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Abstract
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We study an overlapping
domain decomposition method for solving the coupled nonlinear system of equations
arising from the discretization of inverse elliptic problems. Most algorithms
for solving inverse problems take advantage of the fact that the optimality
system has a natural splitting into three components: the state equation for
the constraints, the adjoint equation for the Lagrange multipliers, and the
equation for the parameter to be identified. Such algorithms often involve
interiterations between the three separate solvers, and the intercomponent iteration is
sequential. Several fully coupled or so-called one-shot approaches exist, and
the main challenges in these approaches are that the system has stronger
nonlinearity, and the corresponding Jacobian system is more ill-conditioned, in
addition to being three times larger. Here we investigate a class of overlapping
Newton–Krylov–Schwarz algorithms for solving such coupled systems, obtained with
a pointwise ordering of the variables, and show numerically that, with a
reasonably large overlap, the algorithm is capable of finding the solution even
with noise and discontinuous coefficients. More importantly, we show that
this approach is fully parallel and scalable with respect to the size of the
problems.
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Keywords
inverse problems, domain decomposition,
parallel computing, partial differential equations
constrained optimiztion, inexact Newton
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Mathematical Subject Classification
Primary: 65N21, 65N55, 65Y05
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Milestones
Received: 24 March 2008
Accepted: 15 March 2009
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