Wolfgang Bangerth
A framework for the adaptive finite element
solution of large inverse
problems. I. Basic techniques
ICES Report 2004-39, 2004.
Since problems involving the estimation of distributed coefficients in
partial differential equations are numerically very challenging, efficient
methods are indispensable. In this first part of a series, we will introduce
a framework for the efficient solution of such problems. This comprises the
use of adaptive finite element schemes, efficient solvers for the large
linear systems arising from discretization, and methods to
treat additional information in the form of inequality constraints on the
parameter sought. The methods to be developed will be based on an
\textit{all-at-once} approach, in which the inverse problem is solved
through a Lagrangian formulation. In order to allow for discretizations that
are adaptively refined as nonlinear iterations proceed, all algorithms are
formulated in a continuous function-space setting. Numerical examples will
demonstrate the applicability of the method for problems with several
million unknowns and more than 10,000 parameters.
This article also defines the notation and basic methods used in subsequent
parts to develop a posteriori error estimates, upon which optimal
discretizations will be based.
Wolfgang Bangerth
Wed May 28 14:37:56 CDT 2008