Citation
Kravaris, Constantine (1984) Identification of SpatiallyVarying Parameters in Distributed Parameter Systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/m0kvm285. https://resolver.caltech.edu/CaltechETD:etd01092007104956
Abstract
Identification of spatiallyvarying parameters in distributed parameter systems given an observation of the state is as a rule an illposed problem in the sense of Hadamard. Even in case when the solution is unique, it does not depend continuously on the data. The identification problem that motivated this work arises in the description of petroleum reservoirs and subsurface aquifers; it consists of identifying the spatiallyvarying parameter α(x,y) in the diffusion equation u_{t} = (αu_{x})_{x} + (αu_{y})_{y} + f given an observation of u at a discrete set of spatial locations.
The question of uniqueness of α (identifiability problem) is first investigated. The analysis is restricted to the onedimensional version of the above equation i.e. to u_{t} = (αu_{x})_{x} + f and an observation of u at a single point. The identifiability problem is formulated as an inverse SturmLiouville problem for (αy')' + λy = 0. It is proved that the eigenvalues and the normalizing constants determine the above SturmLiouville operator uniquely. Identifiability and nonidentifiability results are obtained for three special cases.
The problem of constructing stable approximate solutions to identification problems in distributed parameter systems is next investigated. The concept of regularization, widely used in solving linear Fredholm integral equations, is developed for the solution of such problems. A general regularization identification theory is presented and applied to the identification of parabolic systems. Two alternative numerical approaches for the minimization of the smoothing functional are investigated: (i) classical Banach space gradient methods and (ii) discretized minimization methods. The latter use finitedimensional convergent approximations in Sobolev spaces and are based on an appropriate convergence theorem. The performance of the regularization identification method is evaluated by numerical experiments on the identification of spatiallyvarying diffusivity α in the diffusion equation.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Chemical Engineering 
Degree Grantor:  California Institute of Technology 
Division:  Chemistry and Chemical Engineering 
Major Option:  Chemical Engineering 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  9 December 1983 
Additional Information:  Author also known as Costas Kravaris. 
Record Number:  CaltechETD:etd01092007104956 
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd01092007104956 
DOI:  10.7907/m0kvm285 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  85 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  09 Jan 2007 
Last Modified:  19 Apr 2021 22:40 
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