Browsing by Author "Hokanson, Jeffrey M."

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    Magnetic damping of an elastic conductor
    (2009) Hokanson, Jeffrey M.; Embree, Mark; Cox, Steven J.
    Many applications call for a design that maximizes the rate of energy decay. Typical problems of this class include one dimensional damped wave operators, where energy dissipation is caused by a damping operator acting on the velocity. Two damping operators are well understood: a multiplication operator (known as viscous damping) and a scaled Laplacian (known as Kelvin---Voigt damping). Paralleling the analysis of viscous damping, this thesis investigates energy decay for a novel third operator known as magnetic damping, where the damping is expressed via a rank-one self-adjoint operator, dependent on a function a. This operator describes a conductive monochord embedded in a spatially varying magnetic field perpendicular to the monochord and proportional to a. Through an analysis of the spectrum, this thesis suggests that unless a has a singularity at one boundary for any finite time, there exist initial conditions that give arbitrarily small energy decay at any time.
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    Numerically Stable and Statistically Efficient Algorithms for Large Scale Exponential Fitting
    (2014-05) Hokanson, Jeffrey M.
    The exponential fitting problem appears in diverse applications such as magnetic resonance spectroscopy, mechanical resonance, chemical reactions, system identification, and radioactive decay. In each application, the exponential fitting problem decomposes measurements into a sum of exponentials with complex coefficients plus noise. Although exponential fitting algorithms have existed since the invention of Prony's Method in 1795, the modern challenge is to build algorithms that stably recover statistically optimal estimates of these complex coefficients while using millions of measurements in the presence of noise. Existing variants of Prony's Method prove either too expensive, most scaling cubically in the number of measurements, or too unstable. Nonlinear least squares methods scale linearly in the number of measurements, but require well-chosen initial estimates lest these methods converge slowly or find a spurious local minimum. We provide an analysis connecting the many variants of Prony's Method that have been developed in different fields over the past 200 years. This provides a unified framework that extends our understanding of the numerical and statistical properties of these algorithms. We also provide two new algorithms for exponential fitting that overcome several practical obstacles. The first algorithm is a modification of Prony's Method that can recover a few exponential coefficients from measurements containing thousands of exponentials, scaling linearly in the number of measurements. The second algorithm compresses measurements onto a subspace that minimizes the covariance of the resulting estimates and then recovers the exponential coefficients using an existing nonlinear least squares algorithm restricted to this subspace. Numerical experiments suggest that small compression spaces can be effective; typically we need fewer than 20 compressed measurements per exponential to recover the parameters with 90% efficiency. We demonstrate the efficacy of this approach by applying these algorithms to examples from magnetic resonance spectroscopy and mechanical vibration. Finally, we use these new algorithms to help answer outstanding questions about damping in mechanical systems. We place a steel string inside vacuum chamber and record the free response at multiple pressures. Analyzing these measurements with our new algorithms, we recover eigenvalue estimates as a function of pressure that illuminate the mechanism behind damping.
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    One Can Hear the Composition of a String: Experiments with an Inverse Eigenvalue Problem
    (2008-07) Cox, Steven J.; Embree, Mark; Hokanson, Jeffrey M.
    To what extent do the vibrations of a mechanical system reveal its composition? Despite innumerable applications and mathematical elegance, this question often slips through those cracks that separate courses in mechanics, differential equations, and linear algebra. We address this omission by detailing a classical nite dimensional example: the use of frequencies of vibration to recover positions and masses of beads vibrating on a string. First we derive the equations of motion, then compare the eigenvalues of the resulting linearized model against vibration data measured from our laboratory's monochord. More challenging is the recovery of masses and positions of the beads from spectral data, a problem elegantly solved, through application of continued fractions, by Mark Krein. After presenting Krein's algorithm in a manner suitable for advanced undergraduates, we confirm its effcacy through physical experiment. We encourage readers to conduct their own explorations using data sets we provide on the web.