Venkateswaran Krishnan

My main research interest is in inverse problems. Roughly speaking, an inverse problem may be described as a problem of determination of model parameters of an object based on some measured or observed data from the exterior/boundary of the object. This requires sending waves such as X-rays, light waves, pressure waves or sound waves, measuring the response at the boundary, and analyzing it to determine the object parameters. The parameters modeling the object frequently appear as coefficients of a partial differential equation (PDE). The forward problem is the problem of analysis of solutions based on a priori knowledge of the object (the coefficients of the PDE). In contrast, the inverse problem is the determination of object parameters based on boundary measurements. In practice, an inverse problem is far more difficult to solve. For example, even if the PDE model is linear, the inverse problem can be non-linear.

Inverse problems appear in several applications; here we name a few.

• Medical Imaging -- A brain CT scan image involves the solution of an inverse problem.
• Cancer Detection -- Given the invasive nature of cancer treatment and the difficulty that a patient has to go through, there is obvious human interest in making non-invasive cancer diagnosis. This involves solutions of inverse problems.
• Seismic Imaging -- Given the high cost of drilling to extract oil, one is obviously interested in making surface measurements to determine whether a region beneath the surface has oil reserves.
• Non-destructive Material Testing -- Let's say one has material parts that appear in airplanes, rockets etc. Finding out whether such materials have developed subsurface cracks, which are invisible to the human eye, involves the solution of inverse problems.

I study inverse problems from a theoretical point of view. For instance, analyzing questions such as whether a given boundary data of a medium uniquely determines the model parameters in the interior, and if so, whether there is stability of determination are a couple of central themes of study. In real life boundary data is corrupted with noise and this is one reason why stability questions are important. 

The principal tools that I rely on to understand problems in these fields are the theory of pseudodifferential and Fourier integral operators (FIOs), microlocal analysis, Fourier analysis, Riemannian geometry and integral geometry techniques.