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We investigate the applicability of the method of maximum entropy regularization (MER) including convergence and convergence rates of regularized solutions to the specific inverse problem (SIP) of calibrating a purely time-dependent volatility function. In this context, we extend the results of [16] and [17] in some details. Due to the explicit structure of the forward operator based on a generalized Black-Scholes formula the ill-posedness character of the nonlinear inverse problem (SIP) can be verified. Numerical case studies illustrate the chances and limitations of (MER) versus Tikhonov regularization (TR) for smooth solutions and solutions with a sharp peak.

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Imaging speed is very important in magnetic resonance imaging (MRI), especially in dynamic cardiac applications, which involve respiratory motion and heart motion. With the introduction of reduced-data MR imaging methods, increasing acquisition speed has become possible without requiring a higher gradient system. But these reduced-data imaging methods carry a price for higher imaging speed. This may be a signal-to-noise ratio (SNR) penalty, reduced resolution, or a combination of both. Many methods sacrifice edge information in favor of SNR gain, which is not preferable for applications which require accurate detection of myocardial boundaries. The central goal of this thesis is to develop novel reduced-data imaging methods to improve reconstructed image performance. This thesis presents a novel reduced-data imaging method, PINOT (Parallel Imaging and NOquist in Tandem), to accelerate MR imaging. As illustrated by a variety of computer simulated and real cardiac MRI data experiments, PINOT preserves the edge details, with flexibility of improving SNR by regularization. Another contribution is to exploit the data redundancy from parallel imaging, rFOV and partial Fourier methods. A Gerchberg Reduced Iterative System (GRIS), implemented with the Gerchberg-Papoulis (GP) iterative algorithm is introduced. Under the GRIS, which utilizes a temporal band-limitation constraint in the image reconstruction, a variant of Noquist called iterative implementation iNoquist (iterative Noquist) is proposed. Utilizing a different source of prior information, first combining iNoquist and Partial Fourier technique (phase-constrained iNoquist) and further integrating with parallel imaging methods (PINOT-GRIS) are presented to achieve additional acceleration gains.

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Emerging Ultra Wideband (UWB) Orthogonal Frequency Division Multiplexing (OFDM) systems hold the promise of delivering wireless data at high speeds, exceeding hundreds of megabits per second over typical distances of 10 meters or less. The purpose of this Thesis is to estimate the timing accuracies required with such systems in order to achieve Bit Error Rates (BER) of the order of magnitude of 10-12 and thereby avoid overloading the correction of irreducible errors due to misaligned timing errors to a small absolute number of bits in error in real-time relative to a data rate of hundreds of megabits per second. Our research approach involves managing bit error rates through identifying maximum timing synchronization errors. Thus, it became our research goal to determine the timing accuracies required to avoid operation of communication systems within the asymptotic region of BER flaring at low BERs in the resultant BER curves. We propose pushing physical layer bit error rates to below 10-12 before using forward error correction (FEC) codes. This way, the maximum reserve is maintained for the FEC hardware to correct for burst as well as recurring bit errors due to corrupt bits caused by other than timing synchronization errors.

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For many scientific applications, inverse problems have played a key role in solving important problems by enabling researchers to estimate desired parameters of a system from observed measurements. For example, large-scale inverse problems arise in many global problems and medical imaging problems such as greenhouse gas tracking and computational tomography reconstruction. This dissertation describes advancements in computational tools for solving large-scale inverse problems and for uncertainty quantification. Oftentimes, inverse problems are ill-posed and large-scale. Iterative projection methods have dramatically reduced the computational costs of solving large-scale inverse problems, and regularization methods have been critical in obtaining stable estimations by applying prior information of unknowns via Bayesian inference. However, by combining iterative projection methods and variational regularization methods, hybrid projection approaches, in particular generalized hybrid methods, create a powerful framework that can maximize the benefits of each method. In this dissertation, we describe various advancements and extensions of hybrid projection methods that we developed to address three recent open problems. First, we develop hybrid projection methods that incorporate mixed Gaussian priors, where we seek more sophisticated estimations where the unknowns can be treated as random variables from a mixture of distributions. Second, we describe hybrid projection methods for mean estimation in a hierarchical Bayesian approach. By including more than one prior covariance matrix (e.g., mixed Gaussian priors) or estimating unknowns and hyper-parameters simultaneously (e.g., hierarchical Gaussian priors), we show that better estimations can be obtained. Third, we develop computational tools for a respirometry system that incorporate various regularization methods for both linear and nonlinear respirometry inversions. For the nonlinear systems, blind deconvolution methods are developed and prior knowledge of nonlinear parameters are used to reduce the dimension of the nonlinear systems. Simulated and real-data experiments of the respirometry problems are provided. This dissertation provides advanced tools for computational inversion and uncertainty quantification. / Doctor of Philosophy / For many scientific applications, inverse problems have played a key role in solving important problems by enabling researchers to estimate desired parameters of a system from observed measurements. For example, large-scale inverse problems arise in many global problems such as greenhouse gas tracking where the problem of estimating the amount of added or removed greenhouse gas at the atmosphere gets more difficult. The number of observations has been increased with improvements in measurement technologies (e.g., satellite). Therefore, the inverse problems become large-scale and they are computationally hard to solve. Another example of an inverse problem arises in tomography, where the goal is to examine materials deep underground (e.g., to look for gas or oil) or reconstruct an image of the interior of the human body from exterior measurements (e.g., to look for tumors). For tomography applications, there are typically fewer measurements than unknowns, which results in non-unique solutions. In this dissertation, we treat unknowns as random variables with prior probability distributions in order to compensate for a deficiency in measurements. We consider various additional assumptions on the prior distribution and develop efficient and robust numerical methods for solving inverse problems and for performing uncertainty quantification. We apply our developed methods to many numerical applications such as greenhouse gas tracking, seismic tomography, spherical tomography problems, and the estimation of CO2 of living organisms.