AN ANALYSIS OF THE OPTIMIZATION FORMULATION OF ELASTIC INVERSE PROBLEMS

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. We consider in particular formulating this inverse problem as a constrained optimization problem in which we minimize the objective functional (the data mismatch) with the plane stress incom-pressible elasticity equations as constraints. The Lagrange multiplier method is applied resulting in a two-eld variational formulation that is often called a mixed formula-tion. We analyze the well-posedness of this problem. Under some relatively strong mathematical conditions, we prove that the continuous optimization formulation of this inverse problem is well-posed. In practice, however, when we solve this problem approximately by using classical discretization techniques, problems like oscillations and spurious non-physical solutions appear. Finally, we introduce stabilization in the discrete optimization problem to control the spurious solutions. In the context of in-verse problems, stabilization must be contrasted to regularization. The latter is used to render a continuous problem well-posed. Stabilization is used to make well-behaved a particular discretization of a well-posed mathematical problem. We expect this research to improve computational approaches to nearly all inverse problems solved via optimization methods. In the context of elastography, this research will improve our ability to screen, diagnose, and treat breast cancer, prostate cancer, and other tissue pathologies