T1: Functional analysis, differential equations, and inverse problems
Format
Block course, 2 weeks
Contents
Duality and orthogonality; Fredholm alternative; Sobolev spaces; solution methods for PDE; linear regularisation theory; iterative regularisation; parameter choice rules; non-linear Tikhonov regularisation and sparsity promotion; problem sessions.
Concepts
Participants will acquire fundamental skills in dealing with linear ill-posed problems formulated as operator equations, together with a glance on sparsity-promoting regularisation techniques. They will further apply this theoretic knowledge to inverse problems for elliptic and parabolic differential equations, both analytically and numerically. As we assume that all PhD students have taken basic courses on functional analysis and partial differential equations during their studies, the course merely repeats basics from these fields. In contrast, analytic and numeric solution methods for differential equations will be considered in more detail. A central aim of the course is to provide enough analytical insights into regularisation theory and numerical techniques for the discrete solution of ill-posed problems such that participants can tackle such problems on their own after the course. For that reason, participants are required to work both on theoretic tutorials and on numerical exercises during the problem sessions. Concerning sparsity-promoting regularisation schemes, the course offers an introduction to non-linear Tikhonov regularisation with applications to ℓ1-minimisation, together with applications in inverse source problems for elliptic differential equations.
Basic texts
- L.C. Evans. Partial Differential Equations: Second Edition. American Mathematical Society, Providence, 2010.
- H.W. Engl, M. Hanke, and A. Neubauer. Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht, 1996.