4,668 research outputs found
Robust Optimization of PDEs with Random Coefficients Using a Multilevel Monte Carlo Method
This paper addresses optimization problems constrained by partial
differential equations with uncertain coefficients. In particular, the robust
control problem and the average control problem are considered for a tracking
type cost functional with an additional penalty on the variance of the state.
The expressions for the gradient and Hessian corresponding to either problem
contain expected value operators. Due to the large number of uncertainties
considered in our model, we suggest to evaluate these expectations using a
multilevel Monte Carlo (MLMC) method. Under mild assumptions, it is shown that
this results in the gradient and Hessian corresponding to the MLMC estimator of
the original cost functional. Furthermore, we show that the use of certain
correlated samples yields a reduction in the total number of samples required.
Two optimization methods are investigated: the nonlinear conjugate gradient
method and the Newton method. For both, a specific algorithm is provided that
dynamically decides which and how many samples should be taken in each
iteration. The cost of the optimization up to some specified tolerance
is shown to be proportional to the cost of a gradient evaluation with requested
root mean square error . The algorithms are tested on a model elliptic
diffusion problem with lognormal diffusion coefficient. An additional nonlinear
term is also considered.Comment: This work was presented at the IMG 2016 conference (Dec 5 - Dec 9,
2016), at the Copper Mountain conference (Mar 26 - Mar 30, 2017), and at the
FrontUQ conference (Sept 5 - Sept 8, 2017
Optimal Control of Convective FitzHugh-Nagumo Equation
We investigate smooth and sparse optimal control problems for convective
FitzHugh-Nagumo equation with travelling wave solutions in moving excitable
media. The cost function includes distributed space-time and terminal
observations or targets. The state and adjoint equations are discretized in
space by symmetric interior point Galerkin (SIPG) method and by backward Euler
method in time. Several numerical results are presented for the control of the
travelling waves. We also show numerically the validity of the second order
optimality conditions for the local solutions of the sparse optimal control
problem for vanishing Tikhonov regularization parameter. Further, we estimate
the distance between the discrete control and associated local optima
numerically by the help of the perturbation method and the smallest eigenvalue
of the reduced Hessian
Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs
This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic PDE. We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on with , and that the PDE operator may depend non-affinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [CDS1, CDS2], we show that the state and the control are analytic as functions depending on these parameters . We
establish sparsity of generalized polynomial chaos (gpc) expansions of both, state and control, in terms of the stochastic coordinate sequence of the random inputs, and prove convergence rates of best -term truncations of these expansions. Such truncations are the key for subsequent computations since they do {\em not} assume that the stochastic input data has a finite expansion. In the follow-up paper [KS2], we explain two methods how such best -term truncations can practically be computed, by greedy-type algorithms
as in [SG, Gi1], or by multilevel Monte-Carlo methods as in
[KSS]. The sparsity result allows in conjunction with adaptive wavelet Galerkin schemes for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK, GK, K], see [KS2]
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