TY - THES A1 - Samaras, Stefanos T1 - Microphysical retrieval of non-spherical aerosol particles using regularized inversion of multi-wavelength lidar data T1 - Retrieval der Mikrophysik von nichtkugelförmigen Aerosolpartikeln durch regularisierte Inversion von Mehrwellenlängen-Lidardaten N2 - Numerous reports of relatively rapid climate changes over the past century make a clear case of the impact of aerosols and clouds, identified as sources of largest uncertainty in climate projections. Earth’s radiation balance is altered by aerosols depending on their size, morphology and chemical composition. Competing effects in the atmosphere can be further studied by investigating the evolution of aerosol microphysical properties, which are the focus of the present work. The aerosol size distribution, the refractive index, and the single scattering albedo are commonly used such properties linked to aerosol type, and radiative forcing. Highly advanced lidars (light detection and ranging) have reduced aerosol monitoring and optical profiling into a routine process. Lidar data have been widely used to retrieve the size distribution through the inversion of the so-called Lorenz-Mie model (LMM). This model offers a reasonable treatment for spherically approximated particles, it no longer provides, though, a viable description for other naturally occurring arbitrarily shaped particles, such as dust particles. On the other hand, non-spherical geometries as simple as spheroids reproduce certain optical properties with enhanced accuracy. Motivated by this, we adapt the LMM to accommodate the spheroid-particle approximation introducing the notion of a two-dimensional (2D) shape-size distribution. Inverting only a few optical data points to retrieve the shape-size distribution is classified as a non-linear ill-posed problem. A brief mathematical analysis is presented which reveals the inherent tendency towards highly oscillatory solutions, explores the available options for a generalized solution through regularization methods and quantifies the ill-posedness. The latter will improve our understanding on the main cause fomenting instability in the produced solution spaces. The new approach facilitates the exploitation of additional lidar data points from depolarization measurements, associated with particle non-sphericity. However, the generalization of LMM vastly increases the complexity of the problem. The underlying theory for the calculation of the involved optical cross sections (T-matrix theory) is computationally so costly, that would limit a retrieval analysis to an unpractical point. Moreover the discretization of the model equation by a 2D collocation method, proposed in this work, involves double integrations which are further time consuming. We overcome these difficulties by using precalculated databases and a sophisticated retrieval software (SphInX: Spheroidal Inversion eXperiments) especially developed for our purposes, capable of performing multiple-dataset inversions and producing a wide range of microphysical retrieval outputs. Hybrid regularization in conjunction with minimization processes is used as a basis for our algorithms. Synthetic data retrievals are performed simulating various atmospheric scenarios in order to test the efficiency of different regularization methods. The gap in contemporary literature in providing full sets of uncertainties in a wide variety of numerical instances is of major concern here. For this, the most appropriate methods are identified through a thorough analysis on an overall-behavior basis regarding accuracy and stability. The general trend of the initial size distributions is captured in our numerical experiments and the reconstruction quality depends on data error level. Moreover, the need for more or less depolarization points is explored for the first time from the point of view of the microphysical retrieval. Finally, our approach is tested in various measurement cases giving further insight for future algorithm improvements. N2 - Zahlreiche Berichte von relativ schnellen Klimaveränderungen im vergangenen Jahrhundert liefern überzeugende Argumente über die Auswirkungen von Aerosolen und Wolken auf Wetter und Klima. Aerosole und Wolken wurden als Quellen größter Unsicherheit in Klimaprognosen identifiziert. Die Strahlungsbilanz der Erde wird verändert durch die Partikelgröße, ihre Morphologie und ihre chemische Zusammensetzung. Konkurrierende Effekte in der Atmosphäre können durch die Bestimmung von mikrophysikalischen Partikeleigenschaften weiter untersucht werden, was der Fokus der vorliegenden Arbeit ist. Die Aerosolgrößenverteilung, der Brechungsindex der Partikeln und die Einzel-Streu-Albedo sind solche