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Author: Adam, Lukáš × End date: 2019 × Start date: 2010 × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik ×

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Now showing 1 - 4 of 4
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    A semismooth Newton method with analytical path-following for the H1-projection onto the Gibbs simplex
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Adam, Lukáš; Hintermüller, Michael; Surowiec, Thomas M.
    An efficient, function-space-based second-order method for the H1-projection onto the Gibbs-simplex is presented. The method makes use of the theory of semismooth Newton methods in function spaces as well as Moreau-Yosida regularization and techniques from parametric optimization. A path-following technique is considered for the regularization parameter updates. A rigorous first and second-order sensitivity analysis of the value function for the regularized problem is provided to justify the update scheme. The viability of the algorithm is then demonstrated for two applications found in the literature: binary image inpainting and labeled data classification. In both cases, the algorithm exhibits meshindependent behavior.
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    Optimization of a multiphysics problem in semiconductor laser design
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Adam, Lukáš; Hintermüller, Michael; Peschka, Dirk; Surowiec, Thomas M.
    A @multimaterial topology optimization framework is suggested for the simultaneous optimization of mechanical and optical properties to be used in the development of optoelectronic devices. Based on the physical aspects of the underlying device, a nonlinear multiphysics model for the elastic and optical properties is proposed. Rigorous proofs are provided for the sensitivity of the fundamental mode of the device with respect to the changes in the underlying topology. After proving existence and optimality results, numerical experiments leading to an optimal material distribution for maximizing the strain in a Ge-on-Si microbridge are given. The highly favorable electronic properties of this design are demonstrated by steady-state simulations of the corresponding van Roosbroeck (drift-diffusion) system.
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    On M-stationarity conditions in MPECs and the associated qualification conditions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Adam, Lukáš; Henrion, René; Outrata, Jir̆í
    Depending on whether a mathematical program with equilibrium constraints (MPEC) is considered in its original or its enhanced (via KKT conditions) form, the assumed constraint qualifications (CQs) as well as the derived necessary optimality conditions may differ significantly. In this paper, we study this issue when imposing one of the weakest possible CQs, namely the calmness of the perturbation mapping associated with the respective generalized equations in both forms of the MPEC. It is well known that the calmness property allows one to derive socalled M-stationarity conditions. The strength of assumptions and conclusions in the two forms of the MPEC is strongly related with the CQs on the lower level imposed on the set whose normal cone appears in the generalized equation. For instance, under just the Mangasarian-Fromovitz CQ (a minimum assumption required for this set), the calmness properties of the original and the enhanced perturbation mapping are drastically different. They become identical in the case of a polyhedral set or when adding the Full Rank CQ. On the other hand, the resulting optimality conditions are affected too. If the considered set even satisfies the Linear Independence CQ, both the calmness assumption and the derived optimality conditions are fully equivalent for the original and the enhanced form of the MPEC. A compilation of practically relevant consequences of our analysis in the derivation of necessary optimality conditions is provided in the main Theorem 4.3. The obtained results are finally applied to MPECs with structured equilibria.
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    A PDE-constrained optimization approach for topology optimization of strained photonic devices
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Adam, Lukáš; Hintermüller, Michael; Surowiec, Thomas M.
    Recent studies have demonstrated the potential of using tensile-strained, doped Germanium as a means of developing an integrated light source for (amongst other things) future microprocessors. In this work, a multi-material phase-field approach to determine the optimal material configuration within a so-called Germanium-on-Silicon microbridge is considered. Here, an optimal configuration is one in which the strain in a predetermined minimal optical cavity within the Germanium is maximized according to an appropriately chosen objective functional. Due to manufacturing requirements, the emphasis here is on the cross-section of the device; i.e. a socalled aperture design. Here, the optimization is modeled as a non-linear optimization problem with partial differential equation (PDE) and manufacturing constraints. The resulting problem is analyzed and solved numerically. The theory portion includes a proof of existence of an optimal topology, differential sensitivity analysis of the displacement with respect to the topology, and the derivation of first and second-order optimality conditions. For the numerical experiments, an array of first and second-order solution algorithms in function-space are adapted to the current setting, tested, and compared. The numerical examples yield designs for which a significant increase in strain (as compared to an intuitive empirical design) is observed.