Numerical solution of optimal control problems with constant control delays

We investigate a class of optimal control problems that exhibit constant exogenously given delays in the control in the equation of motion of the differential states. Therefore, we formulate an exemplary optimal control problem with one stock and one control variable and review some analytic properties of an optimal solution. However, analytical considerations are quite limited in case of delayed optimal control problems. In order to overcome these limits, we reformulate the problem and apply direct numerical methods to calculate approximate solutions that give a better understanding of this class of optimization problems. In particular, we present two possibilities to reformulate the delayed optimal control problem into an instantaneous optimal control problem and show how these can be solved numerically with a stateof- the-art direct method by applying Bock’s direct multiple shooting algorithm. We further demonstrate the strength of our approach by two economic examples.

The Nature of Scientific Proof in the Age of Simulations. Is numerical mimicry a third way of establishing truth?

Is numerical mimicry a third way of establishing truth? Kevin Heng received his M.S. and Ph.D. in astrophysics from the Joint Institute for Laboratory Astrophysics (JILA) and the University of Colorado at Boulder. He joined the Institute for Advanced Study in Princeton from 2007 to 2010, first as a Member and later as the Frank & Peggy Taplin Member. From 2010 to 2012 he was a Zwicky Prize Fellow at ETH Z¨urich (the Swiss Federal Institute of Technology). In 2013, he joined the Center for Space and Habitability (CSH) at the University of Bern, Switzerland, as a tenure-track assistant professor, where he leads the Exoplanets and Exoclimes Group. He has worked on, and maintains, a broad range of interests in astrophysics: shocks, extrasolar asteroid belts, planet formation, fluid dynamics, brown dwarfs and exoplanets. He coordinates the Exoclimes Simulation Platform (ESP), an open-source set of theoretical tools designed for studying the basic physics and chemistry of exoplanetary atmospheres and climates (www.exoclime.org). He is involved in the CHEOPS (Characterizing Exoplanet Satellite) space telescope, a mission approved by the European Space Agency (ESA) and led by Switzerland. He spends a fair amount of time humbly learning the lessons gleaned from studying the Earth and Solar System planets, as related to him by atmospheric, climate and planetary scientists. He received a Sigma Xi Grant-in-Aid of Research in 2006

Experimental, theoretical and numerical investigation of the nonlinear micromechanical properties of bone

Aging societies suffer from an increasing incidence of bone fractures. Bone strength depends on the amount of mineral measured by clinical densitometry, but also on the micromechanical properties of the bone hierarchical organization. A good understanding has been reached for elastic properties on several length scales, but up to now there is a lack of reliable postyield data on the lower length scales. In order to be able to describe the behavior of bone at the microscale, an anisotropic elastic-viscoplastic damage model was developed using an eccentric generalized Hill criterion and nonlinear isotropic hardening. The model was implemented as a user subroutine in Abaqus and verified using single element tests. A FE simulation of microindentation in lamellar bone was finally performed show-ing that the new constitutive model can capture the main characteristics of the indentation response of bone. As the generalized Hill criterion is limited to elliptical and cylindrical yield surfaces and the correct shape for bone is not known, a new yield surface was developed that takes any convex quadratic shape. The main advantage is that in the case of material identification the shape of the yield surface does not have to be anticipated but a minimization results in the optimal shape among all convex quadrics. The generality of the formulation was demonstrated by showing its degeneration to classical yield surfaces. Also, existing yield criteria for bone at multiple length scales were converted to the quadric formulation. Then, a computational study to determine the influence of yield surface shape and damage on the in-dentation response of bone using spherical and conical tips was performed. The constitutive model was adapted to the quadric criterion and yield surface shape and critical damage were varied. They were shown to have a major impact on the indentation curves. Their influence on indentation modulus, hardness, their ratio as well as the elastic to total work ratio were found to be very well described by multilinear regressions for both tip shapes. For conical tips, indentation depth was not a significant fac-tor, while for spherical tips damage was insignificant. All inverse methods based on microindentation suffer from a lack of uniqueness of the found material properties in the case of nonlinear material behavior. Therefore, monotonic and cyclic micropillar com-pression tests in a scanning electron microscope allowing a straightforward interpretation comple-mented by microindentation and macroscopic uniaxial compression tests were performed on dry ovine bone to identify modulus, yield stress, plastic deformation, damage accumulation and failure mecha-nisms. While the elastic properties were highly consistent, the postyield deformation and failure mech-anisms differed between the two length scales. A majority of the micropillars showed a ductile behavior with strain hardening until failure by localization in a slip plane, while the macroscopic samples failed in a quasi-brittle fashion with microcracks coalescing into macroscopic failure surfaces. In agreement with a proposed rheological model, these experiments illustrate a transition from a ductile mechanical behavior of bone at the microscale to a quasi-brittle response driven by the growth of preexisting cracks along interfaces or in the vicinity of pores at the macroscale. Subsequently, a study was undertaken to quantify the topological variability of indentations in bone and examine its relationship with mechanical properties. Indentations were performed in dry human and ovine bone in axial and transverse directions and their topography measured by AFM. Statistical shape modeling of the residual imprint allowed to define a mean shape and describe the variability with 21 principal components related to imprint depth, surface curvature and roughness. The indentation profile of bone was highly consistent and free of any pile up. A few of the topological parameters, in particular depth, showed significant correlations to variations in mechanical properties, but the cor-relations were not very strong or consistent. We could thus verify that bone is rather homogeneous in its micromechanical properties and that indentation results are not strongly influenced by small de-viations from the ideal case. As the uniaxial properties measured by micropillar compression are in conflict with the current literature on bone indentation, another dissipative mechanism has to be present. The elastic-viscoplastic damage model was therefore extended to viscoelasticity. The viscoelastic properties were identified from macroscopic experiments, while the quasistatic postelastic properties were extracted from micropillar data. It was found that viscoelasticity governed by macroscale properties has very little influence on the indentation curve and results in a clear underestimation of the creep deformation. Adding viscoplasticity leads to increased creep, but hardness is still highly overestimated. It was possible to obtain a reasonable fit with experimental indentation curves for both Berkovich and spherical indenta-tion when abandoning the assumption of shear strength being governed by an isotropy condition. These results remain to be verified by independent tests probing the micromechanical strength prop-erties in tension and shear. In conclusion, in this thesis several tools were developed to describe the complex behavior of bone on the microscale and experiments were performed to identify its material properties. Micropillar com-pression highlighted a size effect in bone due to the presence of preexisting cracks and pores or inter-faces like cement lines. It was possible to get a reasonable fit between experimental indentation curves using different tips and simulations using the constitutive model and uniaxial properties measured by micropillar compression. Additional experimental work is necessary to identify the exact nature of the size effect and the mechanical role of interfaces in bone. Deciphering the micromechanical behavior of lamellar bone and its evolution with age, disease and treatment and its failure mechanisms on several length scales will help preventing fractures in the elderly in the future.

