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Nonlocal Inverse Problems
Last Updated: 2026-02-05 15:54:11
Abstract
This course is an introduction to the Calderón problem and nonlocal inverse problems for the fractional Schrödinger equation. These are examples of nonlinear inverse problems. The classical Calderón problem models electrical impedance tomography (EIT) and fractional operators appear, for example, in some mathematical models in finance.
Objective
Students become familiar with the Calderón problem and some nonlocal phenomena related to the fractional Laplacian. Advanced students should be able to read research articles on the fractional Calderón problems after the course.
Content
In the beginning of the course, we will introduce some basic theory for the classical Calderón problem. The focus of the course will be in the study of nonlocal inverse problems for the fractional Schrödinger equation with lower order perturbations. We discuss necessary preliminaries on Sobolev spaces, Fourier analysis, functional analysis and theory of PDEs. Our scope will be in the uniqueness properties. Classical Calderón problem (about 1/3): Conductivity and Schrödinger equations, Dirichlet-to-Neumann maps, Cauchy data, and related boundary value inverse problems. The methods include, for example, complex geometric optics (CGO) solutions. Fractional Calderón problem (about 2/3): Nonlocal unique continuation principles (UCP), Runge approximation properties, and uniqueness for the fractional Calderón problem. The methods include, for example, Caffarelli-Silvestre extensions, the fractional Poincaré inequality and Riesz transforms.
Resources
Lecture Notes
Lecture notes and exercises
Literature
1. M. Salo: Calderón problem. Lecture notes, University of Helsinki (2008). (Available at http://users.jyu.fi/~salomi/index.html .) 2. T. Ghosh, M. Salo, G. Uhlmann: The Calderón problem for the fractional Schrödinger equation. Analysis & PDE 13 (2020), no. 2, 455-475. 3. A. Rüland, M. Salo: The fractional Calderón problem: low regularity and stability. Nonlinear Analysis 193 (2020), special issue "Nonlocal and Fractional Phenomena", 111529. 4. Other literature will be specified in the course.
General Information
- Language
- English
- Levels
- DR , MSC
Examination
- Type
- session examination
- Mode
- oral 20 minutes
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture | Nonlocal Inverse Problems |
|
2 h weekly |
Offered In
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Electives (For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.)
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Doctoral Department of Mathematics (More Information at: The list of courses (together with the allocated credit points) eligible for doctoral students is published each semester in the newsletter of the ZGSM. WARNING: Do not mistake ECTS credits for credit points for doctoral studies!)
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Graduate School (Official website of the Zurich Graduate School in Mathematics:)
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