Inverse problems

The aim of the course is to learn how to use adjoint equations and formulate an optimal control problem for systems with distributed parameters. Industrial applications of control problems governed by partial differential equations are numerous. Many problems can be reduced to control problems. Optimizing the shape of antennas, mechanical structures or aerodynamic bodies is a typical example of control by boundary conditions. For evolutionary problems, an example is the calculation of optimal perturbations to suppress or increase the instability of a system. Finally, the adjoint equation method is also used for modeling problems, by solving an appropriate inverse problem. This course lays the foundations for numerical approximations of the above-mentioned problems.
Program: Direct and adjoint linear operators. Adjoint operators in spectral problems. Adjoint equations and linear functionals. Adjoint equations and perturbation theory. Non-linear problems: the case of compressible non-viscous fluid flows. Adjoint equations and inverse problems. Adjoint equations for non-stationary problems.