## Fachbereich Mathematik

### Filtern

#### Erscheinungsjahr

#### Dokumenttyp

- Dissertation (226) (entfernen)

#### Schlagworte

- Algebraische Geometrie (6)
- Finanzmathematik (5)
- Optimization (5)
- Portfolio Selection (5)
- Stochastische dynamische Optimierung (5)
- Navier-Stokes-Gleichung (4)
- Numerische Mathematik (4)
- Portfolio-Optimierung (4)
- portfolio optimization (4)
- Computeralgebra (3)
- Elastizität (3)
- Erwarteter Nutzen (3)
- Finite-Volumen-Methode (3)
- Gröbner-Basis (3)
- Homogenisierung <Mathematik> (3)
- Inverses Problem (3)
- Numerische Strömungssimulation (3)
- Optionspreistheorie (3)
- Portfoliomanagement (3)
- Transaction Costs (3)
- Tropische Geometrie (3)
- Wavelet (3)
- optimales Investment (3)
- Asymptotic Expansion (2)
- Asymptotik (2)
- Bewertung (2)
- Derivat <Wertpapier> (2)
- Elasticity (2)
- Endliche Geometrie (2)
- Erdmagnetismus (2)
- Filtergesetz (2)
- Filtration (2)
- Finite Pointset Method (2)
- Geometric Ergodicity (2)
- Hamilton-Jacobi-Differentialgleichung (2)
- Hochskalieren (2)
- IMRT (2)
- Kreditrisiko (2)
- Level-Set-Methode (2)
- Lineare Elastizitätstheorie (2)
- Local smoothing (2)
- Mehrskalenanalyse (2)
- Mehrskalenmodell (2)
- Modulraum (2)
- Partial Differential Equations (2)
- Partielle Differentialgleichung (2)
- Portfolio Optimization (2)
- Poröser Stoff (2)
- Regularisierung (2)
- Schnitttheorie (2)
- Stochastic Control (2)
- Stochastische Differentialgleichung (2)
- Transaktionskosten (2)
- Upscaling (2)
- Vektorwavelets (2)
- White Noise Analysis (2)
- curve singularity (2)
- domain decomposition (2)
- duality (2)
- finite volume method (2)
- geomagnetism (2)
- homogenization (2)
- illiquidity (2)
- interface problem (2)
- isogeometric analysis (2)
- mesh generation (2)
- optimal investment (2)
- splines (2)
- "Slender-Body"-Theorie (1)
- 3D image analysis (1)
- A-infinity-bimodule (1)
- A-infinity-category (1)
- A-infinity-functor (1)
- Ableitungsfreie Optimierung (1)
- Advanced Encryption Standard (1)
- Algebraic dependence of commuting elements (1)
- Algebraic geometry (1)
- Algebraische Abhängigkeit der kommutierende Elementen (1)
- Algebraischer Funktionenkörper (1)
- Analysis (1)
- Annulus (1)
- Anti-diffusion (1)
- Antidiffusion (1)
- Approximationsalgorithmus (1)
- Arbitrage (1)
- Arc distance (1)
- Archimedische Kopula (1)
- Asiatische Option (1)
- Asympotic Analysis (1)
- Asymptotic Analysis (1)
- Asymptotische Entwicklung (1)
- Ausfallrisiko (1)
- Automorphismengruppe (1)
- Autoregressive Hilbertian model (1)
- B-Spline (1)
- Barriers (1)
- Basket Option (1)
- Bayes-Entscheidungstheorie (1)
- Beam models (1)
- Beam orientation (1)
- Beschichtungsprozess (1)
- Beschränkte Krümmung (1)
- Betrachtung des Schlimmstmöglichen Falles (1)
- Bildsegmentierung (1)
- Binomialbaum (1)
- Biorthogonalisation (1)
- Biot Poroelastizitätgleichung (1)
- Biot-Savart Operator (1)
- Biot-Savart operator (1)
- Boltzmann Equation (1)
- Bondindizes (1)
- Bootstrap (1)
- Boundary Value Problem / Oblique Derivative (1)
- Brinkman (1)
- Brownian Diffusion (1)
- Brownian motion (1)
- Brownsche Bewegung (1)
- CDO (1)
- CDS (1)
- CDSwaption (1)
- CFD (1)
- CHAMP (1)
- CPDO (1)
- Castelnuovo Funktion (1)
- Castelnuovo function (1)
- Cauchy-Navier-Equation (1)
- Cauchy-Navier-Gleichung (1)
- Censoring (1)
- Center Location (1)
- Change Point Analysis (1)
- Change Point Test (1)
- Change-point Analysis (1)
- Change-point estimator (1)
- Change-point test (1)
- Charakter <Gruppentheorie> (1)
- Chi-Quadrat-Test (1)
- Cholesky-Verfahren (1)
- Chow Quotient (1)
- Circle Location (1)
- Coarse graining (1)
- Cohen-Lenstra heuristic (1)
- Combinatorial Optimization (1)
- Commodity Index (1)
- Computer Algebra (1)
- Computer Algebra System (1)
- Computer algebra (1)
- Computeralgebra System (1)
- Conditional Value-at-Risk (1)
- Consistencyanalysis (1)
- Consistent Price Processes (1)
- Construction of hypersurfaces (1)
- Copula (1)
- Coupled PDEs (1)
- Crash (1)
- Crash Hedging (1)
- Crash modelling (1)
- Crashmodellierung (1)
- Credit Default Swap (1)
- Credit Risk (1)
- Curvature (1)
- Curved viscous fibers (1)
- DSMC (1)
- Darstellungstheorie (1)
- Das Urbild von Ideal unter einen Morphismus der Algebren (1)
- Debt Management (1)
- Defaultable Options (1)
- Deformationstheorie (1)
