Multiobjective Optimal Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling

In a wide range of applications it is desirable to optimally control a dynamical system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem where, instead of computing a single optimal solution, the set of optimal compromises, the so-called Pareto set, has to be approximated. When the problem under consideration is described by a partial differential equation (PDE), as is the case for fluid flow, the computational cost rapidly increases and makes its direct treatment infeasible. Reduced order modeling is a very popular method to reduce the computational cost, in particular in a multi query context such as uncertainty quantification, parameter estimation or optimization. In this article, we show how to combine reduced order modeling and multiobjective optimal control techniques in order to efficiently solve multiobjective optimal control problems constrained by PDEs. We consider a global, derivative free optimization method as well as a local, gradient-based approach for which the optimality system is derived in two different ways. The methods are compared with regard to the solution quality as well as the computational effort and they are illustrated using the example of the flow around a cylinder and a backward-facing-step channel flow.

     

Teachers' beliefs on teacher training contents and related characteristics of implementation—the example of introducing the topic study method in mathematics classrooms

So-called “bottom-up” strategies for implementation based on mathematics teachers' own developmental activities are considered to be a powerful approach when encouraging teachers to introduce alternative instructional practices. For evaluational research of in-service teacher training programs using “bottom-up” implementation strategies, the way how teachers implement contents of the teacher training is at the centre of interest. As the teachers' active role in the implementation process is necessary, their individual beliefs on the contents of the teacher training and their expectancies might influence the teachers' implementational activities. These beliefs can be considered as components of professional knowledge and pedagogical contents knowledge (Shulman, 1986) in particular. For this reason, the study focuses on the development of beliefs on contents of a teacher training program throughout the training on the one hand and relationships with characteristics of implementation on the other hand. We consider the example of introducing a student-centred learning environment, the so-called topic study method, in the teachers' classrooms. The results indicate that there are interdependencies between beliefs on the teacher training contents and characteristics of implementation.     

The structures of German mathematics textbooks

From a socio-cultural perspective it is argued that the modality of artefacts has, structuring effects on the activities in which the artefact is involved. The mathematics textbook is an artefact that has a major influence on the activity of learning mathematics. Against this setting, the structures of the units in German mathematics textbooks for different grades and ability levels have been analysed. Firstly, the different structural elements have been examined with regard to: characteristics in terms of content; linguistic characteristics; visual characteristics; their pedagogical functions within the learning process; and situative conditions. Secondly, the orders of the structural elements within the units of the different textbooks have been compared. The findings reveal that the structure of the units is very similar in different mathematics textbooks. The units are not only composed of analogous structural elements, but these elements are also arranged in almost the same sequence. In order to develop a deeper understanding of these findings the structure of the units has been compared to the influential learning theories of J. F. Herbart and H. Roth. On this basis it is argued that the structure of the units seems to reflect the phases of idealised learning processes in general. The issue is raised if this is an appropriate structure in order to provide opportunities to learn mathematics.     

Manifolds with almost nonnegative curvature operator and principal bundles

We study closed manifolds with almost nonnegative curvature operator (ANCO) and derive necessary and/or sufficient conditions for the total spaces of principal bundles over (A)NCO manifolds to admit ANCO connection metrics. In particular, we provide first examples of closed simply connected ANCO manifolds which do not admit metrics with nonnegative curvature operator.     

A deflated conjugate gradient method for multiple right hand sides and multiple shifts

We consider the task of computing solutions of linear systems that only differ by a shift with the identity matrix as well as linear systems with several different right-hand sides. In the past, Krylov subspace methods have been developed which exploit either the need for solutions to multiple right-hand sides (e.g. deflation type methods and block methods) or multiple shifts (e.g. shifted CG) with some success. In this paper we present a block Krylov subspace method which, based on a block Lanczos process, exploits both features—shifts and multiple right-hand sides—at once. Such situations arise, for example, in lattice quantum chromodynamics (QCD) simulations within the Rational Hybrid Monte Carlo (RHMC) algorithm. We present numerical evidence that our method is superior compared to applying other iterative methods to each of the systems individually as well as, in typical situations, to shifted or block Krylov subspace methods.     

