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 (Thursday) 02/02 (Tuesday) 04/02 (Thursday) 08/02
(Monday)
Description and aim10/02 (Wednesday) 12/02 (Friday) 15/02 (Monday) 17/02 (Wednesday) 19/02 (Friday) 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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