- Delaunay (1)
- Delaunay triangulation (1)
- Delaunay triangulierung (1)
- Differenzenverfahren (1)
- Differenzmenge (1)
- Diffusion (1)
- Diffusion processes (1)
- Diffusionsprozess (1)
- Discriminatory power (1)
- Diskrete Fourier-Transformation (1)
- Dispersionsrelation (1)
- Dissertation (1)
- Druckkorrektur (1)
- Dünnfilmapproximation (1)
- EM algorithm (1)
- Edwards Model (1)
- Effective Conductivity (1)
- Efficiency (1)
- Effizienter Algorithmus (1)
- Effizienz (1)
- Eikonal equation (1)
- Elastische Deformation (1)
- Elastoplastizität (1)
- Elektromagnetische Streuung (1)
- Eliminationsverfahren (1)
- Elliptische Verteilung (1)
- Elliptisches Randwertproblem (1)
- Endliche Gruppe (1)
- Endliche Lie-Gruppe (1)
- Entscheidungsbaum (1)
- Entscheidungsunterstützung (1)
- Enumerative Geometrie (1)
- Erdöl Prospektierung (1)
- Erwartungswert-Varianz-Ansatz (1)
- Expected shortfall (1)
- Exponential Utility (1)
- Exponentieller Nutzen (1)
- Extrapolation (1)
- Extreme Events (1)
- Extreme value theory (1)
- FEM (1)
- FFT (1)
- FPM (1)
- Faden (1)
- Fatigue (1)
- Feedfoward Neural Networks (1)
- Feynman Integrals (1)
- Feynman path integrals (1)
- Fiber suspension flow (1)
- Financial Engineering (1)
- Finanzkrise (1)
- Finanznumerik (1)
- Finite-Elemente-Methode (1)
- Finite-Punktmengen-Methode (1)
- Firmwertmodell (1)
- First Order Optimality System (1)
- Flachwasser (1)
- Flachwassergleichungen (1)
- Fluid dynamics (1)
- Fluid-Feststoff-Strömung (1)
- Fluid-Struktur-Wechselwirkung (1)
- Foam decay (1)
- Fokker-Planck-Gleichung (1)
- Forward-Backward Stochastic Differential Equation (1)
- Fourier-Transformation (1)
- Fredholmsche Integralgleichung (1)
- Functional autoregression (1)
- Functional time series (1)
- Funktionenkörper (1)
- GARCH (1)
- GARCH Modelle (1)
- Galerkin-Methode (1)
- Gamma-Konvergenz (1)
- Garbentheorie (1)
- Gebietszerlegung (1)
- Gebietszerlegungsmethode (1)
- Gebogener viskoser Faden (1)
- Geodesie (1)
- Geometrische Ergodizität (1)
- Gewichteter Sobolev-Raum (1)
- Gittererzeugung (1)
- Gleichgewichtsstrategien (1)
- Granular flow (1)
- Granulat (1)
- Gravitationsfeld (1)
- Gromov Witten (1)
- Gromov-Witten-Invariante (1)
- Große Abweichung (1)
- Gruppenoperation (1)
- Gruppentheorie (1)
- Gröbner bases (1)
- Gröbner-basis (1)
- Gyroscopic (1)
- Hadamard manifold (1)
- Hadamard space (1)
- Hadamard-Mannigfaltigkeit (1)
- Hadamard-Raum (1)
- Hamiltonian Path Integrals (1)
- Handelsstrategien (1)
- Harmonische Analyse (1)
- Harmonische Spline-Funktion (1)
- Hazard Functions (1)
- Heavy-tailed Verteilung (1)
- Hedging (1)
- Helmholtz Type Boundary Value Problems (1)
- Heston-Modell (1)
- Hidden Markov models for Financial Time Series (1)
- Hierarchische Matrix (1)
- Homogenization (1)
- Homologische Algebra (1)
- Hub Location Problem (1)
- Hydrostatischer Druck (1)
- Hyperelliptische Kurve (1)
- Hyperflächensingularität (1)
- Hyperspektraler Sensor (1)
- Hysterese (1)
- ITSM (1)
- Idealklassengruppe (1)
- Illiquidität (1)
- Image restoration (1)
- Immiscible lattice BGK (1)
- Immobilienaktie (1)
- Inflation (1)
- Infrarotspektroskopie (1)
- Intensität (1)
- Internationale Diversifikation (1)
- Inverse Problem (1)
- Irreduzibler Charakter (1)
- Isogeometrische Analyse (1)
- Ito (1)
- Jacobigruppe (1)
- Kanalcodierung (1)
- Karhunen-Loève expansion (1)
- Kategorientheorie (1)
- Kelvin Transformation (1)
- Kirchhoff-Love shell (1)
- Kiyoshi (1)
- Kombinatorik (1)
- Kommutative Algebra (1)
- Konjugierte Dualität (1)
- Konstruktion von Hyperflächen (1)
- Kontinuum <Mathematik> (1)
- Kontinuumsphysik (1)
- Konvergenz (1)
- Konvergenzrate (1)
- Konvergenzverhalten (1)
- Konvexe Optimierung (1)
- Kopplungsmethoden (1)
- Kopplungsproblem (1)
- Kopula <Mathematik> (1)
- Kreitderivaten (1)
- Kryptoanalyse (1)
- Kryptologie (1)
- Krümmung (1)
- Kullback-Leibler divergence (1)
- Kurvenschar (1)
- LIBOR (1)
- Lagrangian relaxation (1)
- Laplace transform (1)
- Lattice Boltzmann (1)
- Lattice-BGK (1)
- Lattice-Boltzmann (1)
- Leading-Order Optimality (1)
- Level set methods (1)
- Lie-Typ-Gruppe (1)
- Lineare partielle Differentialgleichung (1)
- Lippmann-Schwinger equation (1)
- Liquidität (1)
- Locally Supported Zonal Kernels (1)
- Location (1)
- MBS (1)
- MKS (1)
- Macaulay’s inverse system (1)
