The Experts below are selected from a list of 237627 Experts worldwide ranked by ideXlab platform

Lizuo Liu - One of the best experts on this subject based on the ideXlab platform.

  • a phase shift deep neural network for high frequency Approximation and wave problems
    SIAM Journal on Scientific Computing, 2020
    Co-Authors: Wei Cai, Lizuo Liu
    Abstract:

    In this paper, we propose a phase shift deep neural network (PhaseDNN), which provides a uniform wideband convergence in approximating high frequency functions and solutions of wave equations. The ...

  • a phase shift deep neural network for high frequency Approximation and wave problems
    arXiv: Learning, 2019
    Co-Authors: Wei Cai, Lizuo Liu
    Abstract:

    In this paper, we propose a phase shift deep neural network (PhaseDNN), which provides a uniform wideband convergence in approximating high frequency functions and solutions of wave equations. The PhaseDNN makes use of the fact that common DNNs often achieve convergence in the low frequency range first, and a series of moderately-sized DNNs are constructed and trained for selected high frequency ranges. With the help of phase shifts in the frequency domain, each of the DNNs will be trained to approximate the function's higher frequency content over a specific range at the the speed of convergence as in the low frequency range. As a result, the proposed PhaseDNN is able to convert high frequency learning to low frequency one, allowing a uniform learning to wideband functions. The PhaseDNN will then be applied to find the solution of high frequency wave equations in inhomogeneous media through both differential and integral equation formulations with least square residual loss functions. Numerical results have demonstrated the capability of the PhaseDNN in learning high frequency functions and oscillatory solutions of interior and exterior Helmholtz equations.

Wei Cai - One of the best experts on this subject based on the ideXlab platform.

  • a phase shift deep neural network for high frequency Approximation and wave problems
    SIAM Journal on Scientific Computing, 2020
    Co-Authors: Wei Cai, Lizuo Liu
    Abstract:

    In this paper, we propose a phase shift deep neural network (PhaseDNN), which provides a uniform wideband convergence in approximating high frequency functions and solutions of wave equations. The ...

  • a phase shift deep neural network for high frequency Approximation and wave problems
    arXiv: Learning, 2019
    Co-Authors: Wei Cai, Lizuo Liu
    Abstract:

    In this paper, we propose a phase shift deep neural network (PhaseDNN), which provides a uniform wideband convergence in approximating high frequency functions and solutions of wave equations. The PhaseDNN makes use of the fact that common DNNs often achieve convergence in the low frequency range first, and a series of moderately-sized DNNs are constructed and trained for selected high frequency ranges. With the help of phase shifts in the frequency domain, each of the DNNs will be trained to approximate the function's higher frequency content over a specific range at the the speed of convergence as in the low frequency range. As a result, the proposed PhaseDNN is able to convert high frequency learning to low frequency one, allowing a uniform learning to wideband functions. The PhaseDNN will then be applied to find the solution of high frequency wave equations in inhomogeneous media through both differential and integral equation formulations with least square residual loss functions. Numerical results have demonstrated the capability of the PhaseDNN in learning high frequency functions and oscillatory solutions of interior and exterior Helmholtz equations.

Maarten V De Hoop - One of the best experts on this subject based on the ideXlab platform.

  • microlocal analysis of seismic inverse scattering in anisotropic elastic media
    Communications on Pure and Applied Mathematics, 2002
    Co-Authors: Christiaan C Stolk, Maarten V De Hoop
    Abstract:

    Seismic data is modeled in the High-Frequency Approximation, using the techniques of microlocal analysis. We consider general, anisotropic elastic media. Our methods are designed to allow for the formation of caustics. The data is modeled in two ways. First, we give a microlocal treatment of the Kirchhoff Approximation, where the medium is assumed to be piecewise smooth, and reflection and transmission occur at interfaces. Second, we give a refined view on the Born Approximation based upon a linearization of the scattering process in the medium parameters around a smooth background medium. The joint formulation of Born and Kirchhoff scattering allows us to take into account general scatterers as well as the nonlinear dependence of reflection coefficients on the medium parameters. The latter allows the treatment of scattering up to grazing angles. The outcome of the analysis is a characterization of the singular part of seismic data. We obtain a set of pseudodifferential operators that annihilate the data. In the process we construct a Fourier integral operator and a reflectivity function such that the data can be represented by this operator acting on the reflectivity function. In our construction this Fourier integral operator becomes invertible. We give the conditions for invertibility for general acquisition geometry. The result is also of interest for inverse scattering in acoustic media. c 2002 John Wiley & Sons, Inc.

