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    Multiple interval methods for ODEs with an optimization constraint

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    Genre
    Thesis/Dissertation
    Date
    2020
    Author
    Yu, Xinli
    Advisor
    Szyld, Daniel
    Klapper, Isaac
    Committee member
    Queisser, Gillian
    Buttaro, Bettina A.
    Department
    Mathematics
    Subject
    Applied Mathematics
    Biology
    Asynchronous Method
    Biofilm
    Domain Decomposition Method
    Flux Balance Analysis
    Numerical Methods
    Optimization
    Permanent link to this record
    http://hdl.handle.net/20.500.12613/4086
    
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    DOI
    http://dx.doi.org/10.34944/dspace/4068
    Abstract
    We are interested in numerical methods for the optimization constrained second order ordinary differential equations arising in biofilm modelling. This class of problems is challenging for several reasons. One of the reasons is that the underlying solution has a steep slope, making it difficult to resolve. We propose a new numerical method with techniques such as domain decomposition and asynchronous iterations for solving certain types of ordinary differential equations more efficiently. In fact, for our class of problems after applying the techniques of domain decomposition with overlap we are able to solve the ordinary differential equations with a steep slope on a larger domain than previously possible. After applying asynchronous iteration techniques, we are able to solve the problem with less time.~We provide theoretical conditions for the convergence of each of the techniques. The other reason is that the second order ordinary differential equations are coupled with an optimization problem, which can be viewed as the constraints. We propose a numerical method for solving the coupled problem and show that it converges under certain conditions. An application of the proposed methods on biofilm modeling is discussed. The numerical method proposed is adopted to solve the biofilm problem, and we are able to solve the problem with larger thickness of the biofilm than possible before as is shown in the numerical experiments.
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