- Magnetoelastic coupling (1)
- Magnetoelasticity (1)
- Magnetostriction (1)
- Marangoni-Effekt (1)
- Markov Chain (1)
- Markov Kette (1)
- Markov-Ketten-Monte-Carlo-Verfahren (1)
- Markov-Prozess (1)
- Marktmanipulation (1)
- Marktrisiko (1)
- Martingaloptimalitätsprinzip (1)
- Mathematical Finance (1)
- Mathematik (1)
- Mathematisches Modell (1)
- Matrixkompression (1)
- Matrizenfaktorisierung (1)
- Matrizenzerlegung (1)
- Maximal Cohen-Macaulay modules (1)
- Maximale Cohen-Macaulay Moduln (1)
- Maximum Likelihood Estimation (1)
- Maximum-Likelihood-Schätzung (1)
- Maxwell's equations (1)
- McKay-Conjecture (1)
- McKay-Vermutung (1)
- Mehrdimensionale Bildverarbeitung (1)
- Mehrdimensionales Variationsproblem (1)
- Mehrkriterielle Optimierung (1)
- Mehrskalen (1)
- Mie- and Helmholtz-Representation (1)
- Mie- und Helmholtz-Darstellung (1)
- Mikroelektronik (1)
- Mikrostruktur (1)
- Mixed integer programming (1)
- Modellbildung (1)
- Molekulardynamik (1)
- Momentum and Mas Transfer (1)
- Monte Carlo (1)
- Monte-Carlo-Simulation (1)
- Moreau-Yosida regularization (1)
- Morphismus (1)
- Mosco convergence (1)
- Multi Primary and One Second Particle Method (1)
- Multi-Asset Option (1)
- Multicriteria optimization (1)
- Multileaf collimator (1)
- Multiperiod planning (1)
- Multiphase Flows (1)
- Multiresolution Analysis (1)
- Multiscale modelling (1)
- Multiskalen-Entrauschen (1)
- Multispektralaufnahme (1)
- Multispektralfotografie (1)
- Multivariate Analyse (1)
- Multivariate Wahrscheinlichkeitsverteilung (1)
- Multivariates Verfahren (1)
- NURBS (1)
- Networks (1)
- Netzwerksynthese (1)
- Neural Networks (1)
- Neuronales Netz (1)
- Nicht-Desarguessche Ebene (1)
- Nichtglatte Optimierung (1)
- Nichtkommutative Algebra (1)
- Nichtkonvexe Optimierung (1)
- Nichtkonvexes Variationsproblem (1)
- Nichtlineare Approximation (1)
- Nichtlineare Diffusion (1)
- Nichtlineare Optimierung (1)
- Nichtlineare Zeitreihenanalyse (1)
- Nichtlineare partielle Differentialgleichung (1)
- Nichtpositive Krümmung (1)
- Niederschlag (1)
- No-Arbitrage (1)
- Non-commutative Computer Algebra (1)
- Nonlinear Optimization (1)
- Nonlinear time series analysis (1)
- Nonparametric time series (1)
- Nulldimensionale Schemata (1)
- Numerical Flow Simulation (1)
- Numerical methods (1)
- Numerische Mathematik / Algorithmus (1)
- Numerisches Verfahren (1)
- Oberflächenmaße (1)
- Oberflächenspannung (1)
- Optimal Control (1)
- Optimale Kontrolle (1)
- Optimale Portfolios (1)
- Optimierung (1)
- Optimization Algorithms (1)
- Option (1)
- Option Valuation (1)
- Optionsbewertung (1)
- Order (1)
- Ovoid (1)
- Gedruckte Kopie bestellen (1)
- Papiermaschine (1)
- Parallel Algorithms (1)
- Paralleler Algorithmus (1)
- Partikel Methoden (1)
- Patchworking Methode (1)
- Patchworking method (1)
- Pathwise Optimality (1)
- Pedestrian FLow (1)
- Pfadintegral (1)
- Planares Polynom (1)
- Poisson noise (1)
- Poisson-Gleichung (1)
- PolyBoRi (1)
- Population Balance Equation (1)
- Portfolio Optimierung (1)
- Portfoliooptimierung (1)
- Preimage of an ideal under a morphism of algebras (1)
- Projektionsoperator (1)
- Projektive Fläche (1)
- Prox-Regularisierung (1)
- Punktprozess (1)
- QMC (1)
- QVIs (1)
- Quadratischer Raum (1)
- Quantile autoregression (1)
- Quasi-Variational Inequalities (1)
- RKHS (1)
- Radial Basis Functions (1)
- Radiotherapy (1)
- Randwertproblem (1)
- Randwertproblem / Schiefe Ableitung (1)
- Rank test (1)
- Rarefied gas (1)
- Reflexionsspektroskopie (1)
- Regime Shifts (1)
- Regime-Shift Modell (1)
- Regressionsanalyse (1)
- Regularisierung / Stoppkriterium (1)
- Regularization / Stop criterion (1)
- Regularization methods (1)
- Reliability (1)
- Restricted Regions (1)
- Riemannian manifolds (1)
- Riemannsche Mannigfaltigkeiten (1)
- Rigid Body Motion (1)
- Risikomanagement (1)
- Risikomaße (1)
- Risikotheorie (1)
- Risk Measures (1)
- Robust smoothing (1)
- Rohstoffhandel (1)
- Rohstoffindex (1)
- Räumliche Statistik (1)
- SWARM (1)
- Scale function (1)
- Schaum (1)
- Schaumzerfall (1)
- Schiefe Ableitung (1)
- Schwache Formulierung (1)
- Schwache Konvergenz (1)
- Schwache Lösu (1)
- Second Order Conditions (1)
- Semi-Markov-Kette (1)
- Sequenzieller Algorithmus (1)
- Serre functor (1)
- Shallow Water Equations (1)