Fe valence state at the surface of the Fe0.5Cu0.5Cr2S4 spinel

Here we present X‐ray absorption measurements of a vacuum cleaved Fe_0.5Cu_0.5Cr₂S₄ single crystal. Measurements at different positions on the cleaved sample surface clearly reveal a difference between the valence state of the Fe ions in the sur‐ face layers and the valency of the Fe ions present in the bulk. These results confirm the findings of recent measurements outlined previously. (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Aldehyde‐Assisted Ruthenium(II)‐Catalyzed CH Oxygenations

Versatile ruthenium(II) complexes allow for site‐selective C? H oxygenations with weakly‐coordinating aldehydes. The challenging C? H functionalizations proceed with high chemoselectivity by rate‐determining C? H metalation. The new method features an ample substrate scope, which sets the stage for the step‐economical preparation of various bioactive heterocycles.     

The NiII,HgII and CuII complexes of 12‐membered‐ring mixed‐donor macrocycles

The structures of di­aqua(1,7‐dioxa‐4‐thia‐10‐aza­cyclo­do­decane)­nickel dinitrate, [Ni(C₈H₁₇NO₂S)(H₂O)₂](NO₃)₂, (I), bis­(nitrato‐O,O′)(1,4,7‐trioxa‐10‐aza­cyclo­do­decane)­mercury, [Hg(NO₃)₂(C₈H₁₇NO₃)], (II), and aqua­(nitrato‐O)(1‐oxa‐4,7,10‐tri­aza­cyclo­do­decane)copper nitrate, [Cu(NO₃)(C₈H₁₉N₃O)(H₂O)]NO₃, (III), reveal each macrocycle binding in a tetradentate manner. The conformations of the ligands in (I) and (III) are the same and distinct from that identified for (II). These differences are in agreement with molecular‐mechanics predictions of ligand conformation as a function of metal‐ion size.     

Molybdenum and tungsten structural differences are dependent on ndz(2)/(n + 1)s mixing: comparisons of (silox)3MX/R (M = Mo, W; silox = (t)Bu3SiO)

Treatment of trans-(Et 2O) 2MoCl 4 with 2 or 3 equiv of Na(silox) (i.e., NaOSi (t) Bu 3) afforded (silox) 3MoCl 2 ( 1-Mo) or (silox) 3MoCl ( 2-Mo). Purification of 2-Mo was accomplished via addition of PMe 3 to precipitate (silox) 3ClMoPMe 3 ( 2-MoPMe 3), followed by thermolysis to remove phosphine. Use of MoCl 3(THF) 3 with various amounts of Na(silox) produced (silox) 2ClMoMoCl(silox) 2 ( 3-Mo). Alkylation of 2-Mo with MeMgBr or EtMgBr afforded (silox) 3MoR (R = Me, 2-MoMe; Et, 2-MoEt). 2-MoEt was also synthesized from C 2H 4 and (silox) 3MoH, which was prepared from 2-Mo and NaBEt 3H. Thermolysis of WCl 6 with HOSi ( t )Bu 3 afforded (silox) 2WCl 4 ( 4-W), and sequential treatment of 4-W with Na/Hg and Na(silox) provided (silox) 3WCl 2 ( 1-W, tbp, X-ray), which was alternatively prepared from trans-(Et 2S) 2WCl 4 and 3 equiv of Tl(silox). Na/Hg reduction of 1-W generated (silox) 3WCl ( 2-W). Alkylation of 2-W with MeMgBr produced (silox) 3WMe ( 2-WMe), which dehydrogenated to (silox) 3WCH ( 6-W) with Delta H (double dagger) = 14.9(9) kcal/mol and Delta S (double dagger) = -26(2) eu. Magnetism and structural studies revealed that 2-Mo and 2-MoEt have triplet ground states (GS) and distorted trigonal monopyramid (tmp) and tmp structures, respectively. In contrast, 2-W and 2-WMe possess squashed-T d (distorted square planar) structures, and the former has a singlet GS. Quantum mechanics/molecular mechanics studies of the S = 0 and S = 1 states for full models of 2-Mo, 2-MoEt, 2-W, and 2-WMe corroborate the experimental findings and are consistent with the greater nd z (2) /( n + 1)s mixing in the third-row transition-metal species being the dominant feature in determining the structural disparity between molybdenum and tungsten.