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Title: Bayesian estimation of Karhunen–Loève expansions; A random subspace approach

Authors:
;
Publication Date:
Sponsoring Org.:
USDOE
OSTI Identifier:
1324851
Grant/Contract Number:
AC04-94-AL85000
Resource Type:
Journal Article: Publisher's Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 319; Journal Issue: C; Related Information: CHORUS Timestamp: 2017-07-04 09:36:31; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English

Citation Formats

Chowdhary, Kenny, and Najm, Habib N. Bayesian estimation of Karhunen–Loève expansions; A random subspace approach. United States: N. p., 2016. Web. doi:10.1016/j.jcp.2016.02.056.
Chowdhary, Kenny, & Najm, Habib N. Bayesian estimation of Karhunen–Loève expansions; A random subspace approach. United States. doi:10.1016/j.jcp.2016.02.056.
Chowdhary, Kenny, and Najm, Habib N. 2016. "Bayesian estimation of Karhunen–Loève expansions; A random subspace approach". United States. doi:10.1016/j.jcp.2016.02.056.
@article{osti_1324851,
title = {Bayesian estimation of Karhunen–Loève expansions; A random subspace approach},
author = {Chowdhary, Kenny and Najm, Habib N.},
abstractNote = {},
doi = {10.1016/j.jcp.2016.02.056},
journal = {Journal of Computational Physics},
number = C,
volume = 319,
place = {United States},
year = 2016,
month = 8
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record at 10.1016/j.jcp.2016.02.056

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  • Resolving weak spectral variations in the dynamic response of materials that are either dominated or excited by stochastic processes remains a challenge. Responses that are thermal in origin are particularly relevant examples due to the delocalized nature of heat. Despite its inherent properties in dealing with stochastic processes, the Karhunen–Loève expansion has not been fully exploited in measurement of systems that are driven solely by random forces or can exhibit large thermally driven random fluctuations. Here in this paper, we present experimental results and analysis of the archetypes (a) the resonant excitation and transient response of an atomic force microscopemore » probe by the ambient random fluctuations and nanoscale photothermal sample response, and (b) the photothermally scattered photons in pump–probe spectroscopy. In each case, the dynamic process is represented as an infinite series with random coefficients to obtain pertinent frequency shifts and spectral peaks and demonstrate spectral enhancement for a set of compounds including the spectrally complex biomass. The considered cases find important applications in nanoscale material characterization, biosensing, and spectral identification of biological and chemical agents.« less
  • While parametric uncertainty quantification for NDE models has been addressed in recent years, the problem of stochastic field parameters such as spatially distributed electrical conductivity has only been investigated minimally in the last year. In that work, the authors treated the field as a one-dimensional random process and Karhunen-Loeve methods were used to discretize this process to make it amenable to UQ methods such as ANOVA expansions. In the present work, we will treat the field as a two dimensional random process, and the eigenvalues and eigenfunctions of the integral operator will be determined via Galerkin methods. The Karhunen-Loeve methodsmore » is extended to two dimensions and implemented to represent this process. Several different choices for basis functions will be discussed, as well as convergence criteria for each. The methods are applied to correlation functions collected over electron backscatter data from highly micro textured Ti-7Al.« less
  • An important issue in nonlinear dynamics is the optimal estimation and detection of the partially observed states of a system at low signal-to-noise ratios. In this paper, we will outline a Bayesian-based approach that allows for the optimal determination of the state probability density function in time as a function of the observations. This leads to optimal detector designs based on the notion of generalized innovation sequences. Here, the density functions are defined over a computational grid which is designed to capture the phase space dynamics of the nonlinear system. Partial measurements are used to update the projected system statemore » and density function. Estimation and detection decisions are based on the propagated density functions. {copyright} {ital 1996 American Institute of Physics.}« less
  • The mixing of groundwaters with different ages in aquifers, groundwater age is more appropriately represented by a distribution rather than a scalar number. To infer a groundwater age distribution from environmental tracers, a mathematical form is often assumed for the shape of the distribution and the parameters of the mathematical distribution are estimated using deterministic or stochastic inverse methods. We found that the prescription of the mathematical form limits the exploration of the age distribution to the shapes that can be described by the selected distribution. In this paper, the use of freeform histograms as groundwater age distributions is evaluated.more » A Bayesian Markov Chain Monte Carlo approach is used to estimate the fraction of groundwater in each histogram bin. This method was able to capture the shape of a hypothetical gamma distribution from the concentrations of four age tracers. The number of bins that can be considered in this approach is limited based on the number of tracers available. The histogram method was also tested on tracer data sets from Holten (The Netherlands; 3H, 3He, 85Kr, 39Ar) and the La Selva Biological Station (Costa-Rica; SF 6, CFCs, 3H, 4He and 14C), and compared to a number of mathematical forms. According to standard Bayesian measures of model goodness, the best mathematical distribution performs better than the histogram distributions in terms of the ability to capture the observed tracer data relative to their complexity. Among the histogram distributions, the four bin histogram performs better in most of the cases. The Monte Carlo simulations showed strong correlations in the posterior estimates of bin contributions, indicating that these bins cannot be well constrained using the available age tracers. The fact that mathematical forms overall perform better than the freeform histogram does not undermine the benefit of the freeform approach, especially for the cases where a larger amount of observed data is available and when the real groundwater distribution is more complex than can be represented by simple mathematical forms.« less