- Shape optimization, gradient based optimization, adjoint method (1)
- Singular <Programm> (1)
- Singularity theory (1)
- Singularität (1)
- Singularitätentheorie (1)
- Slender body theory (1)
- Sobolev spaces (1)
- Sobolev-Raum (1)
- Spannungs-Dehn (1)
- Spatial Statistics (1)
- Spectral theory (1)
- Spektralanalyse <Stochastik> (1)
- Spherical Fast Wavelet Transform (1)
- Spherical Location Problem (1)
- Sphärische Approximation (1)
- Spline-Approximation (1)
- Split Operator (1)
- Splitoperator (1)
- Sprung-Diffusions-Prozesse (1)
- Stabile Vektorbundle (1)
- Stable vector bundles (1)
- Standard basis (1)
- Standortprobleme (1)
- Steuer (1)
- Stochastic Impulse Control (1)
- Stochastic Processes (1)
- Stochastische Inhomogenitäten (1)
- Stochastische Processe (1)
- Stochastische Zinsen (1)
- Stochastische optimale Kontrolle (1)
- Stochastischer Prozess (1)
- Stokes-Gleichung (1)
- Stop- und Spieloperator (1)
- Stoßdämpfer (1)
- Strahlentherapie (1)
- Strahlungstransport (1)
- Strukturiertes Finanzprodukt (1)
- Strukturoptimierung (1)
- Strömungsdynamik (1)
- Strömungsmechanik (1)
- Success Run (1)
- Survival Analysis (1)
- Systemidentifikation (1)
- Sägezahneffekt (1)
- Tail Dependence Koeffizient (1)
- Test for Changepoint (1)
- Thermophoresis (1)
- Thin film approximation (1)
- Tichonov-Regularisierung (1)
- Time Series (1)
- Time-Series (1)
- Time-delay-Netz (1)
- Topologieoptimierung (1)
- Topology optimization (1)
- Traffic flow (1)
- Transaction costs (1)
- Trennschärfe <Statistik> (1)
- Tropical Grassmannian (1)
- Tropical Intersection Theory (1)
- Tube Drawing (1)
- Two-phase flow (1)
- Unreinheitsfunktion (1)
- Untermannigfaltigkeit (1)
- Upwind-Verfahren (1)
- Utility (1)
- Value at Risk (1)
- Value-at-Risk (1)
- Variationsrechnung (1)
- Vectorfield approximation (1)
- Vektorfeldapproximation (1)
- Vektorkugelfunktionen (1)
- Verschwindungsatz (1)
- Viskoelastische Flüssigkeiten (1)
- Viskose Transportschemata (1)
- Volatilität (1)
- Volatilitätsarbitrage (1)
- Vorkonditionierer (1)
- Vorwärts-Rückwärts-Stochastische-Differentialgleichung (1)
- Wave Based Method (1)
- Wavelet-Theorie (1)
- Wavelet-Theory (1)
- Weißes Rauschen (1)
- White Noise (1)
- Wirbelabtrennung (1)
- Wirbelströmung (1)
- Worst-Case (1)
- Wärmeleitfähigkeit (1)
- Yaglom limits (1)
- Zeitintegrale Modelle (1)
- Zeitreihe (1)
- Zentrenprobleme (1)
- Zero-dimensional schemes (1)
- Zopfgruppe (1)
- Zufälliges Feld (1)
- Zweiphasenströmung (1)
- abgeleitete Kategorie (1)
- algebraic attack (1)
- algebraic correspondence (1)
- algebraic function fields (1)
- algebraic geometry (1)
- algebraic number fields (1)
- algebraic topology (1)
- algebraische Korrespondenzen (1)
- algebraische Topologie (1)
- algebroid curve (1)
- alternating minimization (1)
- alternating optimization (1)
- analoge Mikroelektronik (1)
- angewandte Mathematik (1)
- angewandte Topologie (1)
- anisotropen Viskositätsmodell (1)
- anisotropic viscosity (1)
- applied mathematics (1)
- archimedean copula (1)
- asian option (1)
- basket option (1)
- benders decomposition (1)
- bending strip method (1)
- binomial tree (1)
- blackout period (1)
- bocses (1)
- boundary value problem (1)
- canonical ideal (1)
- canonical module (1)
- changing market coefficients (1)
- closure approximation (1)
- combinatorics (1)
- composites (1)
- computational finance (1)
- computer algebra (1)
- computeralgebra (1)
- convergence behaviour (1)
- convex constraints (1)
- convex optimization (1)
- correlated errors (1)
- coupling methods (1)
- crash (1)
- crash hedging (1)
- credit risk (1)
- curvature (1)
- decision support (1)
- decision support systems (1)
- decoding (1)
- default time (1)
- degenerations of an elliptic curve (1)
- dense univariate rational interpolation (1)
- derived category (1)
- diffusion models (1)
- discrepancy (1)
- double exponential distribution (1)
- downward continuation (1)
- efficiency loss (1)
- elastoplasticity (1)
- elliptical distribution (1)
- endomorphism ring (1)
- enumerative geometry (1)
- equilibrium strategies (1)
- equisingular families (1)
- face value (1)
- fiber reinforced silicon carbide (1)
- filtration (1)
- financial mathematics (1)
- finite difference schemes (1)
- finite element method (1)
- first hitting time (1)
- float glass (1)
- flood risk (1)
