A recording of the defense presentation for this dissertation is available at: http://hdl.handle.net/10754/625197 / Steady-state elliptic partial differential equations (PDEs) are frequently used to model a diverse range of physical phenomena. The source and boundary data estimation problems for such PDE systems are of prime interest in various engineering disciplines including biomedical engineering, mechanics of materials and earth sciences. Almost all existing solution strategies for such problems can be broadly classified as optimization-based techniques, which are computationally heavy especially when the problems are formulated on higher dimensional space domains. However, in this dissertation, feedback based state estimation algorithms, known as state observers, are developed to solve such steady-state problems using one of the space variables as time-like. In this regard, first, an iterative observer algorithm is developed that sweeps over regular-shaped domains and solves boundary estimation problems for steady-state Laplace equation. It is well-known that source and boundary estimation problems for the elliptic PDEs are highly sensitive to noise in the data. For this, an optimal iterative observer algorithm, which is a robust counterpart of the iterative observer, is presented to tackle the ill-posedness due to noise. The iterative observer algorithm and the optimal iterative algorithm are then used to solve source localization and estimation problems for Poisson equation for noise-free and noisy data cases respectively. Next, a divide and conquer approach is developed for three-dimensional domains with two congruent parallel surfaces to solve the boundary and the source data estimation problems for the steady-state Laplace and Poisson kind of systems respectively. Theoretical results are shown using a functional analysis framework, and consistent numerical simulation results are presented for several test cases using finite difference discretization schemes.

In many instances arising prominently, but not exclusively, in imaging problems, it is important to condense the salient information so as to obtain a low-dimensional approximant of the data. This thesis is concerned with two basic situations which call for such a dimension reduction. The first of these is the statistical recovery of smooth edges in regression and density surfaces. The edges are understood to be contiguous curves, although they are allowed to meander almost arbitrarily through the plane, and may even split at a finite number of points to yield an edge graph. A novel locally-parametric nonparametric method is proposed which enjoys the benefit of being relatively easy to implement via a `tracking' approach. These topics are discussed in Chapters 2 and 3, with pertaining background material being given in the Appendix. In Chapter 4 we construct concomitant confidence bands for this estimator, which have asymptotically correct coverage probability. The construction can be likened to only a few existing approaches, and may thus be considered as our main contribution. ¶ Chapter 5 discusses numerical issues pertaining to the edge and confidence band estimators of Chapters 2-4. Connections are drawn to popular topics which originated in the fields of computer vision and signal processing, and which surround edge detection. These connections are exploited so as to obtain greater robustness of the likelihood estimator, such as with the presence of sharp corners. ¶ Chapter 6 addresses a dimension reduction problem for spatial data where the ultimate objective of the analysis is the discrimination of these data into one of a few pre-specified groups. In the dimension reduction step, an instrumental role is played by the recently developed methodology of functional data analysis. Relatively standar non-linear image processing techniques, as well as wavelet shrinkage, are used prior to this step. A case study for remotely-sensed navigation radar data exemplifies the methodology of Chapter 6.

Interference and jamming severely disrupt our ability to communicate by decreasing the effective signal-to-noise ratio and by making parameter estimation difficult at the receiver. The objective of this research work is to design robust wireless systems and algorithms to suppress the adverse effects of non-intentional co-channel interference (CCI) or intentional jamming. In particular, we develop chip-combining schemes with timing, channel, and noise-power estimation techniques, all of which mitigate CCI or jamming. We also exploit the spatial diversity and iterative receiver techniques for this purpose. Most of the existing timing estimation algorithms are robust against either large frequency offsets or CCI, but not against both at the same time. Hence, we develop a new frame boundary estimation method that is robust in the presence of severe co-channel interference and large carrier-frequency offsets. To solve the high peak-to-average-power ratio problem of a multicarrier code division multiple access (MC-CDMA) system and enhance its robustness against fading and jamming, we propose a constant-envelope MC-CDMA system employing cyclic delay diversity (CDD) as transmit diversity. We analyze the diversity order, coding gain, and bit-error rate upper bound. We also propose a blind, accurate, and computationally efficient signal-to-noise ratio estimator for the proposed system. We propose a configurable robust anti-jam receiver that estimates the frequency- or time-domain jammer state information (JSI) and uses it for chip combining in the corresponding domain. A soft-JSI-based chip-combining technique is proposed that outperforms conventional hard-JSI-based chip combining. We also derive a chip combiner that provides sufficient statistics to the decoder. Channel estimation is necessary for coherent signal detection and JSI estimation. Conversely, knowledge of the jamming signal power and JSI of different subcarriers can improve the accuracy of the channel estimates. Hence, we propose joint iterative estimation of the multiple-input multiple-output (MIMO) channel coefficients, jamming power, and JSI for a coded MC-CDMA MIMO system operating under jamming and a time-varying frequency-selective fading channel. Finally, we reduce the computational complexity of the JSI-based anti-jam receivers by introducing an expectation-maximization-based joint channel and noise-covariance estimator that does not need either the subcarrier JSI or the individual powers of the AWGN and jamming signal.