Well-posedness in Maxwell systems with distributions of polarization relaxation parameters

Banks, Harvey Thomas; Gibson, Nathan L.

doi:10.1016/j.aml.2004.02.008

Cited by 32 publications

(43 citation statements)

References 11 publications

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“…However, the estimated population data obtained using SPL (5,35,9,35) gave a much better fit to the data for Day 216 than the results obtained using Figure 8: a) Estimated probability density and b) estimated probability distribution for DEL (32,16) SPL (5,35,5,35) and SPL (5,35,7,35). The corresponding residual, J, for SPL (5,35,9,35) is 0.0054, and these results were obtained in 1.8822 × 10 3 seconds, or approximately 32 minutes. In comparison to the estimated population data produced by the estimated growth rate distribution from DEL (32,16), the estimated population data produced by the estimated growth rate distribution from SPL (5,35,9,35) does not give as good a fit to the field data, as is seen in Figure 9.…”

Section: Computational Results For Inverse Problems With Field Datamentioning

confidence: 68%

“…The corresponding residual, J, for SPL (5,35,9,35) is 0.0054, and these results were obtained in 1.8822 × 10 3 seconds, or approximately 32 minutes. In comparison to the estimated population data produced by the estimated growth rate distribution from DEL (32,16), the estimated population data produced by the estimated growth rate distribution from SPL (5,35,9,35) does not give as good a fit to the field data, as is seen in Figure 9. The estimated probability density and distribution are shown in Figure 10.…”

Section: Computational Results For Inverse Problems With Field Datamentioning

confidence: 81%

“…While no useful information can be obtained from Figure 8a, the estimated probability distribution in Figure 8b appears to be Bi-Gaussian in both b and γ, which is expected based on results from [9] and [10]. (32,16) The best results obtained using SPL (M 1 , N 1 , M 2 , N 2 ) for the estimation of the growth rate distribution of the field data are shown in Figure 9 for SPL (5,35,9,35). We note that the results obtained using SPL (5,35,5,35) and SPL (5,35,7,35) were approximately the same as those obtained using SPL (5,35,9,35) for Day 202 and Day 209.…”

Section: Computational Results For Inverse Problems With Field Datamentioning

confidence: 88%

“…(32,16) The best results obtained using SPL (M 1 , N 1 , M 2 , N 2 ) for the estimation of the growth rate distribution of the field data are shown in Figure 9 for SPL (5,35,9,35). We note that the results obtained using SPL (5,35,5,35) and SPL (5,35,7,35) were approximately the same as those obtained using SPL (5,35,9,35) for Day 202 and Day 209. However, the estimated population data obtained using SPL (5,35,9,35) gave a much better fit to the data for Day 216 than the results obtained using Figure 8: a) Estimated probability density and b) estimated probability distribution for DEL (32,16) SPL (5,35,5,35) and SPL (5,35,7,35).…”

Section: Computational Results For Inverse Problems With Field Datamentioning

confidence: 95%

“…No dynamics are available for individual trajectories x(t, q) for a given q ∈ Q. Such problems arise in viscoelasticity and electromagnetics as well as biology [3,5,6,12,23]. While the approximations we discuss in this paper are applicable to all three types of problems, we shall illustrate the computational results in the context of size-structured marine populations where the inverse problems are of Type II.…”

Section: Introduction and Theoretical Backgroundmentioning

confidence: 99%

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A comparison of approximation methods for the estimation of probability distributions on parameters

Banks

Davis

2007

Applied Numerical Mathematics

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In this paper, we compare two computationally efficient approximation methods for the estimation of growth rate distributions in size-structured population models. After summarizing the underlying theoretical framework, we present several numerical examples as validation of the theory. Furthermore, we compare the results from a spline based approximation method and a delta function based approximation method for the inverse problem involving the estimation of the distributions of growth rates in size-structured mosquitofish populations. Convergence as well as sensitivity of the estimates with respect to noise in the data are discussed for both approximation methods. 1 Report Documentation PageForm Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

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Section: Computational Results For Inverse Problems With Field Datamentioning

confidence: 68%

Section: Computational Results For Inverse Problems With Field Datamentioning

confidence: 81%

Section: Computational Results For Inverse Problems With Field Datamentioning

confidence: 88%

Section: Computational Results For Inverse Problems With Field Datamentioning

confidence: 95%

Section: Introduction and Theoretical Backgroundmentioning

confidence: 99%

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A comparison of approximation methods for the estimation of probability distributions on parameters

Banks

Davis

2007

Applied Numerical Mathematics

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Analysis of stability and dispersion in a finite element method for Debye and Lorentz dispersive media

Banks

Bokil

Gibson

2008

Numerical Methods Partial

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Abstract:We study the stability properties of, and the phase error present in, a finite element scheme for Maxwell's equations coupled with Debye or Lorentz polarization. In one dimension we consider a second order formulation for the electric field with an ordinary differential equation for the polarization added as an auxiliary constraint. The finite element method uses linear finite elements in space for the electric field as well as the polarization, and a theta scheme for the time discretization. Numerical experiments suggest the method is unconditionally stable for both Debye and Lorentz models. We compare the stability and phase error properties of the method presented here with those of finite difference methods that have been analyzed in the literature. We also conduct numerical simulations that verify the stability and dispersion properties of the scheme. Keywords Report Documentation Page Form Approved OMB No. 0704-0188Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

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A finite element method for Maxwell polynomial chaos Debye model

Yao

Zhou

Jia

2018

Applied Mathematics and Computation

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Well-posedness in Maxwell systems with distributions of polarization relaxation parameters

Abstract: We present existence, uniqueness and continuous dependence (with respect to probability distributions on polarization parameters) of solutions in Maxwell systems. This provides a theoretical and computational foundation for associated inverse problems.

Cited by 32 publications

References 11 publications

A comparison of approximation methods for the estimation of probability distributions on parameters

A comparison of approximation methods for the estimation of probability distributions on parameters

Analysis of stability and dispersion in a finite element method for Debye and Lorentz dispersive media

A finite element method for Maxwell polynomial chaos Debye model

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