häufig verwendeten Parameter, die mit dem Aerosoltyp und dem Strahlungsantrieb verbunden sind. Hoch entwickelte Lidare (Light Detection and Ranging) haben die Aerosolüberwachung und die optische Profilierung zu einem Routineprozess gemacht. Lidar-Daten wurden verwendet um die Größenverteilung zu bestimmen, was durch die Inversion des sogenannten Lorenz-Mie-Modells (LMM) gelingt. Dieses Modell bietet eine angemessene Behandlung für sphärisch angenäherte Partikeln, es stellt aber keine brauchbare Beschreibung für andere natürlich auftretende beliebig geformte Partikeln -wie z.B. Staubpartikeln- bereit. Andererseits stellt die Einbeziehung einer nicht kugelförmigen Geometrie –wie z.B. einfache Sphäroide- bestimmte optische Eigenschaften mit verbesserter Genauigkeit dar. Angesichts dieser Tatsache erweitern wir das LMM durch die Approximation von Sphäroid-Partikeln. Dazu ist es notwendig den Begriff einer zweidimensionalen Größenverteilung einzuführen. Die Inversion einer sehr geringen Anzahl optischer Datenpunkte zur Bestimmung der Form der Größenverteilung ist als ein nichtlineares schlecht gestelltes Problem bekannt. Eine kurze mathematische Analyse wird vorgestellt, die die inhärente Tendenz zu stark oszillierenden Lösungen zeigt. Weiterhin werden Optionen für eine verallgemeinerte Lösung durch Regularisierungsmethoden untersucht und der Grad der Schlechtgestelltheit quantifiziert. Letzteres wird unser Verständnis für die Hauptursache der Instabilität bei den berechneten Lösungsräumen verbessern. Der neue Ansatz ermöglicht es uns, zusätzliche Lidar-Datenpunkte aus Depolarisationsmessungen zu nutzen, die sich aus der Nicht-sphärizität der Partikeln assoziieren. Die Verallgemeinerung des LMMs erhöht erheblich die Komplexität des Problems. Die zugrundeliegende Theorie für die Berechnung der beteiligten optischen Querschnitte (T-Matrix-Ansatz) ist rechnerisch so aufwendig, dass eine Neuberechnung dieser nicht sinnvoll erscheint. Darüber hinaus wird ein zweidimensionales Kollokationsverfahren für die Diskretisierung der Modellgleichung vorgeschlagen. Dieses Verfahren beinhaltet Doppelintegrationen, die wiederum zeitaufwendig sind. Wir überwinden diese Schwierigkeiten durch Verwendung vorgerechneter Datenbanken sowie einer hochentwickelten Retrieval-Software (SphInX: Spheroidal Inversion eXperiments). Diese Software wurde speziell für unseren Zweck entwickelt und ist in der Lage mehrere Datensatzinversionen gleichzeitig durchzuführen und eine große Auswahl von mikrophysikalischen Retrieval-Ausgaben bereitzustellen. Eine hybride Regularisierung in Verbindung mit einem Minimierungsverfahren wird als Grundlage für unsere Algorithmen verwendet. Synthetische Daten-Inversionen werden mit verschiedenen atmosphärischen Szenarien durchgeführt, um die Effizienz verschiedener Regularisierungsmethoden zu untersuchen. Die Lücke in der gegenwärtigen wissenschaftlichen Literatur gewisse Unsicherheiten durch breitgefächerte numerische Fälle bereitzustellen, ist ein Hauptanliegen dieser Arbeit. Motiviert davon werden die am besten geeigneten Verfahren einer gründlichen Analyse in Bezug auf ihr Gesamtverhalten, d.h. Genauigkeit und Stabilität, unterzogen. Der allgemeine Trend der Anfangsgrößenverteilung wird in unseren numerischen Experimenten erfasst. Zusätzlich hängt die Rekonstruktionsqualität vom Datenfehler ab. Darüber hinaus wird die Anzahl der notwendigen Depolarisationspunkte zum ersten Mal aus der Sicht des mikrophysikalischen Parameter-Retrievals erforscht. Abschließend verwenden wir unsere Software für verschiedene Messfälle, was weitere Einblicke für künftige Verbesserungen des Algorithmus gibt. KW - microphysics KW - retrieval KW - lidar KW - aerosols KW - regularization KW - ill-posed KW - inversion KW - Mikrophysik KW - Retrieval KW - Lidar KW - Aerosole KW - Regularisierung KW - schlecht gestellt KW - Inversion Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-396528 ER - TY - JOUR A1 - Pornsawad, Pornsarp A1 - Böckmann, Christine T1 - Modified Iterative Runge-Kutta-Type Methods for Nonlinear Ill-Posed Problems JF - Numerical functional analysis and optimization : an international journal of rapid publication N2 - This work is devoted to the convergence analysis of a modified Runge-Kutta-type iterative regularization method for solving nonlinear ill-posed problems under a priori and a posteriori stopping rules. The convergence rate results of the proposed method can be obtained under a Holder-type sourcewise condition if the Frechet derivative is properly scaled and locally Lipschitz continuous. Numerical results are achieved by using the Levenberg-Marquardt, Lobatto, and Radau methods. KW - Nonlinear ill-posed problems KW - Runge-Kutta methods KW - regularization methods KW - Holder-type source condition KW - stopping rules Y1 - 2016 U6 - https://doi.org/10.1080/01630563.2016.1219744 SN - 0163-0563 SN - 1532-2467 VL - 37 SP - 1562 EP - 1589 PB - Wiley-VCH CY - Philadelphia ER - TY - THES A1 - Pirhayati, Mohammad T1 - Edge operators and boundary value problems Y1 - 2016 ER - TY - JOUR A1 - Lyu, Xiaojing A1 - Schulze, Bert-Wolfgang T1 - Mellin Operators in the Edge Calculus JF - Complex analysis and operator theory N2 - A manifold M with smooth edge Y is locally near Y modelled on X-Delta x Omega for a cone X-Delta := ( (R) over bar (+) x X)/({0} x X) where Xis a smooth manifold and Omega subset of R-q an open set corresponding to a chart on Y. Compared with pseudo-differential algebras, based on other quantizations of edge-degenerate symbols, we extend the approach with Mellin representations on the r half-axis up to r = infinity, the conical exit of X-boolean AND = R+ x X (sic) (r, x) at infinity. The alternative description of the edge calculus is useful for pseudo-differential structures on manifolds with higher singularities. KW - Edge degenerate operators KW - Mellin and Green operators edge symbols Y1 - 2016 U6 - https://doi.org/10.1007/s11785-015-0511-6 SN - 1661-8254 SN - 1661-8262 VL - 10 SP - 965 EP - 1000 PB - Springer CY - Basel ER - TY - THES A1 - Lyu, Xiaojing T1 - Operators on singular manifolds T1 - Operatoren auf singuläre Mannigfaltigkeiten N2 - We study the interplay between analysis on manifolds with singularities and complex analysis and develop new structures of operators based on the Mellin transform and tools for iterating the calculus for higher singularities. We refer to the idea of interpreting boundary value problems (BVPs) in terms of pseudo-differential operators with a principal symbolic hierarchy, taking into account that BVPs are a source of cone and edge operator algebras. The respective cone and edge pseudo-differential algebras in turn are the starting point of higher corner theories. In addition there are deep relationships between corner operators and complex analysis. This will be illustrated by the Mellin symbolic calculus. N2 - Wir studieren den Zusammenhang zwischen Analysis auf Mannigfaltigkeiten mit Singularitäten und komplexer Analysis und entwickeln neue Strukturen von Operatoren basierend auf der Mellin-Transformation und Hilfsmitteln für die Iteration des Kalküls für höhere Singularitäten. Wir beziehen uns auf die Idee von der Interpretation von Randwert-Problemen (BVPs) durch Pseudo-Differential-operatoren und Hauptsymbol-Hierarchien, unter Berüksichtigung der Tatsache, dass BVPs eine Quelle von Konus- und Kanten-Operator- algebren sind. Die betreffenden Konus- und Kanten-Pseudo-differentiellen Algebren sind wiederum der Startpunkt von höheren Eckentheorien. Zusätzlich bestehen tiefe Beziehungen zwischen Ecken-Operatoren und komplexer Analysis. Dies wird illustiert durch den Mellin-Symbol Kalkül. KW - order filtration KW - Mellin-Symbols KW - singular manifolds KW - Ordnungs-Filtrierung KW - Mellin-Symbole KW - singuläre Mannigfaltigkeiten Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-103643 ER - TY - JOUR A1 - Hedayat Mahmoudi, Mahdi A1 - Schulze, Bert-Wolfgang T1 - Corner boundary value problems JF - Asian-European journal of mathematics N2 - The paper develops some crucial steps in extending the first-order cone or edge calculus to higher singularity orders. We focus here on order 2, but the ideas are motivated by an iterative approach for higher singularities. KW - Mellin operators KW - Mellin oscillatory integrals KW - exit calculus KW - weighted Sobolev spaces Y1 - 2016 U6 - https://doi.org/10.1142/S1793557117500541 SN - 1793-5571 SN - 1793-7183 VL - 10 IS - 1 PB - World Scientific CY - Singapore ER - TY - JOUR A1 - Flad, H. -J. A1 - Harutyunyan, Gohar A1 - Schulze, Bert-Wolfgang T1 - Asymptotic parametrices of elliptic edge operators JF - Journal of pseudo-differential operators and applications N2 - We study operators on singular manifolds, here of conical or edge type, and develop a new general approach of representing asymptotics of solutions to elliptic equations close to the singularities. We introduce asymptotic parametrices, using tools from cone and edge pseudo-differential algebras. Our structures are motivated by models of many-particle physics with singular Coulomb potentials that contribute higher order singularities in Euclidean space, determined by the number of particles. KW - Cone and edge pseudo-differential operators KW - Ellipticity of edge-degenerate operators KW - Meromorphic operator-valued symbols KW - Asymptotics of solutions Y1 - 2016 U6 - https://doi.org/10.1007/s11868-016-0159-7 SN - 1662-9981 SN - 