3. Postdoctoral Program - MSRI

Below is the list of topics considered in this minicourse.
1. Introduction: Examples of dynamic inverse problems from biological models
2. Stochastic models as expression of uncertainty. Probabilities and densities
3. Probability densities, samples and histograms
4. Propagation of probability densities, propagation of uncertainties
5. Updating densities using data. Resampling
6. Bayesian filtering
Each lecture was accompanied by computational examples and exercises that the students were
encouraged to solve with Matlab.
• An Introduction to the Calderón Problem
Oleg Imanuvilov (Colorado State University) and Gunther Uhlmann (University of Washington)
In this minicourse it was considered the problem of determining a complex-valued potential q
in a bounded two dimensional domain from the Cauchy data measured on an arbitrary open
subset of the boundary for the associated Schrödinger equation ∆ + q. A motivation comes
from the classical inverse problem of electrical impedance tomography problem. In this inverse
problem one attempts to determine the electrical conductivity of a body by measurements of
voltage and current on the boundary of the body. This problem was proposed by Calderón
and is also known as Calderón’s problem. It was also discussed the case where the electrical
measurements were made on part of the boundary. There were problem sessions for this and
the related minicourse of Lassi Päivärinta and Mikko Salo on the two dimensional case.
• Transforms of Radon Type in Computed Tomography
Lecturers: Peter Kuchment (Texas A&M) and Leonid Kunyansky (University of Arizona)
The mini-course was devoted to some areas of mathematics underlying many contemporary
methods of medical, industrial, and geophysical imaging. More specifically, the integral geometric
transforms and their applications in medical (as well as industrial and geophysical)
imaging were studied. The main emphasis was on the X-ray transform that integrates a function
over lines, Radon transform (integrating functions over hyperplanes) and their weighted
and curvilinear versions (e.g., integrals over certain sets of circles or spheres). The issues of
uniqueness of reconstruction of a function from its transform, inversion formulas, stability of
inversion, incomplete data issues, etc. were addressed. It was also shown how these transforms
arise and are applied in the X-ray CAT scan, MRI, emission tomography, and some
novel imaging methods, such as thermoacoustics.i the students worked on computer labs using
matlab programs.
• The Calderón Problem; the Two Dimensional Case
Lecturers: Lassi Päivärinta (U. Helsinki) and Mikko Salo (U. Helsinki)
Abstract: The recent developments mathematical theory of electrical impedance tomography
in the two dimensional case were discussed. This was a follow up to the minicourse by O.
Imanuvilov and G. Uhlmann that gave an introduction to the problem and discussed the three
dimensional problem. The applications of EIT include monitoring heart and lungs of unconscious
patients, detecting pulmonary edema and enhancing ECG and EEG. In two dimensions
the tools of complex analysis come handy. Especially methods of analytic and quasi-conformal
mappings turn out to be central and some of them were presented in the lectures.
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