Inverse Problems: Summer school

Published
31.7.2024

Courses on inverse problems in JSS35.

General information

qr-code for JSS inverse problems courses

IP1: Electrical Impedance Tomography: Computation and Applications

  • Time: 1-3 pm every day
  • Lecturer(s): Melody Alsaker (Gonzaga University, United States), email: alsaker@gonzaga.edu
  • Code: MATJ5129
  • Modes of study: Lectures (and project work for those who want credits)
  • Credits: 2 ECTS
  • Evaluation: Project work pass/fail
  • Completion: Return exercises to lecturer by the end of August as instructed during the lectures
  • Contents: This course focuses on the applied and computational aspects of electrical impedance tomography (EIT), including modeling, EIT systems, reconstruction algorithms, and hands-on MATLAB implementation. We will examine how EIT is used in biomedical, industrial, and geophysical settings, how modeling choices influence image quality, and how to interpret reconstructed conductivity images. Modern reconstruction methods, including the use of Complex Geometrical Optics solutions in the direct D-bar method, will be explored both conceptually and through MATLAB demonstrations. It is recommended that participants bring a laptop with MATLAB installed, although a computer lab will be available. Recommended to take together with Mathematics of Electrical Impedance Tomography.
  • Learning outcomes: Insight into practical EIT modeling, applications, and reconstruction algorithms.
  • Prerequisites: Basics of linear algebra and numerical methods, introductory exposure to PDEs, and basic programming skills (preferably in MATLAB)
  • Material: 

IP2: Mathematics of Electrical Impedance Tomography

  • Time: 9-11 am every day
  • Lecturer(s): Samuli Siltanen (University of Helsinki, Finland), email: samuli.siltanen@helsinki.fi
  • Code: MATJ5130
  • Modes of study: Lectures (and project work for those who want credits)
  • Credits: 2 ECTS
  • Evaluation: Project work pass/fail
  • Completion: Return exercises to lecturer by the end of August as instructed during the lectures
  • Contents: This course focuses on mathematical aspects of electrical impedance tomography (EIT). A simple pixel-based diffusion model serves as a gentle introduction to the principle of EIT measurement, illustrating key challenges. Calderón’s inverse conductivity problem is then derived from Maxwell’s equations, and basic properties of the conductivity equation are discussed. Some knowledge of elliptic partial differential equations and Fourier transforms is useful here, but there is a strong effort to make the material as self-contained as possible. Analytic expressions are computed for the Dirichlet-to-Neumann map in case of rotationally symmetric conductivities. This makes it possible to study in concrete terms (i) Alessandrini’s example showing the ill-posedness of EIT, (ii) Calder’on’s original reconstruction approach, and (iii) Ikehata’s enclosure method. The rest of the course is devoted to the use of Complex Geometric Optics solutions for uniqueness proofs and reconstruction via the D-bar method. Recommended to take together with Electrical Impedance Tomography: Computation and Applications
  • Learning outcomes: Insight into the theory of EIT, including nonlinearity, ill-posedness, and reconstruction approaches.
  • Prerequisites: Introductory exposure to PDEs and Fourier transforms.
  • Course material & Exercises:

Previous courses

Below are listed the courses in inverse problems for Jyväskylä Summer School for previous years.