Browsing by Author "Geldermans, Peter"

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    Accelerated PDE Constrained Optimization using Direct Solvers
    (2018-04-17) Geldermans, Peter; Gillman, Adrianna; Heinkenschloss, Matthias
    In this thesis, I propose a method to reduce the cost of computing solutions to optimization problems governed by partial differential equations (PDEs). Standard second order methods such as Newton require the solution of two PDEs per iteration of the Newton system, which can be prohibitively expensive for iterative solvers. In contrast, this work takes advantage a recently developed high order discretization method that comes with an efficient direct solver. The new technique precomputes a solution operator that can be reused for any body load, which is applied whenever a PDE solve is required. Thus the precomputation cost is amortized over many PDE solves. This approach will make second order optimization algorithms computationally affordable for practical applications such as photoacoustic tomography and optimal design problems.
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    Robust and efficient numerical algorithms for the discrete prolate spheroidal wave functions
    (2019-04-16) Geldermans, Peter; Gillman, Adrianna
    This thesis presents novel algorithms for the numerical evaluation of the discrete prolate spheroidal wave functions (DPSWFs) and their associated integral operator eigenvalues. The DPSWFs and associated eigenvalues arise in a variety of science and engineering applications including signal processing, communications technology, paleoclimatology, fluid dynamics, and wave phenomena. Existing algorithms compute the integral operator eigenvalues to high relative accuracy when the eigenvalues are not close to zero. However, the integral operator eigenvalues computed by these algorithms lose all digits of relative accuracy when the eigenvalues are small. The new numerical algorithms compute the eigenvalues to high relative accuracy independent of their mangitude. The proposed algorithms exploit the fact that the integral operator commutes with a second order linear differential operator. While this differential operator was identified in 1978, it has not been used in numerical algorithms to evaluate the DPSWFs nor the associated eigenvalues until this work. Numerical experiments demonstrate the accuracy of the proposed algorithms. The design of the proposed algorithms exploits several properties of the DPSWFs to reduce computational cost. Furthermore, the use of high order numerical methods ensures that the algorithms are efficient. In addition to algorithms for computing the eigenvalues to high relative accuracy, several new properties of the DPSWFs are derived.