Peng Xu - One of the best experts on this subject based on the ideXlab platform.

  • on the geometrical optics hagfors law and physical optics Approximations for scattering from exponentially correlated surfaces
    IEEE Transactions on Geoscience and Remote Sensing, 2007
    Co-Authors: J T Johnson, Karl F Warnick, Peng Xu
    Abstract:

    High-Frequency Approximations to the physical optics (PO) theory of scattering from exponentially correlated rough surfaces are examined and used to interpret the expected accuracy of the PO theory. As an introduction, a review of the PO theory for Gaussian-correlated surfaces is provided, and in this process an analytical summation of the PO series for specular scattering from Gaussian-correlated surfaces is obtained. A similar form is then derived for specular scattering from exponentially correlated surfaces and contrasted to the Gaussian case. These series allow the accuracy of the leading order term (i.e., the geometrical optics limit) in the High-Frequency Approximation of PO scattering for Gaussian or exponentially correlated surfaces to be investigated analytically. The leading order term in the High-Frequency expansion for general PO scattering from exponentially correlated surfaces (Hagfors' Law) is then reviewed and interpreted in terms of a recently published theory of PO for surfaces with infinite rms slopes. The approximate ldquocutoffrdquo wavenumber from Hagfors' Law at which the High-Frequency portion of the spectrum of an exponentially correlated surface can be truncated without producing large errors in PO predicted scattering is also discussed. Using this cutoff wavenumber, an approximate region of validity of the complete PO theory for exponentially correlated surfaces is obtained. The validity condition indicates that, for fixed surface statistics, the PO method produces accurate predictions of true surface scattering only up to a specific frequency, and that PO is inaccurate in the High-Frequency limit. Comparisons of PO predictions with those of a Monte Carlo numerical simulation are used to show that the validity condition derived appears to provide a reasonable indication of PO accuracy. These results have important implications for current investigations of scattering from exponentially correlated surfaces and for the use of Hagfors' Law, as it is traditional to accept PO as the appropriate High-Frequency limit in most existing approximate models of surface scattering.

Blas M Vinagre - One of the best experts on this subject based on the ideXlab platform.

  • a new iir type digital fractional order differentiator
    Signal Processing, 2003
    Co-Authors: Yangquan Chen, Blas M Vinagre
    Abstract:

    A new infinite impulse response (IIR)-type digital fractional order differentiator (DFOD) is proposed by using a new family of first-order digital differentiators expressed in the second-order IIR filter form. The integer first-order digital differentiators are obtained by the stable inversion of the weighted sum of Simpson integration rule and the trapezoidal integration rule. The distinguishing point of the proposed DFOD lies in an additional tuning knob to compromise the High-Frequency Approximation accuracy.

  • a new discretization method for fractional order differentiators via continued fraction expansion
    ASME 2003 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2003
    Co-Authors: Yangquan Chen, Blas M Vinagre, Igor Podlubny
    Abstract:

    In this contribution, to discretize the fractional order differentiators in continuous time domain, a new IIR (infinite impulse response) type digital fractional order differentiator (DFOD) is proposed by using a new family of first order digital differentiators expressed in the second order IIR filter form. The integer first order digital differentiators are obtained by the stable inversion of the weighted sum of Simpson integration rule and the trapezoidal integration rule. The distinguishing point of the proposed DFOD lies in an additional tuning knob to compromise the high frequency Approximation accuracy.