European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
13:30   MS30: Numerical methods for PDE-constrained and controlled optimization problems with appplications (Part 2)
Chair: Maya Neytcheva
25 mins
Two-sided bounds for cost functionals of time-periodic parabolic optimal control problems
Monika Wolfmayr
Abstract: In this talk, a new technique is presented on deriving computable, guaranteed lower bounds of functional type (minorants) applied on two different cost functionals subject to a parabolic time-periodic boundary value problem. The technique was earlier derived in [2] for elliptic optimal control problems. Together with previous results on upper bounds (majorants) for one of the cost functionals (see [1]), both minorants and majorants lead to two-sided estimates of functional type for the optimal control problem. Moreover, a different cost functional is discussed with the same PDE-constraints. Both upper and lower bounds are derived. The time-periodic optimal control problems are discretized by the multiharmonic finite element method leading to large systems of linear equations having a saddle point structure. The derivation of preconditioners for the minimal residual method for the new optimal control problem is discussed. Finally, several numerical experiments for both optimal control problems are presented confirming the theoretical results obtained. This work was supported by the Academy of Finland under the grant 295897. References: [1] U. Langer, S. Repin, and M. Wolfmayr, Functional a posteriori error estimates for time-periodic parabolic optimal control problems, Numerical Functional Analysis and Optimization, 37 (2016), pp. 1267–1294, [2] M.Wolfmayr, A note on functional a posteriori estimates for elliptic optimal control problems, Numer Meth Part Differ Equat, 33 (2017), pp. 403–424, [3] M. Wolfmayr, Two-sided bounds for cost functionals of time-periodic parabolic optimal control problems , arXiv preprint arXiv:1901.09924 (2019),
25 mins
Hierarchical DWR Error Estimates for the Navier Stokes Equation: h and p Enrichment
Bernhard Endtmayer, Ulrich Langer, Jan Philipp Thiele, Thomas Wick
Abstract: In this work, we further develop multigoal-oriented a posteriori error estimation for the nonlinear, stationary, incompressible Navier-Stokes equations. It is an extension of our previous work [B. Endtmayer, U. Langer, T. Wick: Two-side a posteriori error estimates for the DWR method, SISC, 2019, accepted]. We now focus on h mesh refinement and p enrichment for the error estimator. These advancements are demonstrated with the help of a numerical example.
25 mins
PDE-constrained optimization for importance sampling of rare events
George Biros, Siddhant Wahal
Abstract: We consider the problem of estimating rare event probabilities, focusing on systems whose evolution is governed by differential equations with uncertain input parameters. If the system dynamics is expensive to compute, standard sampling algorithms such as the Monte Carlo method may require infeasible running times to accurately evaluate these probabilities. We propose an importance sampling scheme (christened BIMC) that relies on solving a “fictitious” Bayesian inverse problem. The solution of the inverse problem yields a posterior PDF, a local Gaussian approximation to which serves as the importance sampling density. We apply BIMC to several problems and demonstrate that it can lead to computational savings of several orders of magnitude over the Monte Carlo method. We delineate conditions under which BIMC is optimal, as well as conditions when it can fail to yield an effective important sampling density. Keyworkds: Rare events, Monte Carlo Methods, Importance Sampling
25 mins
On the use of Gauss–Newton directions for design optimization
Martin Berggren, Anders Bernland
Abstract: Numerical design optimization problems—shape and topology optimization—are typically large-scale problems; a recent topology optimization benchmark involves up to a billion design variables (Aage et al, Nature, 550:84–86, 2017). The extreme design flexibility emerging from the large scales is essential to allow for unexpected and surprising shapes of high performance. On the other hand, the diversity of computationally tractable optimization algorithms is severely limited for large-scale problems. The use of adjoint equations allows for objective-function derivatives to be efficiently computed. Optimization methods requiring second derivatives can sometimes successfully be used (Evgrafov, Comput. Methods Appl. Mech. Engrg., 278:272–290, 2014), but in general it is computationally very expensive to compute full Hessian information. However, for design problems featuring a least-squares objective, it is sometimes possible to efficiently compute Gauss–Newton directions, which will give partial second-order information that can significantly speed up the convergence compared to strictly first-order methods. An example of design problems that sometimes allows efficient computation of Gauss–Newton directions are acoustic or electromagnetic devices optimized with respect to the so-called scattering parameters. In particular, for such a problem—an acoustic design problem in frequency-domain—the Gauss–Newton direction was simple to compute, and when it was used in in the context of a Levenberg–Marquardt algorithm, we observed a significant improvement of the computational efficiency compared to when BFGS secant approximations of the Hessian were used. An ongoing investigation concerns a generalization to corresponding problems in time domain, where the least-squares functional consists of an integral in time. A naive approach to the calculation of the Jacobians needed for the Gauss–Newton direction would here require an infinite number of adjoint equations, one for each time. However, there are problems with a regular enough structure so that only one adjoint equation is needed to compute all the Jacobians.