A new method for obtaining velocity and diffusivity from timedependent distributions of a tracer via the maximum likelihood estimator for the advectiondiffusion equation
Abstract
An inverse problem for the advectiondiffusion equation is considered, and a method of maximum likelihood (ML) estimation is developed to derive velocity and diffusivity from timedependent distributions of a tracer. Piterbarg and Rozovskii showed theoretically that the ML estimator for diffusivity is consistent ever in an asymptotic case of infinite number of observational spatial modes. In the present work, the ML estimator is studied based on numerical experiments with a tracer in a twodimensional flow under the condition of a limited number of observations in space. The numerical experiments involve the direct and the inverse problems. For the former, the time evolution of a tracer is simulated using the Galerkintype methodas a response of the conservation equation to stochastic forcing. In the inverse problem, the advectiondiffusion equation is fitted to the simulated data employing the ML estimator. It is shown that the ML method allows us a method to estimate diffusion coefficient components D{sub x} and D{sub y} based on a short time series of tracer observations. The estimate of the diffusion anistropy, D{sub x}/D{sub y}, is shown to be even more robust than the estimate of the diffusivity itself. A comparison with an estimation technique based on the finitedifferencemore »
 Authors:
 Kyushu Univ., Kasuga (Japan)
 Univ. of Southern California, Los Angeles, CA (United States)
 Publication Date:
 OSTI Identifier:
 535390
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Computational Physics; Journal Volume: 133; Journal Issue: 2; Other Information: PBD: 15 May 1997
 Country of Publication:
 United States
 Language:
 English
 Subject:
 66 PHYSICS; 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; DIFFUSION; PARTIAL DIFFERENTIAL EQUATIONS; ADVECTION; NUMERICAL SOLUTION; TIME DEPENDENCE; MAXIMUMLIKELIHOOD FIT
Citation Formats
Ostrovskii, A.G., and Piterbarg, L.I. A new method for obtaining velocity and diffusivity from timedependent distributions of a tracer via the maximum likelihood estimator for the advectiondiffusion equation. United States: N. p., 1997.
Web. doi:10.1006/jcph.1997.5674.
Ostrovskii, A.G., & Piterbarg, L.I. A new method for obtaining velocity and diffusivity from timedependent distributions of a tracer via the maximum likelihood estimator for the advectiondiffusion equation. United States. doi:10.1006/jcph.1997.5674.
Ostrovskii, A.G., and Piterbarg, L.I. 1997.
"A new method for obtaining velocity and diffusivity from timedependent distributions of a tracer via the maximum likelihood estimator for the advectiondiffusion equation". United States.
doi:10.1006/jcph.1997.5674.
@article{osti_535390,
title = {A new method for obtaining velocity and diffusivity from timedependent distributions of a tracer via the maximum likelihood estimator for the advectiondiffusion equation},
author = {Ostrovskii, A.G. and Piterbarg, L.I.},
abstractNote = {An inverse problem for the advectiondiffusion equation is considered, and a method of maximum likelihood (ML) estimation is developed to derive velocity and diffusivity from timedependent distributions of a tracer. Piterbarg and Rozovskii showed theoretically that the ML estimator for diffusivity is consistent ever in an asymptotic case of infinite number of observational spatial modes. In the present work, the ML estimator is studied based on numerical experiments with a tracer in a twodimensional flow under the condition of a limited number of observations in space. The numerical experiments involve the direct and the inverse problems. For the former, the time evolution of a tracer is simulated using the Galerkintype methodas a response of the conservation equation to stochastic forcing. In the inverse problem, the advectiondiffusion equation is fitted to the simulated data employing the ML estimator. It is shown that the ML method allows us a method to estimate diffusion coefficient components D{sub x} and D{sub y} based on a short time series of tracer observations. The estimate of the diffusion anistropy, D{sub x}/D{sub y}, is shown to be even more robust than the estimate of the diffusivity itself. A comparison with an estimation technique based on the finitedifference approximation demonstrates advantages of the ML estimator. Finally, the ML method is employed for analysis of heat balance in the upper layer of the North Pacific in the winter. This application focuses on the heat diffusion anisotropy at the ocean mesoscale. 29 refs., 14 figs.},
doi = {10.1006/jcph.1997.5674},
journal = {Journal of Computational Physics},
number = 2,
volume = 133,
place = {United States},
year = 1997,
month = 5
}

The advectiondiffusion equation with time dependent velocity and anisotropic time dependent diffusion tensor is examined in regard to its nonclassical transport features and to the use of a nonorthogonal coordinate system. Although this equation appears in diverse physical problems, particularly in particle transport in stochastic velocity fields and in underground porous media, a detailed analysis of its solutions is lacking. In order to study the effects of the timedependent coefficients and the anisotropic diffusion on transport, we solve analytically the equation for an initial Dirac delta pulse. Here, we discuss the solutions to three cases: one based on powerlaw correlationmore »

What is the best method to fit timeresolved data? A comparison of the residual minimization and the maximum likelihood techniques as applied to experimental timecorrelated, singlephoton counting data
The need for measuring fluorescence lifetimes of species in subdiffractionlimited volumes in, for example, stimulated emission depletion (STED) microscopy, entails the dual challenge of probing a small number of fluorophores and fitting the concomitant sparse data set to the appropriate excitedstate decay function. This need has stimulated a further investigation into the relative merits of two fitting techniques commonly referred to as “residual minimization” (RM) and “maximum likelihood” (ML). Fluorescence decays of the wellcharacterized standard, rose bengal in methanol at room temperature (530 ± 10 ps), were acquired in a set of five experiments in which the total number ofmore »Cited by 2 
Efficient LevenbergMarquardt minimization of the maximum likelihood estimator for Poisson deviates
Histograms of counted events are Poisson distributed, but are typically fitted without justification using nonlinear least squares fitting. The more appropriate maximum likelihood estimator (MLE) for Poisson distributed data is seldom used. We extend the use of the LevenbergMarquardt algorithm commonly used for nonlinear least squares minimization for use with the MLE for Poisson distributed data. In so doing, we remove any excuse for not using this more appropriate MLE. We demonstrate the use of the algorithm and the superior performance of the MLE using simulations and experiments in the context of fluorescence lifetime imaging. Scientists commonly form histograms ofmore » 
A New MaximumLikelihood Change Estimator for TwoPass SAR Coherent Change Detection.
Abstract not provided.