Akhtar A. Khan: Research


My research interests lie broadly in the theory, computation, and applications of inverse problems. Besides this, I am also actively engaged in set-valued optimization, quasi-variational inequalities, optimal control, and uncertainty quantification. In the following, I briefly discuss some of my main research contributions in these areas.


Elliptic Inverse Problems

Soft tissue cancers in the interior of the human body reign among the deadliest forms of the disease, making up a majority of the 7.6 million estimated cancer deaths in 2008 [Cancer (Fact Sheet No. 297), WHO, 2013]. The effectiveness of most treatments hinges upon early detection but the process of finding tumors inside the body remains a difficult one. It is known that the stiffness of soft tissue can vary widely based on its molecular makeup and differing macroscopic/microscopic structure and that such changes in stiffness correlate with changes in tissue health. Palpation allows doctors to feel directly for changes in tissue stiffness and can detect such stiff lesions but the practice is subjective and is usually limited to finding exceptionally hard nodules near the skin's surface. Ultrasound can also be used to diagnose tumors further within the body as well as quantify their stiffness, but even hard growths can lack the necessary acoustical properties for effective detection. Elasticity imaging inverse problem extends the practice of palpation, making use of the varying elastic properties of healthy and diseased tissue to identify likely tumors. A relatively small external quasistatic compression force is applied to the tissue and then the tissue's axial displacement field is measured either directly or indirectly through the comparison of an undeformed and deformed image. A tumor can be identified by solving the inverse problem of determining the tissue's underlying elastic properties from this measurement. Although elasticity imaging inverse problem is a well established approach commonly used in a clinical setting, the underlying mathematical formulation is in terms of a non-convex optimization framework and hence it cannot ensure that a global solution can be found. It has been a long standing problem whether it is possible to remedy this drawback.

From a mathematical stand point this inverse problem seeks the elasticity parameter Mu from a measurement of the displacement vector under the assumption that the parameter Lambda is very large. This is due to the fact that in most of the existing literature on elasticity imaging inverse problem, the human body is modelled as an incompressible elastic object. Although this assumption simplifies the identification process as there is only one parameter to identify, it significantly complicates the computational process as the classical finite element methods become quite ineffective due to the so-called locking effect. One of the few techniques to handle this problem is by resorting to mixed finite element formulation. I have developed various optimization based approaches for this inverse problems.

Set Optimization: Optimality Conditions and Applications

During my stay in Germany I worked on a research project sponsored by German Research Foundation (DFG), devoted to a detailed study of optimization problems involving set-valued maps. In this field, one major difficulty is of differentiating the set-valued maps. (The study of optimization methods for non-differentiable functions also fall in this category.) We established usefulness of the notion of epiderivative of set-valued mapping. This notion is a cornerstone of set optimization and was coined by Johannes Jahn, who was the main investigator of the project. The basic idea is to use certain tangent cones to define the epiderivatives and use them to state optimality conditions in set optimization. Set optimization subsumes nonsmooth optimization which, due to many real-world applications, was one of the most active branch of optimization before the birth of set-optimization. Currently, I am working on general higher-order optimality conditions in set optimization, and on applications of set optimization in robotics and optimal control.

Ill-posed Variational Inequalities

The theory of Variational and quasi-variational inequalities is now an independent branch of applied mathematics and has far reaching applications. My current interest in this field is in developing regularization techniques for ill-posed quasi-variational inequalities. I have published papers both of theoretical nature and papers dealing with iterative schemes.



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