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Levenberg-Marquardt Methods for Parameter Estimation Problems in the Radiative Transfer Equation
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics. (DPC - Digital printing center)ORCID iD: 0000-0002-0529-1009
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
2007 (English)In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 23, no 3, 879-891 p., 002Article in journal (Refereed) Published
Abstract [en]

A discrete ordinate method is developed for solving the radiative transfer equation, and the corresponding parameter estimation problem is given a least-squares formulation. Two Levenberg-Marquardt methods, a feasible-path approach and an SQP type method, are analyzed and compared. A sensitivity analysis is given, and it is shown how it can be used for designing measurements with minimal impact of measurement noise. Numerical experiments are performed to exemplify the usefulness of the theory.

Place, publisher, year, edition, pages
2007. Vol. 23, no 3, 879-891 p., 002
Keyword [en]
Radiative transfer, parameter estimation
National Category
Mathematics
Identifiers
URN: urn:nbn:se:miun:diva-4111DOI: 10.1088/0266-5611/23/3/002ISI: 000246789100002Scopus ID: 2-s2.0-34249701208Local ID: 4718OAI: oai:DiVA.org:miun-4111DiVA: diva2:29143
Available from: 2008-09-30 Created: 2008-09-30 Last updated: 2016-10-05Bibliographically approved
In thesis
1. Mathematical modeling and numerical tools for simulation and design of light scattering in paper and print
Open this publication in new window or tab >>Mathematical modeling and numerical tools for simulation and design of light scattering in paper and print
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This work starts with a real industrial problem - the perceived need for a moredetailed and more accurate model for light scattering in paper and print than theKubelka‐Munk model of today. A careful analysis transfers this problem into aphysical description of the phenomena involved. This is then given a mathematicalformulation, and a detailed analysis leads to numerical solution procedures forspecific sub problems. Methods from scientific computing make it possible to meetindustrial demands made on speed and stability, and implementation in computercode is then followed by analysis of accuracy and stability.A problem formulation and a solution method are outlined for the forwardradiative transfer problem. First, all necessary steps to arrive at a numericallystable solution procedure are treated, and then methods are introduced to increasethe speed by a factor of several thousands or millions compared to a naiveapproach. The method is shown to be unconditionally stable, though the problemwas previously considered numerically intractable, and systematic studies ofnumerical performance are presented.The inverse radiative transfer problem is given a least‐squares formulation, anddifferent solution methods are analyzed and compared. Specifically, a two‐phasemethod for estimation of the scattering and absorption coefficients and theasymmetry factor (σs, σa and g) is presented. A sensitivity analysis is given, and it isshown how it can be used for designing measurements with minimal impact frommeasurement noise.It is shown how the standardized use of Kubelka‐Munk and the d/0°instrument leads to errors, and that the errors arising from an over‐idealized viewof the instrument - due to the fact that instrument readings are incorrectlyinterpreted - can be larger than any errors inherent in the Kubelka‐Munk modelitself. It is argued that the measurement device and the simulation model cannot beviewed as separate instances, which is a widespread implicit practice in appliedreflectance measurements. Rather, given a measurement device, measurement datashould be interpreted through a model that takes into consideration the actualgeometry, function and calibration of the instrument.The resulting tool, DORT2002, is in all aspects the Next Generation Kubelka‐Munk, and provides a greater range of applicability, higher accuracy and increasedunderstanding. It offers better interpretation of measurement data, and facilitatesthe exchange of data between the paper and graphical arts industries. It opens forunderstanding of anisotropic reflectance and for the utilization of the asymmetryfactor to design anisotropy, and thereby for the design of different visualappearance or optical performance in new printed or paper products.

Place, publisher, year, edition, pages
Sundsvall: Mittuniversitetet, 2007. 32 p.
Series
Mid Sweden University doctoral thesis, ISSN 1652-893X ; 22
Keyword
mathematical modeling, radiative transfer, integro-differential equations, inverse problems, parameter estimation, solution method, numerical performance, light scattering, paper industry applications, Kubelka-Munk
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-5908 (URN)5026 (Local ID)978-91-85317-50-9 (ISBN)5026 (Archive number)5026 (OAI)
Public defence
(English)
Available from: 2008-09-30 Created: 2009-05-06 Last updated: 2009-07-13Bibliographically approved

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