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Title: A new method for obtaining velocity and diffusivity from time-dependent distributions of a tracer via the maximum likelihood estimator for the advection-diffusion equation

Abstract

An inverse problem for the advection-diffusion equation is considered, and a method of maximum likelihood (ML) estimation is developed to derive velocity and diffusivity from time-dependent 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 two-dimensional 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 Galerkin-type method-as a response of the conservation equation to stochastic forcing. In the inverse problem, the advection-diffusion 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 finite-differencemore » 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.« less

Authors:
 [1];  [2]
  1. Kyushu Univ., Kasuga (Japan)
  2. 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; MAXIMUM-LIKELIHOOD FIT

Citation Formats

Ostrovskii, A.G., and Piterbarg, L.I. A new method for obtaining velocity and diffusivity from time-dependent distributions of a tracer via the maximum likelihood estimator for the advection-diffusion 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 time-dependent distributions of a tracer via the maximum likelihood estimator for the advection-diffusion 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 time-dependent distributions of a tracer via the maximum likelihood estimator for the advection-diffusion equation". United States. doi:10.1006/jcph.1997.5674.
@article{osti_535390,
title = {A new method for obtaining velocity and diffusivity from time-dependent distributions of a tracer via the maximum likelihood estimator for the advection-diffusion equation},
author = {Ostrovskii, A.G. and Piterbarg, L.I.},
abstractNote = {An inverse problem for the advection-diffusion equation is considered, and a method of maximum likelihood (ML) estimation is developed to derive velocity and diffusivity from time-dependent 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 two-dimensional 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 Galerkin-type method-as a response of the conservation equation to stochastic forcing. In the inverse problem, the advection-diffusion 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 finite-difference 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 advection-diffusion equation with time dependent velocity and anisotropic time dependent diffusion tensor is examined in regard to its non-classical transport features and to the use of a non-orthogonal 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 time-dependent 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 power-law correlationmore » functions where the pulse diffuses faster than the classical rate ~t, a second case specically designed to display slower rate of diffusion than the classical one, and a third case to describe hydrodynamic dispersion in porous media« less
  • The need for measuring fluorescence lifetimes of species in subdiffraction-limited 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 excited-state 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 well-characterized 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 » “photon counts” was approximately 20, 200, 1000, 3000, and 6000 and there were about 2–200 counts at the maxima of the respective decays. Each set of experiments was repeated 50 times to generate the appropriate statistics. Each of the 250 data sets was analyzed by ML and two different RM methods (differing in the weighting of residuals) using in-house routines and compared with a frequently used commercial RM routine. Convolution with a real instrument response function was always included in the fitting. While RM using Pearson’s weighting of residuals can recover the correct mean result with a total number of counts of 1000 or more, ML distinguishes itself by yielding, in all cases, the same mean lifetime within 2% of the accepted value. For 200 total counts and greater, ML always provides a standard deviation of <10% of the mean lifetime, and even at 20 total counts there is only 20% error in the mean lifetime. Here, the robustness of ML advocates its use for sparse data sets such as those acquired in some subdiffraction-limited microscopies, such as STED, and, more importantly, provides greater motivation for exploiting the time-resolved capacities of this technique to acquire and analyze fluorescence lifetime data.« less
  • 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 Levenberg-Marquardt 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 » counted events from their data, and extract parameters by fitting to a specified model. Assuming that the probability of occurrence for each bin is small, event counts in the histogram bins will be distributed according to the Poisson distribution. We develop here an efficient algorithm for fitting event counting histograms using the maximum likelihood estimator (MLE) for Poisson distributed data, rather than the non-linear least squares measure. This algorithm is a simple extension of the common Levenberg-Marquardt (L-M) algorithm, is simple to implement, quick and robust. Fitting using a least squares measure is most common, but it is the maximum likelihood estimator only for Gaussian-distributed data. Non-linear least squares methods may be applied to event counting histograms in cases where the number of events is very large, so that the Poisson distribution is well approximated by a Gaussian. However, it is not easy to satisfy this criterion in practice - which requires a large number of events. It has been well-known for years that least squares procedures lead to biased results when applied to Poisson-distributed data; a recent paper providing extensive characterization of these biases in exponential fitting is given. The more appropriate measure based on the maximum likelihood estimator (MLE) for the Poisson distribution is also well known, but has not become generally used. This is primarily because, in contrast to non-linear least squares fitting, there has been no quick, robust, and general fitting method. In the field of fluorescence lifetime spectroscopy and imaging, there have been some efforts to use this estimator through minimization routines such as Nelder-Mead optimization, exhaustive line searches, and Gauss-Newton minimization. Minimization based on specific one- or multi-exponential models has been used to obtain quick results, but this procedure does not allow the incorporation of the instrument response, and is not generally applicable to models found in other fields. Methods for using the MLE for Poisson-distributed data have been published by the wider spectroscopic community, including iterative minimization schemes based on Gauss-Newton minimization. The slow acceptance of these procedures for fitting event counting histograms may also be explained by the use of the ubiquitous, fast Levenberg-Marquardt (L-M) fitting procedure for fitting non-linear models using least squares fitting (simple searches obtain {approx}10000 references - this doesn't include those who use it, but don't know they are using it). The benefits of L-M include a seamless transition between Gauss-Newton minimization and downward gradient minimization through the use of a regularization parameter. This transition is desirable because Gauss-Newton methods converge quickly, but only within a limited domain of convergence; on the other hand the downward gradient methods have a much wider domain of convergence, but converge extremely slowly nearer the minimum. L-M has the advantages of both procedures: relative insensitivity to initial parameters and rapid convergence. Scientists, when wanting an answer quickly, will fit data using L-M, get an answer, and move on. Only those that are aware of the bias issues will bother to fit using the more appropriate MLE for Poisson deviates. However, since there is a simple, analytical formula for the appropriate MLE measure for Poisson deviates, it is inexcusable that least squares estimators are used almost exclusively when fitting event counting histograms. There have been ways found to use successive non-linear least squares fitting to obtain similarly unbiased results, but this procedure is justified by simulation, must be re-tested when conditions change significantly, and requires two successive fits. There is a great need for a fitting routine for the MLE estimator for Poisson deviates that has convergence domains and rates comparable to the non-linear least squares L-M fitting. We show in this report that a simple way to achieve that goal is to use the L-M fitting procedure not to minimize the least squares measure, but the MLE for Poisson deviates.« less
  • Abstract not provided.