful tool for unveiling properties of a cloud of points, among which, density, support and latent structures. It is already well known that the empirical Moment Matrix encodes a great deal of subtle attributes of the underlying measure. Starting from this object as base of observations we combine ideas from statistics, real algebraic geometry, orthogonal polynomials and approximation theory for opening new insights relevant for Machine Learning (ML) problems with data supported on singular sets. Refined concepts and results from real algebraic geometry and approximation theory are empowering a simple tool (the empirical Moment Matrix) for the task of solving non-trivial questions in data analysis. We provide (1) theoretical support, (2) numerical experiments and, (3) connections to real world data as a validation of the stamina of the empirical Moment Matrix approach.

  • Spectral analysis of Moment data
    2018
    Co-Authors: Edouard Pauwels, Mihai Putinar, Jean B. Lasserre
    Abstract:

    Spectral features of the empirical Moment Matrix constitute a resourceful tool for unveiling properties of a cloud of points, among which, density, support and latent structures. It is already well known that the empirical Moment Matrix encodes a great deal of subtle attributes of the underlying measure. Starting from this object as base of observations we combine ideas from statistics, real algebraic geometry, orthogonal poly-nomials and approximation theory for opening new insights relevant for Machine Learning (ML) problems with data supported on singular sets. Refined concepts and results from real algebraic geometry and approximation theory are empowering a simple tool (the empirical Moment Matrix) for the task of solving non-trivial questions in data analysis. We provide (1) theoretical validation , (2) numerical experiments and, (3) connections to real world data as a validation of the stamina of the empirical Moment Matrix approach.

  • NIPS - Sorting out typicality with the inverse Moment Matrix SOS polynomial
    2016
    Co-Authors: Jean B. Lasserre, Edouard Pauwels
    Abstract:

    We study a surprising phenomenon related to the representation of a cloud of data points using polynomials. We start with the previously unnoticed empirical observation that, given a collection (a cloud) of data points, the sublevel sets of a certain distinguished polynomial capture the shape of the cloud very accurately. This distinguished polynomial is a sum-of-squares (SOS) derived in a simple manner from the inverse of the empirical Moment Matrix. In fact, this SOS polynomial is directly related to orthogonal polynomials and the Christoffel function. This allows to generalize and interpret extremality properties of orthogonal polynomials and to provide a mathematical rationale for the observed phenomenon. Among diverse potential applications, we illustrate the relevance of our results on a network intrusion detection task for which we obtain performances similar to existing dedicated methods reported in the literature.

  • Sorting out typicality with the inverse Moment Matrix SOS polynomial
    arXiv: Learning, 2016
    Co-Authors: Jean B. Lasserre, Edouard Pauwels
    Abstract:

    We study a surprising phenomenon related to the representation of a cloud of data points using polynomials. We start with the previously unnoticed empirical observation that, given a collection (a cloud) of data points, the sublevel sets of a certain distinguished polynomial capture the shape of the cloud very accurately. This distinguished polynomial is a sum-of-squares (SOS) derived in a simple manner from the inverse of the empirical Moment Matrix. In fact, this SOS polynomial is directly related to orthogonal polynomials and the Christoffel function. This allows to generalize and interpret extremality properties of orthogonal polynomials and to provide a mathematical rationale for the observed phenomenon. Among diverse potential applications, we illustrate the relevance of our results on a network intrusion detection task for which we obtain performances similar to existing dedicated methods reported in the literature.

Kenneth A.@article2001-17287-00320010501, Abstract = Examines The Most Common Type Of Improper Solut Bollen - One of the best experts on this subject based on the ideXlab platform.

  • Wiley StatsRef: Statistics Reference Online - Structural Equation Models including Historical Origins
    Wiley StatsRef: Statistics Reference Online, 2014
    Co-Authors: Kenneth A.@article2001-17287-00320010501, Abstract = Examines The Most Common Type Of Improper Solut Bollen
    Abstract:

    Structural equation models refer to general statistical procedures for multiequation systems that include continuous latent variables, multiple indicators of concepts, errors of measurement, errors in equations, and observed variables. An analysis that uses structural equation models has several components. These include (a) model specification, (b) the implied Moment Matrix, (c) identification, (d) estimation, (e) model fit, and (f) respecification. Historical origins of structural equation models are also described. Keywords: structural equation models; factor loading Matrix; path analysis; implied Moment Matrix; model identification; respecification

  • Encyclopedia of Biostatistics - Structural Equation Models
    Oxford Handbooks Online, 2009
    Co-Authors: Kenneth A.@article2001-17287-00320010501, Abstract = Examines The Most Common Type Of Improper Solut Bollen, Sophia Rabe-hesketh, Anders Skrondal
    Abstract:

    Structural equation models refer to general statistical procedures for multiequation systems that include continuous latent variables, multiple indicators of concepts, errors of measurement, errors in equations, and observed variables. An analysis that uses structural equation models has several components. These include (a) model specification, (b) the implied Moment Matrix, (c) identification, (d) estimation, (e) model fit, and (f) respecification. Historical origins of structural equation models are also described. Keywords: structural equation models; factor loading Matrix; path analysis; implied Moment Matrix; model identification; respecification

  • structural equation models
    Encyclopedia of Biostatistics, 2005
    Co-Authors: Kenneth A.@article2001-17287-00320010501, Abstract = Examines The Most Common Type Of Improper Solut Bollen, Sophia Rabehesketh, Anders Skrondal
    Abstract:

    Structural equation models refer to general statistical procedures for multiequation systems that include continuous latent variables, multiple indicators of concepts, errors of measurement, errors in equations, and observed variables. An analysis that uses structural equation models has several components. These include (a) model specification, (b) the implied Moment Matrix, (c) identification, (d) estimation, (e) model fit, and (f) respecification. Historical origins of structural equation models are also described. Keywords: structural equation models; factor loading Matrix; path analysis; implied Moment Matrix; model identification; respecification