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Wolfgang Bangerth

**A framework for the adaptive finite element
solution of large inverse
problems**

SIAM Journal on Scientific
Computing, vol. 30 (2008), pp. 2965-2989.

Since problems involving the estimation of distributed coefficients in
partial differential equations are numerically very challenging,
efficient methods are indispensable. In this paper, we will introduce
a framework for the efficient solution of such problems. This
comprises the use of adaptive finite element schemes, solvers for the
large linear systems arising from discretization, and methods to treat
additional information in the form of inequality constraints on the
parameter to be recovered. The methods to be developed will be based
on an all-at-once approach, in which the inverse problem is solved
through a Lagrangian formulation.
The main feature of the paper is the
use of a continuous (function space) setting to formulate algorithms,
in order to allow for discretizations that are adaptively refined as
nonlinear iterations proceed. This entails that steps such as the
description of a Newton step or a line search are first formulated on
continuous functions and only then evaluated for discrete
functions. On the other hand, this approach avoids the dependence of
finite dimensional norms on the mesh size, making individual steps of
the algorithm comparable even if they used differently refined
meshes.
Numerical examples will demonstrate the applicability and
efficiency of the method for problems with several million unknowns
and more than 10,000 parameters.

*Wolfgang Bangerth*

* Thu Feb 15 22:24:40 MST 2018 *