Despite bold steps taken worldwide for the replacement or the reduction of the world’s dependence on fossil fuels, economic and societal realities suggest that a transition to alternative energy forms will be, at best, gradual. It also appears that exploration for new reserves is becoming increasingly more difficult both from a technical and an economic point of view, despite the advent of new technologies. These trends place renewed emphasis on maximizing oil recovery from known fields. In this sense, low-cost and reliable enhanced oil recovery (EOR) methods have a strong role to play. The goal of this dissertation is to explore, using computational simulations, the feasibility of the, so-called, seismic or elastic-wave EOR method, and to provide the mathematical/computational framework under which the method can be systematically assessed, and its feasibility evaluated, on a reservoir-specific basis. A central question is whether elastic waves can generate sufficient motion to increase oil mobility in previously bypassed reservoir zones, and thus lead to increased production rates, and to the recovery of otherwise unexploited oil. To address the many questions surrounding the feasibility of the elastic-wave EOR method, we formulate an inverse source problem, whereby we seek to determine the excitations (wave sources) one needs to prescribe in order to induce an a priori selected maximization mobility outcome to a previously well-characterized reservoir. In the industry’s parlance, we attempt to address questions of the form: how does one shake a reservoir?, or what is the “resonance” frequency of a reservoir?. We discuss first the case of wellbore wave sources, but conclude that surface sources have a better chance of focusing energy to a given reservoir. We, then, discuss a partial-differential-equation-constrained optimization approach for resolving the inverse source problem associated with surface sources, and present a numerical algorithm that robustly provides the necessary excitations that maximize a mobility metric in the reservoir. To this end, we form a Lagrangian encompassing the maximization goal and the underlying physics of the problem, expressed through the side imposition of the governing partial differential equations. We seek to satisfy the first-order optimality conditions, whose vanishing gives rise to a systematic process that, in turn, leads to the prescription of the wave source signals. We explore different (indirect) mobility metrics (kinetic energy or acceleration field maximization), and report numerical experiments under three different settings: (a) targeted formations within one-dimensional multi-layered elastic solids system of semi-infinite extent; (b) targeted formations embedded in a two-dimensional semi-infinite heterogeneous elastic solid medium; and (c) targeted poroelastic formations embedded within elastic heterogeneous surroundings in one dimension. The numerical experiments, employing hypothetical subsurface formation models subjected to, initially unknown, ground surface wave sources, demonstrate that the numerical optimizer leads robustly to optimal loading signals and the illumination of the target formations. Thus, we demonstrate that the theoretical framework for the elastic wave EOR method developed in this dissertation can systematically address the application of the method on a reservoir-specific basis. From an application point of view and based on the numerical experiments reported herein, for shallow reservoirs there is strong promise for increased production. The case of deeper reservoirs can only be addressed with further research that builds on the findings of this work, as we report in the last chapter. / text

The inverse source problem for the Helmholtz equation is studied. An unknown source is to be identified from the knowledge of its radiated wave. The focus is placed on the effect that multi-frequency data has on establishing uniqueness. In particular, we prove that data obtained from finitely many frequencies is not sufficient. On the other hand, if the frequency varies within an open interval of the positive real line, then the source is determined uniquely. An algorithm is based on an incomplete Fourier transform of the measured data and we establish an error estimate under certain regularity assumptions on the source function. We conclude that multi-frequency data not only leads to uniqueness for the inverse source problem, but in fact it contributes with a stability result for the reconstruction of an unknown source.

This paper studies the inverse source problem for the Helmholtz equation with a point source in a two dimensional domain. Given complete boundary data and appropriate discretization Tikhonov regularization is established to be an effective method at finding the point source. Furthermore, it was found that Tikhonov regularization can locate point sources even given significant noise, as well as incomplete boundary data in complicated domains.

The design of reflectarray surface currents that satisfy both radiation and user-defined antenna feasibility constraints is addressed through a novel paradigm which takes advantage of the non-uniqueness of inverse source (IS) problems. To this end, the synthesis is formulated in the IS framework and its non-measurable solutions are employed as a design DoF. Thanks to the adopted framework, a closed-form expression for the design of reflectarray surface currents is derived which does not require any iterative local/global optimization procedure and which inherently satisfies both the radiation and the feasibility design constraints. The features and potentialities of the proposed strategy are assessed through selected numerical experiments dealing with different reflectarray aperture types/sizes and forbidden region definitions.

In recent years, revolutionary "hybrid" or "multi-physics" methods of medical imaging have emerged. By combining two or three different types of waves these methods overcome limitations of classical tomography techniques and deliver otherwise unavailable, potentially life-saving diagnostic information. Thermoacoustic (and photoacoustic) tomography is the most developed multi-physics imaging modality. Thermo- and photo-acoustic tomography require reconstructing initial acoustic pressure in a body from time series of pressure measured on a surface surrounding the body. For the classical case of free space wave propagation, various reconstruction techniques are well known. However, some novel measurement schemes place the object of interest between reflecting walls that form a de facto resonant cavity. In this case, known methods cannot be used. In chapter 2 we present a fast iterative reconstruction algorithm for measurements made at the walls of a rectangular reverberant cavity with a constant speed of sound. We prove the convergence of the iterations under a certain sufficient condition, and demonstrate the effectiveness and efficiency of the algorithm in numerical simulations. In chapter 3 we consider the more general problem of an arbitrarily shaped resonant cavity with a non constant speed of sound and present the gradual time reversal method for computing solutions to the inverse source problem. It consists in solving back in time on the interval [0, T] the initial/boundary value problem for the wave equation, with the Dirichlet boundary data multiplied by a smooth cutoff function. If T is sufficiently large one obtains a good approximation to the initial pressure; in the limit of large T such an approximation converges (under certain conditions) to the exact solution.