- fluid structure (1)
- fluid structure interaction (1)
- forward-shooting grid (1)
- free surface (1)
- freie Oberfläche (1)
- gebietszerlegung (1)
- gitter (1)
- good semigroup (1)
- graph p-Laplacian (1)
- gravitation (1)
- group action (1)
- großer Investor (1)
- hedging (1)
- heuristic (1)
- hierarchical matrix (1)
- hyperbolic systems (1)
- hyperelliptic function field (1)
- hyperelliptische Funktionenkörper (1)
- hyperspectal unmixing (1)
- idealclass group (1)
- image analysis (1)
- image denoising (1)
- impulse control (1)
- impurity functions (1)
- incompressible elasticity (1)
- infinite-dimensional manifold (1)
- inflation-linked product (1)
- integer programming (1)
- integral constitutive equations (1)
- intensity (1)
- inverse optimization (1)
- inverse problem (1)
- jump-diffusion process (1)
- large investor (1)
- large scale integer programming (1)
- lattice Boltzmann (1)
- level K-algebras (1)
- level set method (1)
- limit theorems (1)
- linear code (1)
- localizing basis (1)
- longevity bonds (1)
- low-rank approximation (1)
- macro derivative (1)
- market manipulation (1)
- markov model (1)
- martingale optimality principle (1)
- mathematical modelling (1)
- mathematical morphology (1)
- matrix problems (1)
- matroid flows (1)
- mean-variance approach (1)
- micromechanics (1)
- mixed convection (1)
- mixed methods (1)
- mixed multiscale finite element methods (1)
- modal derivatives (1)
- model order reduction (1)
- moduli space (1)
- monotone Konvergenz (1)
- monotropic programming (1)
- multi scale (1)
- multi-asset option (1)
- multi-class image segmentation (1)
- multi-level Monte Carlo (1)
- multi-phase flow (1)
- multicategory (1)
- multifilament superconductor (1)
- multigrid method (1)
- multileaf collimator (1)
- multiobjective optimization (1)
- multipatch (1)
- multiplicative noise (1)
- multiscale denoising (1)
- multiscale methods (1)
- multivariate chi-square-test (1)
- network flows (1)
- network synthesis (1)
- netzgenerierung (1)
- nicht-newtonsche Strömungen (1)
- nichtlineare Druckkorrektor (1)
- nichtlineare Modellreduktion (1)
- nichtlineare Netzwerke (1)
- non-desarguesian plane (1)
- non-newtonian flow (1)
- nonconvex optimization (1)
- nonlinear circuits (1)
- nonlinear diffusion filtering (1)
- nonlinear model reduction (1)
- nonlinear pressure correction (1)
- nonlinear term structure dependence (1)
- nonlinear vibration analysis (1)
- nonlocal filtering (1)
- nonnegative matrix factorization (1)
- nonwovens (1)
- normalization (1)
- numerical irreducible decomposition (1)
- numerical methods (1)
- numerische Strömungssimulation (1)
- numerisches Verfahren (1)
- oblique derivative (1)
- optimal capital structure (1)
- optimal consumption and investment (1)
- optiman stopping (1)
- option pricing (1)
- option valuation (1)
- partial differential equation (1)
- partial information (1)
- path-dependent options (1)
- pattern (1)
- penalty methods (1)
- penalty-free formulation (1)
- petroleum exploration (1)
- planar polynomial (1)
- poroelasticity (1)
- porous media (1)
- portfolio (1)
- portfolio decision (1)
- portfolio-optimization (1)
- poröse Medien (1)
- potential (1)
- preconditioners (1)
- pressure correction (1)
- primal-dual algorithm (1)
- probability distribution (1)
- projective surfaces (1)
- proximation (1)
- quadrinomial tree (1)
- quasi-Monte Carlo (1)
- quasi-variational inequalities (1)
- quasihomogeneity (1)
- quasiregular group (1)
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- radiation therapy (1)
- radiotherapy (1)
- rare disasters (1)
- rate of convergence (1)
- raum-zeitliche Analyse (1)
- real quadratic number fields (1)
- redundant constraint (1)
- reflectionless boundary condition (1)
- reflexionslose Randbedingung (1)
- regime-shift model (1)
- regression analysis (1)
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- sawtooth effect (1)
- scalar and vectorial wavelets (1)
- second class group (1)
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- sparse interpolation of multivariate rational functions (1)
- sparse multivariate polynomial interpolation (1)
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- value semigroup (1)
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- Überflutungsrisiko (1)
- Übergangsbedingungen (1)

