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Research Training Group
Inverse Problems and Partial Differential Equations

Department of Mathematics, University of Washington

UW RTG IPDE Summer School 2011

Dates: June 20 - July 8, 2011

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Material from this year's summer school is posted here.


This summer school gave advanced undergraduates and beginning graduate students the opportunity to study with the University of Washington's integrated Inverse Problems/PDE group. Students attended lectures in the morning and problem sessions in small groups with mentors in the afternoon. Two courses were given:


X-Ray Tomography and Transport Theory (Guillaume Bal, Steve McDowall, and Gunther Uhlmann). The first topic of this minicourse is the study of the X-ray transform in two dimensions (or Radon transform) arising in medical imaging, in particular in Computed Tomography (CT), and many other fields. X-ray tomography is one of the basic inverse problems and consists of determining the density of tissue by measuring the attenuation of X-rays passing through the body. The measurements are modeled by the X-ray transform and the inverse problem is to invert this transform.

The second topic is the mathematical study of the attenuated X-ray transform arising in medical imaging in particular in Single Positron Emission Tomography (SPECT). A patient is given a pharmaceutical labeled by a radionuclide, which emits photons. The goal is to recover the function that gives the distribution of the radiation sources. The measurements are modeled by the attenuated X-ray transform and the inverse problem is to invert this transform.

Both the X-ray transform of CT and and the attenuated X-ray transform of SPECT are merely reflections of a deeper mathematical object, the so-called radiative transport equation. This equation also handles many other problems, for instance optical tomography. This inverse boundary problem consists of reconstructing the absorption and scattering coefficient of an inhomogeneous medium by probing it with diffuse light. The problem is modeled by the linear Boltzmann equation. The third and final topic of the class will be to study a direct problem and the corresponding inverse problem associated with this equation.

During the afternoon the participants will have problem sessions and Matlab sessions on simulation and inversion of X-ray transforms and direct and inverse transport theory.

Finite Volume Methods and the Clawpack Software (Randall LeVeque and Donna Calhoun) This minicourse will provide a concentrated introduction to the theory and application of hyperbolic partial differential equations, a broad class of equations that model wave propagation problems arising in nearly all fields of science and engineering. Applications include ultrasound, seismic waves, shock waves, tsunamis, detonation waves, and traffic jams. Solving inverse problems in medical or seismic imaging often requires accurate techniques for solving the forward wave propagation problem, often defined by a hyperbolic system. The minicourse will also provide a hands-on introduction to the software package Clawpack, which implements a popular class of numerical methods for solving such problems, incorporating adaptive mesh refinement for the efficient solution of multidimensional problems

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