This thesis studies the use of the Source Reconstruction Method (SRM) to characterize antennas. The SRM calculates equivalent sources/currents on an arbitrarily shaped reconstruction surface to represent the original antenna. This is done by enforcing that the original antenna and equivalent currents radiate the same field at user selected measurement locations. These equivalent currents spatially characterize the original antenna because they can be used in direct radiation problems to obtain field estimates anywhere outside the reconstruction surface, including the far-field. First a spherical SRM algorithm is implemented and the diagnostic capabilities of the SRM are also synthetically shown through an example with an array of elementary dipoles. It is then shown that the SRM compares well to pre-existing commercial antenna software over different frequencies and can also be used successfully with a partial dataset. It is demonstrated that the equivalent currents can also provide meaningful information with experimental data. Next the hierarchical matrix framework is studied in conjunction with the SRM to decrease the algorithm's memory requirement and increase the speed of execution. It is shown that it is beneficial to use the hierarchical matrix framework with the SRM when using Love's condition or with measured data on a surface very close to the reconstruction surface. The SRM is then used to obtain incident field estimates in microwave imaging systems. Using a 2D transverse magnetic framework, we show that even with the limited data available in typical microwave tomography setups the SRM can produce incident field estimates in the imaging domain. These estimates are then used along with an MR-GNI algorithm to image synthetic and experimental objects with uncalibrated measured data. / October 2016

Molecular imaging plays an important role for development of systems biomedicine, which non-invasively extracts pictorial information on physiological and pathological activities at the cellular and molecular levels. Optical molecular tomography is an emerging area of molecular imaging. It locates and quantifies a 3D molecular probe distribution in vivo from data measured on the external surface of a small animal around the visible and infrared range. This approach can facilitate or enable preclinical applications such as cancer studies, involving angiogenesis, tumor growth, cell motility, metastasis, and interaction with a micro-environment. The reconstruction of diffuse light sources is the central task of optical molecular tomography, and generally ill-posed and rather complex. The key element of optical molecular tomography includes the geometrical model, tissue properties, photon characteristics, transport model, and reconstruction algorithm. This dissertation focuses mainly on the development optical molecular tomography methods based on bioluminescence/fluorescence probes to solve some well-known challenges in this field. Our main results are as follows. We developed a new algorithm for estimation of optical parameters based on the phase-approximation model.  Our iterative algorithm takes advantage of both the global search ability of the differential evolution algorithm and the efficiency of the conjugate gradient method. We published the first paper on multispectral bioluminescence tomography (BLT). The multispectral BLT approach improves the accuracy and stability of the BLT reconstruction even if data are highly noisy. We established a well-posed inverse source model for optical molecular tomography. Based on this model, we proposed a differential evolution-based reconstruction algorithm to determine the source locations and strengths accurately and reliably. Furthermore, to enhance the spatial resolution of fluorescence molecular tomography, we proposed fluorescence micro-tomography to image cells in a tissue scaffold based on Monte Carlo simulation on a massive parallel processing architecture. Each of these methods shows better performance in numerical simulation, phantom experiments, and mouse studies than the conventional methods. / Ph. D.

Cavitation is a phenomenon present during many ultrasound therapies, including the thermal ablation of malignant tissue using high intensity focused ultrasound (HIFU). Inertial cavitation, in particular, has been previously shown to result in increased heat deposition and to be associated with broadband noise emissions that can be readily monitored using a passive receiver without interference from the main ultrasound signal. The present work demonstrates how an array of passive receivers can be used to generate maps of cavitation distribution during HIFU exposure, uncovering a new potential method of monitoring HIFU treatment. Using a commercially available ultrasound system (z.one, Zonare, USA), pulse transmission can be switched off and data from 64 elements of an array can be simultaneously acquired to generate passive maps of acoustic source power. For the present work, a 38 mm aperture 5-10 MHz linear array was used, with the 64 elements chosen to span the entire aperture. Theory and simulations were used to show the spatial resolution of the system, the latter showing that the broadband nature of inertial cavitation makes passive maps robust to interference between cavitating bubbles. Passive source mapping was first applied to wire scatterers, demonstrating the ability of the system to resolve broadband sources. With the array transversely placed to the HIFU axis, high-resolution passive maps are generated, and emissions from several cavitating bubbles are resolved. The sensitivity of passive mapping during HIFU exposure is compared with that of an active cavitation detector following exposure. The array was then placed within a rectangular opening in the centre of the HIFU transducer, providing a geometric setup that could be used clinically to monitor HIFU treatment. Cavitation was instigated in continuous and disjoint regions in agar tissue mimicking gel, with the expected regions of cavitation validating the passive maps obtained. Finally, passive maps were generated for samples of ox liver exposed to HIFU. The onset of inertial cavitation as detected by the passive mapping approach was found to provide a much more robust indicator of lesioning than post-exposure B-mode hyperecho, which is in current clinical use. Passive maps based on the broadband component of the received signal were able to localize the lesions both transversely and axially, however cavitation is generally indicated 5 mm prefocal to the lesions. Further work is needed to establish the source of this discrepancy. It is believed that with use of an appropriately designed cavitation detection array, passive mapping will represent a major advance in ultrasound-guided HIFU therapy. Not only can it be utilized in real-time during HIFU exposure, without the need to turn the therapeutic ultrasound field off, but it has also been shown in the context of the present work to provide a strong indicator of successful lesioning and high signal-to-noise compared to conventional B-mode ultrasound techniques.

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