258 research outputs found
Interior Point Methods and Preconditioning for PDE-Constrained Optimization Problems Involving Sparsity Terms
PDE-constrained optimization problems with control or state constraints are
challenging from an analytical as well as numerical perspective. The
combination of these constraints with a sparsity-promoting term
within the objective function requires sophisticated optimization methods. We
propose the use of an Interior Point scheme applied to a smoothed reformulation
of the discretized problem, and illustrate that such a scheme exhibits robust
performance with respect to parameter changes. To increase the potency of this
method we introduce fast and efficient preconditioners which enable us to solve
problems from a number of PDE applications in low iteration numbers and CPU
times, even when the parameters involved are altered dramatically
Preconditioning of Active-Set Newton Methods for PDE-constrained Optimal Control Problems
We address the problem of preconditioning a sequence of saddle point linear
systems arising in the solution of PDE-constrained optimal control problems via
active-set Newton methods, with control and (regularized) state constraints. We
present two new preconditioners based on a full block matrix factorization of
the Schur complement of the Jacobian matrices, where the active-set blocks are
merged into the constraint blocks. We discuss the robustness of the new
preconditioners with respect to the parameters of the continuous and discrete
problems. Numerical experiments on 3D problems are presented, including
comparisons with existing approaches based on preconditioned conjugate
gradients in a nonstandard inner product
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
Optimization Methods for Inverse Problems
Optimization plays an important role in solving many inverse problems.
Indeed, the task of inversion often either involves or is fully cast as a
solution of an optimization problem. In this light, the mere non-linear,
non-convex, and large-scale nature of many of these inversions gives rise to
some very challenging optimization problems. The inverse problem community has
long been developing various techniques for solving such optimization tasks.
However, other, seemingly disjoint communities, such as that of machine
learning, have developed, almost in parallel, interesting alternative methods
which might have stayed under the radar of the inverse problem community. In
this survey, we aim to change that. In doing so, we first discuss current
state-of-the-art optimization methods widely used in inverse problems. We then
survey recent related advances in addressing similar challenges in problems
faced by the machine learning community, and discuss their potential advantages
for solving inverse problems. By highlighting the similarities among the
optimization challenges faced by the inverse problem and the machine learning
communities, we hope that this survey can serve as a bridge in bringing
together these two communities and encourage cross fertilization of ideas.Comment: 13 page
General-purpose preconditioning for regularized interior point methods
In this paper we present general-purpose preconditioners for regularized augmented systems, and their corresponding normal equations, arising from optimization problems. We discuss positive definite preconditioners, suitable for CG and MINRES. We consider “sparsifications" which avoid situations in which eigenvalues of the preconditioned matrix may become complex. Special attention is given to systems arising from the application of regularized interior point methods to linear or nonlinear convex programming problems.</p
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