1662-999X VL - 7 SP - 321 EP - 363 PB - Springer CY - Basel ER - TY - JOUR A1 - Chang, D. -C. A1 - Viahmoudi, M. Hedayat A1 - Schulze, Bert-Wolfgang T1 - PSEUDO-DIFFERENTIAL ANALYSIS WITH TWISTED SYMBOLIC STRUCTURE JF - Journal of nonlinear and convex analysis : an international journal N2 - This paper is devoted to pseudo-differential operators and new applications. We establish necessary extensions of the standard calculus to specific classes of operator-valued symbols occurring in principal symbolic hierarchies of operators on manifolds with singularities or stratified spaces. KW - Pseudo-differential operators KW - boundary value problems KW - operator valued symbols KW - Fourier transform Y1 - 2016 SN - 1345-4773 SN - 1880-5221 VL - 17 SP - 1889 EP - 1937 PB - Yokohama Publishers CY - Yokohama ER - TY - JOUR A1 - Bomanson, Jori A1 - Janhunen, Tomi A1 - Schaub, Torsten A1 - Gebser, Martin A1 - Kaufmann, Benjamin T1 - Answer Set Programming Modulo Acyclicity JF - Fundamenta informaticae N2 - Acyclicity constraints are prevalent in knowledge representation and applications where acyclic data structures such as DAGs and trees play a role. Recently, such constraints have been considered in the satisfiability modulo theories (SMT) framework, and in this paper we carry out an analogous extension to the answer set programming (ASP) paradigm. The resulting formalism, ASP modulo acyclicity, offers a rich set of primitives to express constraints related to recursive structures. In the technical results of the paper, we relate the new generalization with standard ASP by showing (i) how acyclicity extensions translate into normal rules, (ii) how weight constraint programs can be instrumented by acyclicity extensions to capture stability in analogy to unfounded set checking, and (iii) how the gap between supported and stable models is effectively closed in the presence of such an extension. Moreover, we present an efficient implementation of acyclicity constraints by incorporating a respective propagator into the state-of-the-art ASP solver CLASP. The implementation provides a unique combination of traditional unfounded set checking with acyclicity propagation. In the experimental part, we evaluate the interplay of these orthogonal checks by equipping logic programs with supplementary acyclicity constraints. The performance results show that native support for acyclicity constraints is a worthwhile addition, furnishing a complementary modeling construct in ASP itself as well as effective means for translation-based ASP solving. Y1 - 2016 U6 - https://doi.org/10.3233/FI-2016-1398 SN - 0169-2968 SN - 1875-8681 VL - 147 SP - 63 EP - 91 PB - IOS Press CY - Amsterdam ER - TY - JOUR A1 - Kretschmer, Marlene A1 - Coumou, Dim A1 - Donges, Jonathan A1 - Runge, Jakob T1 - Using Causal Effect Networks to Analyze Different Arctic Drivers of Midlatitude Winter Circulation JF - Journal of climate N2 - In recent years, the Northern Hemisphere midlatitudes have suffered from severe winters like the extreme 2012/13 winter in the eastern United States. These cold spells were linked to a meandering upper-tropospheric jet stream pattern and a negative Arctic Oscillation index (AO). However, the nature of the drivers behind these circulation patterns remains controversial. Various studies have proposed different mechanisms related to changes in the Arctic, most of them related to a reduction in sea ice concentrations or increasing Eurasian snow cover. Here, a novel type of time series analysis, called causal effect networks (CEN), based on graphical models is introduced to assess causal relationships and their time delays between different processes. The effect of different Arctic actors on winter circulation on weekly to monthly time scales is studied, and robust network patterns are found. Barents and Kara sea ice concentrations are detected to be important external drivers of the midlatitude circulation, influencing winter AO via tropospheric mechanisms and through processes involving the stratosphere. Eurasia snow cover is also detected to have a causal effect on sea level pressure in Asia, but its exact role on AO remains unclear. The CEN approach presented in this study overcomes some difficulties in interpreting correlation analyses, complements model experiments for testing hypotheses involving teleconnections, and can be used to assess their validity. The findings confirm that sea ice concentrations in autumn in the Barents and Kara Seas are an important driver of winter circulation in the midlatitudes. Y1 - 2016 U6 - https://doi.org/10.1175/JCLI-D-15-0654.1 SN - 0894-8755 SN - 1520-0442 VL - 29 SP - 4069 EP - 4081 PB - American Meteorological Soc. CY - Boston ER -