#### Fachbereich / Organisatorische Einheit

- Fachbereich Mathematik (226)
- Fraunhofer (ITWM) (2)

The main theme of this thesis is the interplay between algebraic and tropical intersection
theory, especially in the context of enumerative geometry. We begin by exploiting
well-known results about tropicalizations of subvarieties of algebraic tori to give a
simple proof of Nishinou and Siebert’s correspondence theorem for rational curves
through given points in toric varieties. Afterwards, we extend this correspondence
by additionally allowing intersections with psi-classes. We do this by constructing
a tropicalization map for cycle classes on toroidal embeddings. It maps algebraic
cycle classes to elements of the Chow group of the cone complex of the toroidal
embedding, that is to weighted polyhedral complexes, which are balanced with respect
to an appropriate map to a vector space, modulo a naturally defined equivalence relation.
We then show that tropicalization respects basic intersection-theoretic operations like
intersections with boundary divisors and apply this to the appropriate moduli spaces
to obtain our correspondence theorem.
Trying to apply similar methods in higher genera inevitably confronts us with moduli
spaces which are not toroidal. This motivates the last part of this thesis, where we
construct tropicalizations of cycles on fine logarithmic schemes. The logarithmic point of
view also motivates our interpretation of tropical intersection theory as the dualization
of the intersection theory of Kato fans. This duality gives a new perspective on the
tropicalization map; namely, as the dualization of a pull-back via the characteristic
morphism of a logarithmic scheme.

We investigate the long-term behaviour of diffusions on the non-negative real numbers under killing at some random time. Killing can occur at zero as well as in the interior of the state space. The diffusion follows a stochastic differential equation driven by a Brownian motion. The diffusions we are working with will almost surely be killed. In large parts of this thesis we only assume the drift coefficient to be continuous. Further, we suppose that zero is regular and that infinity is natural. We condition the diffusion on survival up to time t and let t tend to infinity looking for a limiting behaviour.

The thesis studies change points in absolute time for censored survival data with some contributions to the more common analysis of change points with respect to survival time. We first introduce the notions and estimates of survival analysis, in particular the hazard function and censoring mechanisms. Then, we discuss change point models for survival data. In the literature, usually change points with respect to survival time are studied. Typical examples are piecewise constant and piecewise linear hazard functions. For that kind of models, we propose a new algorithm for numerical calculation of maximum likelihood estimates based on a cross entropy approach which in our simulations outperforms the common Nelder-Mead algorithm.
Our original motivation was the study of censored survival data (e.g., after diagnosis of breast cancer) over several decades. We wanted to investigate if the hazard functions differ between various time periods due, e.g., to progress in cancer treatment. This is a change point problem in the spirit of classical change point analysis. Horváth (1998) proposed a suitable change point test based on estimates of the cumulative hazard function. As an alternative, we propose similar tests based on nonparametric estimates of the hazard function. For one class of tests related to kernel probability density estimates, we develop fully the asymptotic theory for the change point tests. For the other class of estimates, which are versions of the Watson-Leadbetter estimate with censoring taken into account and which are related to the Nelson-Aalen estimate, we discuss some steps towards developing the full asymptotic theory. We close by applying the change point tests to simulated and real data, in particular to the breast cancer survival data from the SEER study.

Nonwoven materials are used as filter media which are the key component of automotive filters such as air filters, oil filters, and fuel filters. Today, the advanced engine technologies require innovative filter media with higher performances. A virtual microstructure of the nonwoven filter medium, which has similar filter properties as the existing material, can be used to design new filter media from existing media. Nonwoven materials considered in this thesis prominently feature non-overlapping fibers, curved fibers, fibers with circular cross section, fibers of apparently infinite length, and fiber bundles. To this end, as part of this thesis, we extend the Altendorf-Jeulin individual fiber model to incorporate all the above mentioned features. The resulting novel stochastic 3D fiber model can generate geometries with good visual resemblance of real filter media. Furthermore, pressure drop, which is one of the important physical properties of the filter, simulated numerically on the computed tomography (CT) data of the real nonwoven material agrees well (with a relative error of 8%) with the pressure drop simulated in the generated microstructure realizations from our model.
Generally, filter properties for the CT data and generated microstructure realizations are computed using numerical simulations. Since numerical simulations require extensive system memory and computation time, it is important to find the representative domain size of the generated microstructure for a required filter property. As part of this thesis, simulation and a statistical approach are used to estimate the representative domain size of our microstructure model. Precisely, the representative domain size with respect to the packing density, the pore size distribution, and the pressure drop are considered. It turns out that the statistical approach can be used to estimate the representative domain size for the given property more precisely and using less generated microstructures than the purely simulation based approach.
Among the various properties of fibrous filter media, fiber thickness and orientation are important characteristics which should be considered in design and quality assurance of filter media. Automatic analysis of images from scanning electron microscopy (SEM) is a suitable tool in that context. Yet, the accuracy of such image analysis tools cannot be judged based on images of real filter media since their true fiber thickness and orientation can never be known accurately. A solution is to employ synthetically generated models for evaluation. By combining our 3D fiber system model with simulation of the SEM imaging process, quantitative evaluation of the fiber thickness and orientation measurements becomes feasible. We evaluate the state-of-the-art automatic thickness and orientation estimation method that way.

