The factorization method for real elliptic problems

The Factorization Method localizes inclusions inside a body from measurements on its surface. Without a priori knowing the physical parameters inside the inclusions, the points belonging to them can be characterized using the range of an auxiliary operator. The method relies on a range characterization that relates the range of the auxiliary operator to the measurements and is only known for very particular applications. In this work we develop a general framework for the method by considering symmetric and coercive operators between abstract Hilbert spaces. We show that the important range characterization holds if the difference between the inclusions and the background medium satisfies a coerciveness condition which can immediately be translated into a condition on the coefficients of a given real elliptic problem. We demonstrate how several known applications of the Factorization Method are covered by our general results and deduce the range characterization for a new example in linear elasticity.

Laboratory study on the optical properties of absorbing atmospheric aerosols and field applications

The interaction between aerosols and sun light plays an important role in the radiative balance of Earth’s atmosphere. This interaction is obtained by measuring the removal (extinction), redistribution (scattering), and transformation into heat (absorption) of light by the aerosols; i.e. their optical properties. Knowledge of these properties is crucial for our understanding of the atmospheric system. rn Light absorption by aerosols is a major contributor to the direct and indirect effects on our climate system, and an accurate and sensitive measurement method is crucial to further our understanding. A homebuilt photoacoustic sensor (PAS), measuring at a 532nm wavelength, was fully characterized and its functionality validated for measurements of absorbing aerosols. The optical absorption cross-sections of absorbing polystyrene latex spheres, to be used as a standard for aerosol absorption measurements, were measured and compared to literature values. Additionally, a calibration method using absorbing aerosol of known complex refractive index was presented.rn A new approach to retrieve the effective broadband refractive indices (mbroad,eff) of aerosol particles by a white light aerosol spectrometer (WELAS) optical particle counter (OPC) was achieved. Using a tandem differential mobility analyzer (DMA)-OPC system, the nbroad,eff are obtained for both laboratory and field applications. This method was tested in the laboratory using substances with a wide range of optical properties and it was used in ambient measurements to retrieve the nbroad,eff of biomass burning aerosols in a nationwide burning event in Israel. The retrieved effective broadband refractive indices for laboratory generated scattering aerosols were: ammonium sulfate (AS), glutaric acid (GA), and sodium chloride, all within 4% of literature values. For absorbing substances, nigrosine and various mixtures of nigrosine with AS and GA were measured, as well as a lightly absorbing substance, Suwannee river fulvic acid (SRFA). For the ambient measurements, the calibration curves generated from this method were to follow the optical evolution of biomass burning (BB) aerosols. A decrease in the overall aerosol absorption and scattering for aged aerosols during the day after the fires compared to the smoldering phase of the fires was found. rn The connection between light extinction of aerosols, their chemical composition and hygroscopicity for particles with different degrees of absorption was studied. The extinction cross-section (σext) at 532nm for different mobility diameters was measured at 80% and 90% relative humidity (RH), and at an RH<10%. The ratio of the humidified aerosols to the dry ones, fRHext(%RH,Dry), is presented. For purely scattering aerosols, fRHext(%RH,Dry) is inversely proportional with size; this dependence was suppressed for lightly absorbing ones. In addition, the validity of the mixing rules for water soluble absorbing aerosols is explored. The difference between the derived and calculated real parts of the complex RIs were less than 5.3% for all substances, wavelengths, and RHs. The obtained imaginary parts for the retrieved and calculated RIs were in good agreement with each other, and well within the measurement errors of retrieval from pulsed CRD spectroscopy measurements. Finally, a core-shell structure model is also used to explore the differences between the models, for substances with low growth factors, under these hydration conditions. It was found that at 80% RH and for size parameters less than 2.5, there is less than a 5 % difference between the extinction efficiencies calculated with both models. This difference is within measurement errors; hence, there is no significant difference between the models in this case. However, for greater size parameters the difference can be up to 10%. For 90% RH the differences below a size parameter of 2.5 were up to 7%.rn Finally, the fully characterized PAS together with a cavity ring down spectrometer (CRD), were used to study the optical properties of soot and secondary organic aerosol (SOA) during the SOOT-11 project in the AIDA chamber in Karlsruhe, Germany. The fresh fractal-like soot particles were allowed to coagulate for 28 hours before stepwise coating them with SOA. The single scattering albedo for fresh fractal-like soot was measured to be 0.2 (±0.03), and after allowing the soot to coagulate for 28 hours and coating it with SOA, it increased to 0.71(±0.01). An absorption enhancement of the coated soot of up to 1.71 (±0.03) times from the non-coated coagulated soot was directly measured with the PAS. Monodisperse measurements of SOA and soot coated with SOA were performed to derive the complex refractive index (m) of both aerosols. A complex refractive index of m = 1.471(±0.008) + i0.0(±0.002) for the SOA-αO3 was retrieved. For the compact coagulated soot a preliminary complex refractive index of m = 2.04(+0.21/-0.14) + i0.34(+0.18/-0.06) with 10nm(+4/-6) coating thickness was retrieved.rn These detail properties can be use by modelers to decrease uncertainties in assessing climatic impacts of the different species and to improve weather forecasting.rn

Optimized scheduling in real-time environments with column generation

Im Bereich sicherheitsrelevanter eingebetteter Systeme stellt sich der Designprozess von Anwendungen als sehr komplex dar. Entsprechend einer gegebenen Hardwarearchitektur lassen sich Steuergeräte aufrüsten, um alle bestehenden Prozesse und Signale pünktlich auszuführen. Die zeitlichen Anforderungen sind strikt und müssen in jeder periodischen Wiederkehr der Prozesse erfüllt sein, da die Sicherstellung der parallelen Ausführung von größter Bedeutung ist. Existierende Ansätze können schnell Designalternativen berechnen, aber sie gewährleisten nicht, dass die Kosten für die nötigen Hardwareänderungen minimal sind. Wir stellen einen Ansatz vor, der kostenminimale Lösungen für das Problem berechnet, die alle zeitlichen Bedingungen erfüllen. Unser Algorithmus verwendet Lineare Programmierung mit Spaltengenerierung, eingebettet in eine Baumstruktur, um untere und obere Schranken während des Optimierungsprozesses bereitzustellen. Die komplexen Randbedingungen zur Gewährleistung der periodischen Ausführung verlagern sich durch eine Zerlegung des Hauptproblems in unabhängige Unterprobleme, die als ganzzahlige lineare Programme formuliert sind. Sowohl die Analysen zur Prozessausführung als auch die Methoden zur Signalübertragung werden untersucht und linearisierte Darstellungen angegeben. Des Weiteren präsentieren wir eine neue Formulierung für die Ausführung mit fixierten Prioritäten, die zusätzlich Prozessantwortzeiten im schlimmsten anzunehmenden Fall berechnet, welche für Szenarien nötig sind, in denen zeitliche Bedingungen an Teilmengen von Prozessen und Signalen gegeben sind. Wir weisen die Anwendbarkeit unserer Methoden durch die Analyse von Instanzen nach, welche Prozessstrukturen aus realen Anwendungen enthalten. Unsere Ergebnisse zeigen, dass untere Schranken schnell berechnet werden können, um die Optimalität von heuristischen Lösungen zu beweisen. Wenn wir optimale Lösungen mit Antwortzeiten liefern, stellt sich unsere neue Formulierung in der Laufzeitanalyse vorteilhaft gegenüber anderen Ansätzen dar. Die besten Resultate werden mit einem hybriden Ansatz erzielt, der heuristische Startlösungen, eine Vorverarbeitung und eine heuristische mit einer kurzen nachfolgenden exakten Berechnungsphase verbindet.