Importance sampling approximations to various probabilities of ruin of spectrally negative Lévy risk processes

This article provides importance sampling algorithms for computing the probabilities of various types ruin of spectrally negative Lévy risk processes, which are ruin over the infinite time horizon, ruin within a finite time horizon and ruin past a finite time horizon. For the special case of the compound Poisson process perturbed by diffusion, algorithms for computing probabilities of ruins by creeping (i.e. induced by the diffusion term) and by jumping (i.e. by a claim amount) are provided. It is shown that these algorithms have either bounded relative error or logarithmic efficiency, as t,x→∞t,x→∞, where t>0t>0 is the time horizon and x>0x>0 is the starting point of the risk process, with y=t/xy=t/x held constant and assumed either below or above a certain constant.

Numerical study of Anderson localization of terahertz waves in disordered waveguides

We present a numerical study of electromagnetic wave transport in disordered quasi-one-dimensional waveguides at terahertz frequencies. Finite element method calculations of terahertz wave propagation within LiNbO3 waveguides with randomly arranged air-filled circular scatterers exhibit an onset of Anderson localization at experimentally accessible length scales. Results for the average transmission as a function of waveguide length and scatterer density demonstrate a clear crossover from diffusive to localized transport regime. In addition, we find that transmission fluctuations grow dramatically when crossing into the localized regime. Our numerical results are in good quantitative agreement with theory over a wide range of experimentally accessible parameters both in the diffusive and localized regime opening the path towards experimental observation of terahertz wave localization.

Mass Movement-Induced Tsunami Hazard on Perialpine Lake Lucerne (Switzerland): Scenarios and Numerical Experiments

Previous studies of the sediments of Lake Lucerne have shown that massive subaqueous mass movements affecting unconsolidated sediments on lateral slopes are a common process in this lake, and, in view of historical reports describing damaging waves on the lake, it was suggested that tsunamis generated by mass movements represent a considerable natural hazard on the lakeshores. Newly performed numerical simulations combining two-dimensional, depth-averaged models for mass-movement propagation and for tsunami generation, propagation and inunda- tion reproduce a number of reported tsunami effects. Four analysed mass-movement scenarios—three based on documented slope failures involving volumes of 5.5 to 20.8 9 106 m3—show peak wave heights of several metres and maximum runup of 6 to [10 m in the directly affected basins, while effects in neighbouring basins are less drastic. The tsunamis cause large-scale inundation over distances of several hundred metres on flat alluvial plains close to the mass-movement source areas. Basins at the ends of the lake experience regular water-level oscillations with characteristic periods of several minutes. The vulnerability of potentially affected areas has increased dramatically since the times of the damaging historical events, recommending a thorough evaluation of the hazard.

The block numerical range of analytic operator functions

We introduce the block numerical range Wn(L) of an operator function L with respect to a decomposition H = H1⊕. . .⊕Hn of the underlying Hilbert space. Our main results include the spectral inclusion property and estimates of the norm of the resolvent for analytic L . They generalise, and improve, the corresponding results for the numerical range (which is the case n = 1) since the block numerical range is contained in, and may be much smaller than, the usual numerical range. We show that refinements of the decomposition entail inclusions between the corresponding block numerical ranges and that the block numerical range of the operator matrix function L contains those of its principal subminors. For the special case of operator polynomials, we investigate the boundedness of Wn(L) and we prove a Perron-Frobenius type result for the block numerical radius of monic operator polynomials with coefficients that are positive in Hilbert lattice sense.