Following the ideas presented in Dahlhaus (2000) and Dahlhaus and Sahm (2000) for time series, we build a Whittle-type approximation of the Gaussian likelihood for locally stationary random fields. To achieve this goal, we extend a Szegö-type formula, for the multidimensional and local stationary case and secondly we derived a set of matrix approximations using elements of the spectral theory of stochastic processes. The minimization of the Whittle likelihood leads to the so-called Whittle estimator \(\widehat{\theta}_{T}\). For the sake of simplicity we assume known mean (without loss of generality zero mean), and hence \(\widehat{\theta}_{T}\) estimates the parameter vector of the covariance matrix \(\Sigma_{\theta}\).
We investigate the asymptotic properties of the Whittle estimate, in particular uniform convergence of the likelihoods, and consistency and Gaussianity of the estimator. A main point is a detailed analysis of the asymptotic bias which is considerably more difficult for random fields than for time series. Furthemore, we prove in case of model misspecification that the minimum of our Whittle likelihood still converges, where the limit is the minimum of the Kullback-Leibler information divergence.
Finally, we evaluate the performance of the Whittle estimator through computational simulations and estimation of conditional autoregressive models, and a real data application.

In this thesis we explicitly solve several portfolio optimization problems in a very realistic setting. The fundamental assumptions on the market setting are motivated by practical experience and the resulting optimal strategies are challenged in numerical simulations.
We consider an investor who wants to maximize expected utility of terminal wealth by trading in a high-dimensional financial market with one riskless asset and several stocks.
The stock returns are driven by a Brownian motion and their drift is modelled by a Gaussian random variable. We consider a partial information setting, where the drift is unknown to the investor and has to be estimated from the observable stock prices in addition to some analyst’s opinion as proposed in [CLMZ06]. The best estimate given these observations is the well known Kalman-Bucy-Filter. We then consider an innovations process to transform the partial information setting into a market with complete information and an observable Gaussian drift process.
The investor is restricted to portfolio strategies satisfying several convex constraints.
These constraints can be due to legal restrictions, due to fund design or due to client's specifications. We cover in particular no-short-selling and no-borrowing constraints.
One popular approach to constrained portfolio optimization is the convex duality approach of Cvitanic and Karatzas. In [CK92] they introduce auxiliary stock markets with shifted market parameters and obtain a dual problem to the original portfolio optimization problem that can be better solvable than the primal problem.
Hence we consider this duality approach and using stochastic control methods we first solve the dual problems in the cases of logarithmic and power utility.
Here we apply a reverse separation approach in order to obtain areas where the corresponding Hamilton-Jacobi-Bellman differential equation can be solved. It turns out that these areas have a straightforward interpretation in terms of the resulting portfolio strategy. The areas differ between active and passive stocks, where active stocks are invested in, while passive stocks are not.
Afterwards we solve the auxiliary market given the optimal dual processes in a more general setting, allowing for various market settings and various dual processes.
We obtain explicit analytical formulas for the optimal portfolio policies and provide an algorithm that determines the correct formula for the optimal strategy in any case.
We also show optimality of our resulting portfolio strategies in different verification theorems.
Subsequently we challenge our theoretical results in a historical and an artificial simulation that are even closer to the real world market than the setting we used to derive our theoretical results. However, we still obtain compelling results indicating that our optimal strategies can outperform any benchmark in a real market in general.

We introduce and investigate a product pricing model in social networks where the value a possible buyer assigns to a product is influenced by the previous buyers. The selling proceeds in discrete, synchronous rounds for some set price and the individual values are additively altered. Whereas computing the revenue for a given price can be done in polynomial time, we show that the basic problem PPAI, i.e., is there a price generating a requested revenue, is weakly NP-complete. With algorithm Frag we provide a pseudo-polynomial time algorithm checking the range of prices in intervals of common buying behavior we call fragments. In some special cases, e.g., solely positive influences, graphs with bounded in-degree, or graphs with bounded path length, the amount of fragments is polynomial. Since the run-time of Frag is polynomial in the amount of fragments, the algorithm itself is polynomial for these special cases. For graphs with positive influence we show that every buyer does also buy for lower prices, a property that is not inherent for arbitrary graphs. Algorithm FixHighest improves the run-time on these graphs by using the above property.
Furthermore, we introduce variations on this basic model. The version of delaying the propagation of influences and the awareness of the product can be implemented in our basic model by substituting nodes and arcs with simple gadgets. In the chapter on Dynamic Product Pricing we allow price changes, thereby raising the complexity even for graphs with solely positive or negative influences. Concerning Perishable Product Pricing, i.e., the selling of products that are usable for some time and can be rebought afterward, the principal problem is computing the revenue that a given price can generate in some time horizon. In general, the problem is #P-hard and algorithm Break runs in pseudo-polynomial time. For polynomially computable revenue, we investigate once more the complexity to find the best price.
We conclude the thesis with short results in topics of Cooperative Pricing, Initial Value as Parameter, Two Product Pricing, and Bounded Additive Influence.

Non–woven materials consist of many thousands of fibres laid down on a conveyor belt
under the influence of a turbulent air stream. To improve industrial processes for the
production of non–woven materials, we develop and explore novel mathematical fibre and
material models.
In Part I of this thesis we improve existing mathematical models describing the fibres on the
belt in the meltspinning process. In contrast to existing models, we include the fibre–fibre
interaction caused by the fibres’ thickness which prevents the intersection of the fibres and,
hence, results in a more accurate mathematical description. We start from a microscopic
characterisation, where each fibre is described by a stochastic functional differential
equation and include the interaction along the whole fibre path, which is described by a
delay term. As many fibres are required for the production of a non–woven material, we
consider the corresponding mean–field equation, which describes the evolution of the fibre
distribution with respect to fibre position and orientation. To analyse the particular case of
large turbulences in the air stream, we develop the diffusion approximation which yields a
distribution describing the fibre position. Considering the convergence to equilibrium on
an analytical level, as well as performing numerical experiments, gives an insight into the
influence of the novel interaction term in the equations.
In Part II of this thesis we model the industrial airlay process, which is a production method
whereby many short fibres build a three–dimensional non–woven material. We focus on
the development of a material model based on original fibre properties, machine data and
micro computer tomography. A possible linking of these models to other simulation tools,
for example virtual tensile tests, is discussed.
The models and methods presented in this thesis promise to further the field in mathematical
modelling and computational simulation of non–woven materials.

