Numerical solutions to Cauchy problems in parabolic and elliptic
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.
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,
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.
Brief description of a few applications (this will, at least partly, be
done at the beginning of the course).
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
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.