SPECIAL MATHEMATICS COLLOQUIUM
Speaker: Julianne Chung
Abstract. In many physical systems, measurements can only be obtained on the exterior of an object (e.g., the human body or the earth's crust), and the goal is to estimate 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. These are natural examples of inverse problems that arise in fields such as medical imaging, astronomy, geophysics, and molecular biology. The difficulty with ill-posed inverse problems is that small errors may result in significant errors in the computed solutions, so regularization must be used to compute stable solution approximations.
In this talk, we investigate regularization methods for linear and nonlinear inverse problems. We describe recent advances in spectral filtering approaches and hybrid iterative methods for regularization of linear inverse problems. Regularization for nonlinear Poisson based models, such as those arising from digital tomosynthesis reconstruction, is significantly more challenging, but accurate reconstruction is important in many real-life applications. Reconstruction algorithms will be discussed, and numerical experiments illustrate the effectiveness and efficiency of the proposed methods.