Research Works

Above and Beyond

My research interests are primarily focused on inverse problems, regularization theory, and its applications. However, I am also interested in numerical and computational analysis, statistics, mathematical modeling, mathematical finance, data analysis, and, recently, I have started working on machine learning and deep learning problems. While pursuing my master's degree I was more inclined towards mathematical or quantitative fiance, wherein my master's thesis I worked on the mathematical derivation of the Black-Scholes formula. My Ph.D. and M.Sc. thesis are attached here

• Ph.D. Thesis: Inverse Problems, Regularization, and Its Applications.

• M.Sc. Thesis: A Mathematical Approach to Black-Scholes Formula.

An overview of the research area on which I worked during my Ph.D. is as follows. Physical theories allow us to make predictions, that is, given a complete description of a physical system, we can predict the outcome of some measurements. This problem of predicting the result of measurements is called the modeling problem or the forward problem. The inverse problem consists of using the actual result of some measurements to infer the values of the parameters that characterize the system. Inverse problems appear in a vast assortment of applications, including imaging (Computed Tomography (CT), Positron Emission Tomography (PET), Magnetic Resonance Imaging (MRI), Electrical Tomography (ET), microscopic imaging, geophysical imaging, signal and image processing, computer vision, machine learning and (big) data analysis in general, and many others. The complete research statement can be found here.

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Currently, I am working on Computational Imaging and Inverse Problems. The following three projects will give an insight into my works and related theory.

1. Interpretation of Plug-and-Play (PnP) Algorithms from a different angle.

2. Instabilities in Plug-and-Play (PnP) Algorithms from a learned denoiser.

3. Regularizing (Stabilizing) Deep Learning-Based Reconstruction Algorithms

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Some of my previous research projects are presented below:

1.  A new regularization method for numerical differentiation.

2.  A new regularization method for solving inverse problems.

3. Parameter estimation in an elliptic problem.

4. An appropriate family of regularizing functionals for inverse problems.

5. A new regularization method for inverse problems in higher dimensions.

6. Inverse problems in Mathematical Finance: Parameter estimation in asset pricing.
   
   

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