Hygrical shrinkage stresses in tiling systems: Numerical modeling combined with field studies

To track down potential sites of material failure in the tile–mortar–substrate systems, locations and intensities of stress concentrations owing to drying-induced shrinkage are investigated. For this purpose, mechanical properties were measured on real systems and used as input parameters for numerical modeling of the effect of shrinkage of substrate and/or mortar using the finite element code Abaqus. On the base of different geometrical set-ups we demonstrate that stress concentrations in the mortar can become critical when (i) substantial mortar shrinkage occurs, (ii) substrate shrinkage can accumulate over considerable spatial distances, particularly (iii) in situations where the mortar layer is not separated from the substrate by a flexible waterproofing membrane. Hence material failure in the system tile–mortar–substrate can be prevented (or reduced) by (i) an application of the tiles after the major stages of substrate shrinkage, (ii) the use of elasto-plastic deformable tile adhesives which can react elastically on local stress concentrations, (iii) the implementation of flexible membranes, and (iv) a reduction of the field size by the installation of flexible joints.

A numerical model for inert gas transport in the lung based on fractal airway morphology

Patients suffering from cystic fibrosis (CF) show thick secretions, mucus plugging and bronchiectasis in bronchial and alveolar ducts. This results in substantial structural changes of the airway morphology and heterogeneous ventilation. Disease progression and treatment effects are monitored by so-called gas washout tests, where the change in concentration of an inert gas is measured over a single or multiple breaths. The result of the tests based on the profile of the measured concentration is a marker for the severity of the ventilation inhomogeneity strongly affected by the airway morphology. However, it is hard to localize underlying obstructions to specific parts of the airways, especially if occurring in the lung periphery. In order to support the analysis of lung function tests (e.g. multi-breath washout), we developed a numerical model of the entire airway tree, coupling a lumped parameter model for the lung ventilation with a 4th-order accurate finite difference model of a 1D advection-diffusion equation for the transport of an inert gas. The boundary conditions for the flow problem comprise the pressure and flow profile at the mouth, which is typically known from clinical washout tests. The natural asymmetry of the lung morphology is approximated by a generic, fractal, asymmetric branching scheme which we applied for the conducting airways. A conducting airway ends when its dimension falls below a predefined limit. A model acinus is then connected to each terminal airway. The morphology of an acinus unit comprises a network of expandable cells. A regional, linear constitutive law describes the pressure-volume relation between the pleural gap and the acinus. The cyclic expansion (breathing) of each acinus unit depends on the resistance of the feeding airway and on the flow resistance and stiffness of the cells themselves. Special care was taken in the development of a conservative numerical scheme for the gas transport across bifurcations, handling spatially and temporally varying advective and diffusive fluxes over a wide range of scales. Implicit time integration was applied to account for the numerical stiffness resulting from the discretized transport equation. Local or regional modification of the airway dimension, resistance or tissue stiffness are introduced to mimic pathological airway restrictions typical for CF. This leads to a more heterogeneous ventilation of the model lung. As a result the concentration in some distal parts of the lung model remains increased for a longer duration. The inert gas concentration at the mouth towards the end of the expirations is composed of gas from regions with very different washout efficiency. This results in a steeper slope of the corresponding part of the washout profile.

Numerical corrections to the strong coupling effective Polyakov-line action for finite T Yang-Mills theory

We consider a three-dimensional effective theory of Polyakov lines derived previously from lattice Yang-Mills theory and QCD by means of a resummed strong coupling expansion. The effective theory is useful for investigations of the phase structure, with a sign problem mild enough to allow simulations also at finite density. In this work we present a numerical method to determine improved values for the effective couplings directly from correlators of 4d Yang-Mills theory. For values of the gauge coupling up to the vicinity of the phase transition, the dominant short range effective coupling are well described by their corresponding strong coupling series. We provide numerical results also for the longer range interactions, Polyakov lines in higher representations as well as four-point interactions, and discuss the growing significance of non-local contributions as the lattice gets finer. Within this approach the critical Yang-Mills coupling β c is reproduced to better than one percent from a one-coupling effective theory on N τ = 4 lattices while up to five couplings are needed on N τ = 8 for the same accuracy.

Numerical study of the laser-tip coupling in surface plasmon assisted stacked-double-gate field emitter arrays

Recently, sub-wavelength-pitch stacked double-gate metal nanotip arrays have been proposed to realize high current, high brightness electron bunches for ultrabright cathodes for x-ray free-electron laser applications. With the proposed device structure, ultrafast field emission of photoexcited electrons is efficiently driven by vertical incident near infrared laser pulses, via near field coupling of the surface plasmon polariton resonance of the gate electrodes with the nanotip apex. In this work, in order to gain insight in the underlying physical processes, the authors report detailed numerical studies of the proposed device. The results indicate the importance of the interaction of the double-layer surface plasmon polariton, the position of the nanotip, as well as the incident angle of the near infrared laser pulses.