Minicourse on Applied Inverse Problems  
Firenze, October 7-11, 2002

  Abstracts of the lectures.


Lars Eldén  Numerical solutions to Cauchy problems in parabolic and elliptic equations.

1. The sideways heat equation
Ill-posedness, analysis using Fourier transform and and singular value decomposition, logarithmic convexity (Levine), stability estimates for Tikhonov regularization (Carasso), results for variable coefficients.
Numerical solution.
Integral equation formulation and Tikhonov regularization, discretization and efficient solution of the matrix problem. Spectral, wavelet, and mollification approximation of the time derivative, stability estimates. Space marching with a bounded time derivative approximation, efficient implementation (a method of lines). Nonlinear problems. Brief description of alternative methods.

2. Cauchy problems for elliptic equations
Ill-posedness for the Laplace equation in rectangular geometry, stability results.
Numerical solution
Discretization of the PDE and Tikhonov regularization, approximation of one space derivative by a bounded operator (spectral, wavelet, spline), method of lines. Numerical mapping of some (rather) general geometries to rectangular regions. Problems with variable coefficients, determination of an unknown boundary.

3. Applications
Brief description of a few applications (this will, at least partly, be done at the beginning of the course).

William Rundell Reconstruction methods in inverse eigenvalue problems and in inverse scattering.

The classical inverse Sturm Liouville problem.

  • History and some of the main uniqueness results together with their proofs.
  • Reconstruction issues and methods.
  • Other inverse eigenvalue problems.

    Inverse obstacle scattering.

  • Introduction and history.
  • Introduction to potential theoretic methods.
  • Uniqueness results.
  • Numerical methods.
  • Some future problems.

  • Erkki Somersalo Statistical methods in inverse problems.

    Inverse problems can be recast in the form of statistical inference: In contrast to the traditional inversion techniques, one seeks not to find an estimate of the unknowns by regularization but rather by using the measurements along with possible prior information one solves the posterior probability distribution of the unknowns.

    Computationally, the statistical inversion is challenging for several reasons:
    1) Often, the prior information is qualitative and it is not clear how to code it into a prior distribution.
    2) The solution of the inverse problem within the statistical approach is a probability density in a high dimensional space. One has to develop effective stochastic methods to explore such densities.
    3) One need to design carefully effective numerical forward solvers for partial differential equations and/or integral equations in order to be able to generate large ensembles from the posterior densities.

    The course will mostly focus on these computational issues. Material will be distributed during the course.