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    Iterative Observer-based Estimation Algorithms for Steady-State Elliptic Partial Differential Equation Systems

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    Type
    Dissertation
    Authors
    Majeed, Muhammad Usman cc
    Advisors
    Laleg-Kirati, Taous-Meriem cc
    Committee members
    Shamma, Jeff S. cc
    Keyes, David E. cc
    Wu, Ying cc
    Smith, Ralph
    Program
    Applied Mathematics and Computational Science
    KAUST Department
    Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division
    Date
    2017-07-19
    Permanent link to this record
    http://hdl.handle.net/10754/625240
    
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    Abstract
    Steady-state elliptic partial differential equations (PDEs) are frequently used to model a diverse range of physical phenomena. The source and boundary data estimation problems for such PDE systems are of prime interest in various engineering disciplines including biomedical engineering, mechanics of materials and earth sciences. Almost all existing solution strategies for such problems can be broadly classified as optimization-based techniques, which are computationally heavy especially when the problems are formulated on higher dimensional space domains. However, in this dissertation, feedback based state estimation algorithms, known as state observers, are developed to solve such steady-state problems using one of the space variables as time-like. In this regard, first, an iterative observer algorithm is developed that sweeps over regular-shaped domains and solves boundary estimation problems for steady-state Laplace equation. It is well-known that source and boundary estimation problems for the elliptic PDEs are highly sensitive to noise in the data. For this, an optimal iterative observer algorithm, which is a robust counterpart of the iterative observer, is presented to tackle the ill-posedness due to noise. The iterative observer algorithm and the optimal iterative algorithm are then used to solve source localization and estimation problems for Poisson equation for noise-free and noisy data cases respectively. Next, a divide and conquer approach is developed for three-dimensional domains with two congruent parallel surfaces to solve the boundary and the source data estimation problems for the steady-state Laplace and Poisson kind of systems respectively. Theoretical results are shown using a functional analysis framework, and consistent numerical simulation results are presented for several test cases using finite difference discretization schemes.
    Description
    A recording of the defense presentation for this dissertation is available at: http://hdl.handle.net/10754/625197
    Citation
    Majeed, M. U. (2017). Iterative Observer-based Estimation Algorithms for Steady-State Elliptic Partial Differential Equation Systems. KAUST Research Repository. https://doi.org/10.25781/KAUST-90472
    DOI
    10.25781/KAUST-90472
    Additional Links
    http://hdl.handle.net/10754/625197
    ae974a485f413a2113503eed53cd6c53
    10.25781/KAUST-90472
    Scopus Count
    Collections
    Applied Mathematics and Computational Science Program; PhD Dissertations; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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