Ph.D. Student of Prof. Isaac Harari

Inverse problems arise in many fields of physics and often involve the recovery of the

physical characteristics of a material from its response to external loads. In the field of medical

imaging, images of parts of the human body are created for diagnostic purposes. One type of image

that may be useful is that of the mechanical properties of a tissue since these are often related to

tissue pathology. To find the mechanical properties, loads are applied to the tissue and the resulting

displacement field is measured. The displacement field is then used in an inverse problem solved

for the mechanical properties. In this work we develop and analyze novel variational formulations

for computing the solution to inverse problems of elasticity that arise in the context of biomedical

imaging.

Inverse problems are typically solved using iterative optimization techniques. This approach

is robust to noise and can handle partial data, but is computationally expensive. When full field data

are available, direct solution approaches can be considered. Here the computational effort is

relatively low, offering potential applications in real time imaging. The direct approach however is

sensitive to noise, and therefore must be carefully applied.

In this work we consider the adjoint weighted equation (AWE) for direct solution to inverse

problems of elasticity, with emphasis on incompressible elasticity. The AWE is a novel variational

formulation which was first applied to inverse problems of heat conduction (for the conductivities)

and later extended to inverse problems of elasticity (for the elastic moduli).

Inverse problems of elasticity involve partial differential equations (PDE's). To guarantee a

unique solution to a PDE, appropriate boundary conditions should be prescribed. For the inverse

problem however, boundary conditions are often unavailable. Therefore, instead of boundary

conditions, we propose "calibration conditions" which serve to constrain the solution and guarantee

uniqueness. Different methods to impose the calibration conditions are considered. We also

incorporate multiple measurements into the AWE formulation, which reduce the necessary amount

of calibration conditions.

The ability of the AWE method to solve inverse problems of elasticity is evaluated through

computational tests. Good performance is demonstrated for problems involving continuous

solutions and noise-free data. Problems involving discontinuous solution and noisy data however,

are challenging for the AWE method. We show that for these cases, the performance can

dramatically be improved by using relatively simple procedures.