13:30
MS30: Numerical methods for PDE-constrained and controlled optimization problems with appplications (Part 2)
Chair: Maya Neytcheva
13:30
25 mins
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Efficient solvers for time-periodic parabolic optimal control problems using two-sided bounds of cost functionals
Monika Wolfmayr
Abstract: This article is devoted to presenting efficient solvers for time-periodic parabolic optimization problems. The solvers are based on deriving two-sided bounds for the cost functional. Here, we especially employ the time-periodic nature of the problem discussed in order to obtain fully computable and guaranteed upper and lower bounds for the cost functional. We present the multiharmonic finite element method as a proper approach for deriving a discretized solution of the time-periodic problem. The multiharmonic finite element functions can be used as initial guess for the arbitrary functions in the upper and lower bounds, which then can be minimized and maximized, respectively, in order to obtain an approximate solution of any desired accuracy. Finally, new numerical results are presented in order to show the efficiency of the method discussed also in practice.
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13:55
25 mins
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Hierarchical DWR Error Estimates for the Navier-Stokes Equations: 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 on two-side a posteriori error estimates for the DWR method.
We now focus on h enrichment and p enrichment for the error estimator.
These advancements are demonstrated with the help of a numerical example.
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14:20
25 mins
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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
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14:45
25 mins
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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.
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