Masahiro Yamamoto (Tokyo)

Intensive course on inverse problems

prof. yamamoto

Link of the first lecture on Microsoft Teams (Tuesday January 26 at 14:00)

Online PhD course
Period: 21 January - 19 February 2021
Dipartimento di Matematica
Università di Roma "Tor Vergata"

Tentative Schedule:

14:00-16:00 CET on the following dates

26/01 (Tuesday)
28/01 (
02/02 (Tuesday)
04/02 (Thursday)
08/02 (Monday)
10/02 (Wednesday)

12/02 (Friday)

15/02 (Monday)
17/02 (Wednesday)

Description and aim
The course is concerned with mathematical analysis of inverse problems for partial differential equations of evolution.  The inverse problems are physically well motivated and their types are comprehensive, so that the analyses require us various mathematical techniques.  Researches for inverse problems are greatly waiting for more participants.
This course is intended to be designed as a self-contained introduction and prepared also for audience who are interested in inverse problems as research targets.
   We start with some physical backgrounds and short glances at several inverse problems.  The main part is the methodology based on Carleman estimates.
The preliminary knowledge is the basic calculus, which is assumed to be provided in first-and second- years of undergraduate courses, although some parts are picked up from current research outputs by the lecturer and his colleagues.  An abridged lecture note will be offered.   

Chapter 1.  Introduction
1.           What are inverse problems?
2.           What are mathematical issues of inverse problems?
3.           Variety of formulation of inverse problems
      Ex: inverse heat source problems
Chapter 2.  Inverse wave source problem by multiplier method (1D case)
Chapter 3.  Inverse problem by Carleman estimate for first-order transport equation (naïve setting)
1.           Derivation of a Carleman estimate
2.           Stability for inverse source problem (standard argument)
Chapter 4.  Inverse problem by Carleman estimate for hyperbolic equation of second order
1.           Carleman estimate (without proof)
2.           Stability for inverse coefficient problem (standard argument)
3.           Derivation of Carleman estimate in 1D case
Chapter 5.  Inverse problem by Carleman estimate for parabolic equation
1.           Derivation of Carleman estimate for parabolic equation  (sketch of proof)
2.           Application to inverse heat conduction problem (lateral Cauchy problem)
Chapter 6.  Inverse problem by Carleman estimate for first-order transport equation (re-visit)
1.           Sharp version of a Carleman estimate
2.           Improvement : stability for inverse source problem
Chapter 7.  Backward parabolic problems
Chapter 8.  Global type of Carleman estimate and application
Chapter 9.  Inverse problems for other partial differential equation in mathematical
physics by Carleman estimates
 1.           Viscoelastic equation
2.           Linear elasticity equation
3.           Navier-Stokes lectures:

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