## Elliptic Inverse Problems with Application to Elasticity Imaging

Faculty: | Akhtar Khan |

**Summary:**

Numerous mathematical models in applied and industrial mathematics take the form of a partial differential equation involving certain coefficients which describe the properties of the system. When these coefficients are known, then the so-called “direct problem” is simply to solve the partial differential equation. By contrast, when the coefficients are not known, an “inverse problem” asks for the identification of the coefficients given measurement data of a solution to the partial differential equation. The primary research focus in this work is the inverse problem of identifying Lame coefficients in the equations of linear elasticity. Recently, applications such as elasticity imaging have sparked a new interest in inverse elasticity problems. In elasticity imaging, ultrasound is used to measure interior displacements in human tissue (for example, breast tissue). Since cancerous tumors differ markedly in their elastic properties from healthy tissue, it may be possible to discover and locate tumors by solving an inverse problem for the Lame parameters. It is of particular interest to identify coefficients which are either discontinuous or vary rapidly. The so-called BV regularization is used to handle this situation.

**Publications:**

- On the inverse problem of identifying Lame coefficients in linear elasticity, A. Khan, B. Jadamba, and F. Raciti, Computer and Mathematics with Applications, 2008.
- An abstract framework for elliptic inverse problems. Part 1: an output least squares approach, A. Khan and M.S. Gockenbach, Mathematics and Mechanics of Solids, 12, 259-276, 2007.
- An abstract framework for elliptic inverse problems. Part 2: an augmented Lagrangian approach, A. Khan and M.S. Gockenbach, Mathematics and Mechanics of Solids, 2008.
- Identification of Lame parameters in linear elasticity: a fixed point approach, A. Khan and M.S. Gockenbach, Journal of Industrial and Management Optimization, Vol. 1, No. 4, 487–497, 2005.