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Advanced nuclear magnetic resonance (NMR) imaging methodologies have been developed to determine porous media properties associated with fluid flow processes. This dissertation presents the development of NMR experimental and analysis methodologies, called NMR probes, particularly for determination of porosity, permeability, and pore-size distributions of porous media while the developed methodologies can be used for other properties. The NMR relaxation distribution can provide various information about porous systems having NMR active nuclei. The determination of the distribution from NMR relaxation data is an ill-posed inverse problem that requires special care, but conventionally the problem has been solved by ad-hoc methods. We have developed a new method based on sound statistical theory that suitably implements smoothness and equality/inequality constraints. This method is used for determination of porosity distributions. A Carr-Purcell-Meiboom-Gill (CPMG) NMR experiment is designed to measure spatially resolved NMR relaxation data. The determined relaxation distribution provides the estimate of intrinsic magnetization which, in turn, is scaled to porosity. A pulsed-field-gradient stimulated-echo (PFGSTE) NMR velocity imaging experiment is designed to measure the superficial average velocity at each volume element. This experiment measures velocity number distributions as opposed to the average phase shift, which is conventionally measured, to suitably quantify the velocities within heterogeneous porous media. The permeability distributions are determined by solving the inverse problem formulated in terms of flow models and the velocity data. We present new experimental designs associated with flow conditions to enhance the accuracy of the estimates. Efforts have been put forth to further improve the accuracy by introducing and evaluating global optimization methods. The NMR relaxation distribution can be scaled to a pore-size distribution once the surface relaxivity is known. We have developed a new method, which avoids limitations on the range of time for which data may be used, to determine surface relaxivity by the PFGSTE NMR diffusion experiment.

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In this paper, we consider as an inverse problem the simultaneous estimation of the five parameters of a jump diffusion process from return observations of a price trajectory. We show that there occur some ill-posedness phenomena in the parameter estimation problem, because the forward operator fails to be injective and small perturbations in the data may lead to large changes in the solution. We illustrate the instability effect by a numerical case study. To overcome the difficulty coming from ill-posedness we use a multi-parameter regularization approach that finds a trade-off between a least-squares approach based on empircal densities and a fitting of semi-invariants. In this context, a fixed point iteration is proposed that provides good results for the example under consideration in the case study.

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.

We deal with locally ill-posed nonlinear operator equations F(x) = y in L^2(0,1), where the Fréchet derivatives A = F'(x_0) of the nonlinear forward operator F are compact linear integral operators A = M ◦ J with a multiplication operator M with integrable multiplier function m and with the simple integration operator J. In particular, we give examples of nonlinear inverse problems in natural sciences and stochastic finance that can be written in such a form with linearizations that contain multiplication operators. Moreover, we consider the corresponding ill-posed linear operator equations Ax = y and their degree of ill-posedness. In particular, we discuss the fact that the noncompact multiplication operator M has only a restricted influence on this degree of ill-posedness even if m has essential zeros of various order.

In this paper, we consider the inverse problem of calibrating a generalization of the bivariate Ornstein-Uhlenbeck model introduced by Lo and Wang. Even though the generalized Black-Scholes option pricing formula still holds, option prices change in comparison to the classical Black-Scholes model. The time-dependent volatility function and the other (real-valued) parameters in the model are calibrated simultaneously from option price data and from some empirical moments of the logarithmic returns. This gives an ill-posed inverse problem, which requires a regularization approach. Applying the theory of Engl, Hanke and Neubauer concerning Tikhonov regularization we show convergence of the regularized solution to the true data and study the form of source conditions which ensure convergence rates.

In this note, we consider an inverse problem in a jump diffusion model. Using characteristic functions we prove the injectivity of the forward operator mapping the five parameters determining the model to the density function of the return distribution.