| Inverse problems are often formulated as optimization problems. One seeks the parameter distribution which, when used in a forward model of the problem, gives the best match possible to the measured data. We consider the problem of imaging the elastic modulus distributions of soft tissues in this context. These optimization formulations are usually ill-posed and hence some form of regularization is needed in order to guarantee uniqueness and stability for the inverse solution. In this dissertation, however, we present two stabilized optimization formulations for the plane stress inverse problem without any form of regularization. These formulations include novel Galerkin Least Squares (GLS) terms. The Flux-GLS and the Bspline-GLS are formulated and analyzed in this work. These GLS terms improve the stability of the optimization formulation without upsetting consistency. We prove that these two stabilized optimization formulations are well defined under reasonable conditions on the data. We prove further that these two stabilized methods have optimal rates of convergence with mesh refinement. Finally, computational examples are presented to demonstrate the analysis described in this dissertation. These numerical results show the accuracy and stability of these two methods in different conditions. In noisy conditions, we recover unbiased reconstructions of material property distributions without regularization in the presence of high levels noise in the data. |
