MATH 6397 Bayesian Inverse Problems and UQ (Spring 2023)
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Important information for the course Bayesian Inverse Problems and Uncertainty Quantification (MATH 6397 — 20393) will be posted on this page. Please visit it on a regular basis. Check the syllabus regularly for any important updates. It is the students responsibility to be aware of additional course policies presented by the instructor during class.Course Description
Inverse problems are of paramount importance and can be found in virtually all scientific disciplines with applica- tions ranging from medicine, geophysics, to engineering. In many of these applications the forward or simulation problem, i.e., the solution of an underlying mathematical model to yield outputs given some inputs, is already a challenging task. Many applications require us to go beyond evaluating forward operators; we have to address what is often the ultimate goal: prediction and decision-making. This requires us to tackle mathematical chal- lenges that comprise, and, therefore, are more difficult than the forward problem. One example is the solution of inverse problems. Here, we seek model inputs (or parameters) so that the output of the forward model matches observational data.This course will cover the mathematical background needed to analyze and further develop numerical methods for Bayesian (statistical) inverse problems and uncertainty quantification. First, we will revisit some theoretical foundations of inverse problems and strategies to their solution. Subsequently, we will quickly transition to topics surrounding statistical inverse problems. Potential topics include relevant theory from discrete probability; statisti- cal computing; sampling methods; modern regularization techniques; prior modeling; MAP estimation and Laplace approximation; variational inference; optimization (under uncertainty); matrix data and latent factor models; and dimensionality reduction. If time permits, it will investigate explore Bayesian inference in a machine learning context. Students will be assessed in several homework projects covering theoretical and practical implementation aspects. We will consider different applications in computational sciences and engineering.