• Inverse problems deal with determining unknown causes from some partially known effects given by a set of measurements. They involve parameters identification in a given dynamical system. With the appearance of advanced computers, this theory has expanded to most of the scientific fields that are covered by mathematics. Applications include seismology, medical imaging, astronomy, etc.
• In this course, we will explore some theoretical and numerical aspects of inverse problems theory: mathematical formulation, conditional well-posedness, qualitative properties (existence, uniqueness and stability) and the different methods and techniques for solving inverse problems and approximating their solutions. First, students will be introduced to main theoretical foundations of the theory, then several applications will be developed by studying some numerical methods.
• Prerequisites: basics of Functional Analysis and PDEs. Intended audience include engineering and computer science students with some PDEs background.

The following topics will be covered

• Introduction and Examples
• Linear inverse and ill-posed problems
• Theoretical methods to prove uniqueness and stability: Carleman estimates method
• Regularization strategies: Tikhonov regularization
• Singular value decomposition method
• Numerical identification by optimization approach: quasi-solution approach
• Gradient calculation by adjoint state method
• Landweber iteration and Conjugate Gradient algorithms.

Course slides and reading material will be available from the instructor.

• Optional Textbooks:
  • V. Isakov. Inverse Problems for Partial Differential Equations. Springer, Cham, third edition, 2017.
  • A. Hasanov Hasanoglu and V.G. Romanov. Introduction to Inverse Problems for Differential Equations. Springer, second edition, 2021.

1. Midterm Exam: 25%
2. Project: 25%
3. Final Exam: 50%