Statistical Inverse problems

Course overview 

Much of experimental science is analysis of measurements. In practice, this often involves fitting a mathematical model to the measurements and estimating the values of model parameters. Doing this optimally is usually not trivial, as the mathematical model that describes the relationship between the measurements and model parameters can be quite complicated. The framework of statistical inverse problems provides tools to solve complicated measurement problems that often often encountered in physics.

The goal of this course is to learn the theoretical foundation of statistical inverse problems, and to build a capability to apply these concepts in practice. Statistical inverse problems can be seen as a ``logic of science'', where interpretation of every measurement can be seen as problem that is optimally solved as a problem of statistical inference. With good knowledge of inverse problems, one can make sound inferences from data and avoid making conclusions which are not necessarily supported measurements.

The course will allow students to learn how to interpret scientific measurements, assign uncertainties to quantities estimates from measurements, and optimally design experiments in a way that minimizes the uncertainty of the quantities of interest. Students will also learn who to numerically solve large scale statistical inverse problems.

The course will cover linear inverse problems, linear regularization techniques, solution strategies to linear inverse problems, and numerical techniques for solving non-linear inverse problems (gradient based methods and Markov Chain Monte-Carlo). tion and inverse diffusion.

Book

The primary material for this course is the following book: Parameter Estimating and Inverse Problems; R. C. Aster, B. Borchers, C. H. Thurber, Chapters 1-11.

Form of exam

The course will approach each topic through short programming exercises. There are seven mandatory programming tasks, including: polynomial fitting, tomography, singular value decomposition, image deconvolution and inverse diffusion. The final exercise to to solve an inverse problem related to your own research. The form of exam is a written report that describes the solution to the exercises. 

Exercises

Can be found here.

Instructor

Juha Vierinen

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