The main goal of my research is to advance knowledge in the field of computational inverse problems. Inverse problems arise in various scientific applications. The solution of an inverse problem is necessary in order to extract underlying structural information from available indirect and noisy observations. For example, in many physical systems, measurements are obtained on the exterior of an object (e.g., the human body or the earth’s crust), and one solves an inverse problem for the purpose of estimating the internal structures. In other systems, signals measured from machines (e.g., cameras) are distorted, and the aim is to recover the original input signal. In these and other data-intensive applications, there is a great need for efficient methods that can compute solutions accurately and in real-time. However, the main computational challenges that have hindered large-scale inversion and data analysis include ill-posedness of the problem, large parameter dimensions, model inaccuracies, and regularization parameter selection.
My research combines rigorous analysis with robust methodologies to address these challenges. More specifically, I develop new theory and tools for the design, computation, and analysis of solutions to large-scale inverse problems, and I work with application scientists to integrate these tools for further scientific knowledge and broader impact.