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SUMMARY:A “matching strategy” for optimal control problems with PDEs a
s constraints
DTSTART:20221024T140000Z
DTEND:20221024T150000Z
DTSTAMP:20240810T111200Z
UID:indico-event-113@events.dm.unipi.it
DESCRIPTION:Speakers: Santolo Leveque (Scuola Normale Superiore\, Pisa)\n\
nSaddle-point systems arise quite often in many areas of scientific comput
ing. For instance\, this type of system can be found in computational flui
d dynamics\, constrained optimization\, finance\, image reconstruction\, a
nd many more scientific applications. Due to their convenient structure an
d high dimensionality\, preconditioned iterative methods are usually emplo
yed in order to solve this class of problems. By making use of saddle-poin
t theory\, one is able to derive optimal preconditioners for a general sad
dle-point system. However\, the major drawback is that the cost of applyin
g their inverse operator is almost as costly as inverting the original sys
tem. For this reason\, one would rather find easy-to-invert approximation
of the main blocks of the preconditioner considered.In this talk\, we will
present the “matching strategy” derived in [3] for approximating the
Schur complement of a saddle-point system arsing from optimal control prob
lems with PDE as constraints. We will mainly focus on time-dependent PDE\,
when employing a Crank–Nicolson discretization in time. After applying
an optimize-then-discretize approach\, one is faced with continuous first-
order optimality conditions consisting of a coupled system of PDEs. After
discretizing with Crank–Nicolson\, one has to solve for a non-symmetric
system. We apply a carefully tailored invertible transformation for symmet
rizing the latter\, and derive an ideal preconditioner by making use of sa
ddle-point theory. The transformation is then employed within the precondi
tioner in order to derive optimal approximations of the (1\,1)-block and o
f the Schur complement. We prove the latter through bounds on the eigenval
ues\, and test our solver against the widely-used preconditioner derived i
n [2] for the linear system arising from a backward Euler discretization i
n time. These demonstrate the effectiveness and robustness of our solver w
ith respect to all the parameters involved in the problem considered.This
talk is based on the work in [1].References[1] S. Leveque and J. W. Pearso
n\, Fast Iterative Solver for the Optimal Control of Time-Dependent PDEs w
ith Crank–Nicolson Discretization in Time\, Numer. Linear Algebra Appl.
29\, e2419\, 2022.[2] J. W. Pearson\, M. Stoll and A. J. Wathen\, Regulari
zation-Robust Preconditioners for Time-Dependent PDE-Constrained Optimizat
ion Problems\, SIAM J. Matrix Anal. Appl. 33\, 1126–1152\, 2012.[3] J. W
. Pearson and A. J. Wathen\, A New Approximation of the Schur Complement i
n Preconditioners for PDE-Constrained Optimization\, Numer. Linear Algebra
Appl. 19\, 816–829\, 2012.\n\nhttps://events.dm.unipi.it/event/113/
LOCATION:Aula Riunioni (Dipartimento di Matematica)
URL:https://events.dm.unipi.it/event/113/
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