DMS Colloquium: Julianne Chung

Speaker: Julianne Chung, Virginia Tech University

Title: Efficient Methods for Large and Dynamic Inverse Problems

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.

In this talk, we describe efficient methods to compute solutions to large, dynamic inverse problems. We focus on addressing two main challenges. First, since most inverse problems are ill-posed, small errors in the data may result in significant errors in the computed solutions. Thus, regularization must be used to compute stable solution approximations, and regularization parameters must be selected. Second, in many realistic scenarios such as in passive seismic tomography or dynamic photoacoustic tomography, the underlying parameters of interest may change during the measurement procedure. Thus, prior information regarding temporal smoothness must be incorporated for better reconstructions, but this can become computationally intensive, in part, due to the large number of unknown parameters. To address these challenges, we describe efficient, iterative, matrix-free methods based on the generalized Golub-Kahan bidiagonalization that allow automatic regularization parameter and variance estimation. These methods can be more flexible than standard methods, and efficient implementations can exploit structure in the prior, as well as possible structure in the forward model. Numerical examples demonstrate the range of applicability and effectiveness of the described approaches.

Faculty host: Yanzhao Cao