Elastography is an emerging imaging modality that seeks to non-destructively determine the mechanical properties of elastic/viscoleastic media from their response to external forces. It has a wide range of applications in medical diagnostics and non-destructive testing. The relevance and high cost of physical experimentation in such instances has driven the need for novel and efficient numerical algorithms to accelerate the corresponding design and identification process. To date, however, existing models and numerical schemes for their resolution have proven inadequate for this purpose. This research project initiates an integrated program for the remediation of this situation through the development of fast methods of broadly applicable models that are based on rigorous mathematical analysis. The results of the work should have a direct impact in a number of these instances, enhanced by the PI's access to real data and experiments from leading practitioners. Improved reconstruction schemes should lead to more accurate identification; their fast implementation, in turn, will allow for true guidance in the design of improved imaging systems.
The objective of this project is to examine fundamental mathematical issues and develop efficient computational methods for solving the direct and inverse problems in elastography. For the forward modeling, fast and accurate integral equation based solvers shall be developed for solid and solid/fluid interaction problems, aimed at attaining fully efficient and reliable simulation infrastructures for the underlying physical models in elastography. The approach proposed will address a number of significant challenges associated with the design of high-order quadratures for hypersingular integrals, with the accelerated evaluation of long-range potentials, and with the development of suitable preconditioners. To tackle the inverse problems, novel procedures with a combination of the multi-frequency continuation with fast linear approximations at long wavelengths shall be developed to address classical difficulties related to the nonlinear and ill-posed character of the inverse problems. This, in turn, should enable the possibility of true virtual design leading to advances in the area of PDE-constrained optimization.
