176 research outputs found
Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods
Many problems in computational science and engineering are simultaneously
characterized by the following challenging issues: uncertainty, nonlinearity,
nonstationarity and high dimensionality. Existing numerical techniques for such
models would typically require considerable computational and storage
resources. This is the case, for instance, for an optimization problem governed
by time-dependent Navier-Stokes equations with uncertain inputs. In particular,
the stochastic Galerkin finite element method often leads to a prohibitively
high dimensional saddle-point system with tensor product structure. In this
paper, we approximate the solution by the low-rank Tensor Train decomposition,
and present a numerically efficient algorithm to solve the optimality equations
directly in the low-rank representation. We show that the solution of the
vorticity minimization problem with a distributed control admits a
representation with ranks that depend modestly on model and discretization
parameters even for high Reynolds numbers. For lower Reynolds numbers this is
also the case for a boundary control. This opens the way for a reduced-order
modeling of the stochastic optimal flow control with a moderate cost at all
stages.Comment: 29 page
Low-Rank Iterative Solvers for Large-Scale Stochastic Galerkin Linear Systems
Otto-von-Guericke-UniversitĂ€t Magdeburg, FakultĂ€t fĂŒr Mathematik, Dissertation, 2016von Dr. rer. pol. Akwum Agwu OnwuntaLiteraturverzeichnis: Seite 135-14
Matching Schur complement approximations for certain saddle-point systems
The solution of many practical problems described by mathematical models requires approximation methods that give rise to linear(ized) systems of equations, solving which will determine the desired approximation. This short contribution describes a particularly effective solution approach for a certain class of so-called saddle-point linear systems which arises in different contexts
A multigrid method for PDE-constrained optimization with uncertain inputs
We present a multigrid algorithm to solve efficiently the large saddle-point
systems of equations that typically arise in PDE-constrained optimization under
uncertainty. The algorithm is based on a collective smoother that at each
iteration sweeps over the nodes of the computational mesh, and solves a reduced
saddle-point system whose size depends on the number of samples used to
discretized the probability space. We show that this reduced system can be
solved with optimal complexity. We test the multigrid method on three
problems: a linear-quadratic problem for which the multigrid method is used to
solve directly the linear optimality system; a nonsmooth problem with box
constraints and -norm penalization on the control, in which the multigrid
scheme is used within a semismooth Newton iteration; a risk-adverse problem
with the smoothed CVaR risk measure where the multigrid method is called within
a preconditioned Newton iteration. In all cases, the multigrid algorithm
exhibits very good performances and robustness with respect to all parameters
of interest.Comment: 24, 2 figure
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