Linear operators in Hilbert spaces:closed and non closed range operators. Ill-posed problems, generalized solution. Compact operators. Singular system and regularization methods: regularization algorithms in the sense of Tikhonov.
Iterative methods: the Landweber method and the conjugate gradient method. Choice of the regularization parameter.
Problems of image reconstruction and of image deconvolution.Regularization methods are analyzed using the tools already exposed adapted its Fourier analysis
Statistical approach to inverse problems: Maximum Likelihood and Bayes Theorem.
Monte Carlo methods for non-linear inverse problems: importance sampling and Markov Chain Monte Carlo.
Methods for dynamic inverse problems: Kalman and particle filtering.
The course also includes numerical experiments with Matlab.