On a class of two-dimensional inverse problems: wavefield-based shape detection and localization and material profile reconstruction
In this dissertation we discuss the numerical treatment of two classical inverse problems: firstly, we are interested in the shape detection and localization problem that arises when it is desirable to identify the location and shape of an unknown object embedded in a host medium using response measurements at remote stations. Secondly, we are concerned with the reconstruction of a medium’s material profile given, again, scant response data. For both problems we use acoustic (or equivalent) waves, to illuminate the interrogated object/medium; however, the mathematical/numerical treatment presented herein extends directly to other wave types. There is a wide, and ever widening, spectrum of possible applications that stand to benefit: of particular interest here are geotechnical applications that arise during site characterization efforts. To tackle both inverse problems we adopt the systematic framework of governing-equation-constrained optimization. Accordingly, misfit functionals are augmented with appropriate regularization terms, and with the weak imposition of the equations describing the physics of the wave interrogation. The governing equations may be either of the partial-differential or integral kind, subject only to user preference or problem bias. The framework is flexible enough to accommodate various misfit norms and regularization terms. We seek solutions that minimize the augmented functional by requiring that the first-order optimality conditions vanish at the optimum, thereby giving rise to Karush-Kuhn-Tucker-type systems. We then solve the associated state, adjoint, and control problems with a reduced-space approach. To alleviate the theoretical and numerical difficulties inherent to all inverse problems that are present here as well, we seek to narrow the solution feasibility space by adopting special schemes. In the shape detection and localization problem we adopt amplitude-based misfit functionals, and a frequencyand directionality-continuation scheme, somewhat akin to multigrid methods, that, thus far, have lend robustness to the inversion process. The mathematical details are based on integral equations, where, in addition, the control problem is cast in the elegant framework of total or material derivatives that allow computational speed-up when compared to finite-difference-based gradient schemes. Similarly, in the material profile reconstruction problem we adopt a time-dependent regularization scheme that exhibits superior performance to classical Tikhonov-type regularizations and is shown to be capable of recovering both sharp and smooth material distributions, while being relatively insensitive to the choice of initial guesses and regularization factors. These schemes constitute particular contributions of this work. We describe the mathematical framework and report numerical results. Specifically, with respect to the shape detection and localization problem we report on the two-dimensional case of sound-hard objects embedded in fullspace; with respect to the material profile reconstruction problem, we report results on the one-dimensional case of horizontally-layered systems, and on the two-dimensional case of finite or infinite-extent domains. We discuss the algorithmic performance in the presence of both noise-free and noisy data and provide recommendations for possible extensions of this work.