In this dissertation convergence of binomial trees for option pricing is investigated. The focus is on American and European put and call options. For that purpose variations of the binomial tree model are reviewed.
In the first part of the thesis we investigated the convergence behavior of the already known trees from the literature (CRR, RB, Tian and CP) for the European options. The CRR and the RB tree suffer from irregular convergence, so our first aim is to find a way to get the smooth convergence. We first show what causes these oscillations. That will also help us to improve the rate of convergence. As a result we introduce the Tian and the CP tree and we proved that the order of convergence for these trees is \(O \left(\frac{1}{n} \right)\).
Afterwards we introduce the Split tree and explain its properties. We prove the convergence of it and we found an explicit first order error formula. In our setting, the splitting time \(t_{k} = k\Delta t\) is not fixed, i.e. it can be any time between 0 and the maturity time \(T\). This is the main difference compared to the model from the literature. Namely, we show that the good properties of the CRR tree when \(S_{0} = K\) can be preserved even without this condition (which is mainly the case). We achieved the convergence of \(O \left(n^{-\frac{3}{2}} \right)\) and we typically get better results if we split our tree later.

In change-point analysis the point of interest is to decide if the observations follow one model
or if there is at least one time-point, where the model has changed. This results in two sub-
fields, the testing of a change and the estimation of the time of change. This thesis considers
both parts but with the restriction of testing and estimating for at most one change-point.
A well known example is based on independent observations having one change in the mean.
Based on the likelihood ratio test a test statistic with an asymptotic Gumbel distribution was
derived for this model. As it is a well-known fact that the corresponding convergence rate is
very slow, modifications of the test using a weight function were considered. Those tests have
a better performance. We focus on this class of test statistics.
The first part gives a detailed introduction to the techniques for analysing test statistics and
estimators. Therefore we consider the multivariate mean change model and focus on the effects
of the weight function. In the case of change-point estimators we can distinguish between
the assumption of a fixed size of change (fixed alternative) and the assumption that the size
of the change is converging to 0 (local alternative). Especially, the fixed case in rarely analysed
in the literature. We show how to come from the proof for the fixed alternative to the
proof of the local alternative. Finally, we give a simulation study for heavy tailed multivariate
observations.
The main part of this thesis focuses on two points. First, analysing test statistics and, secondly,
analysing the corresponding change-point estimators. In both cases, we first consider a
change in the mean for independent observations but relaxing the moment condition. Based on
a robust estimator for the mean, we derive a new type of change-point test having a randomized
weight function. Secondly, we analyse non-linear autoregressive models with unknown
regression function. Based on neural networks, test statistics and estimators are derived for
correctly specified as well as for misspecified situations. This part extends the literature as
we analyse test statistics and estimators not only based on the sample residuals. In both
sections, the section on tests and the one on the change-point estimator, we end with giving
regularity conditions on the model as well as the parameter estimator.
Finally, a simulation study for the case of the neural network based test and estimator is
given. We discuss the behaviour under correct and mis-specification and apply the neural
network based test and estimator on two data sets.

In this thesis, we consider a problem from modular representation theory of finite groups. Lluís Puig asked the question whether the order of the defect groups of a block \( B \) of the group algebra of a given finite group \( G \) can always be bounded in terms of the order of the vertices of an arbitrary simple module lying in \( B \).
In characteristic \( 2 \), there are examples showing that this is not possible in general, whereas in odd characteristic, no such examples are known. For instance, it is known that the answer to Puig's question is positive in case that \( G \) is a symmetric group, by work of Danz, Külshammer, and Puig.
Motivated by this, we study the cases where \( G \) is a finite classical group in non-defining characteristic or one of the finite groups \( G_2(q) \) or \( ³D_4(q) \) of Lie type, again in non-defining characteristic. Here, we generalize Puig's original question by replacing the vertices occurring in his question by arbitrary self-centralizing subgroups of the defect groups. We derive positive and negative answers to this generalized question.
\[\]
In addition to that, we determine the vertices of the unipotent simple \( GL_2(q) \)-module labeled by the partition \( (1,1) \) in characteristic \( 2 \). This is done using a method known as Brauer construction.

This thesis brings together convex analysis and hyperspectral image processing.
Convex analysis is the study of convex functions and their properties.
Convex functions are important because they admit minimization by efficient algorithms
and the solution of many optimization problems can be formulated as
minimization of a convex objective function, extending much beyond
the classical image restoration problems of denoising, deblurring and inpainting.
\(\hspace{1mm}\)
At the heart of convex analysis is the duality mapping induced within the
class of convex functions by the Fenchel transform.
In the last decades efficient optimization algorithms have been developed based
on the Fenchel transform and the concept of infimal convolution.
\(\hspace{1mm}\)
The infimal convolution is of similar importance in convex analysis as the
convolution in classical analysis. In particular, the infimal convolution with
scaled parabolas gives rise to the one parameter family of Moreau-Yosida envelopes,
which approximate a given function from below while preserving its minimum
value and minimizers.
The closely related proximal mapping replaces the gradient step
in a recently developed class of efficient first-order iterative minimization algorithms
for non-differentiable functions. For a finite convex function,
the proximal mapping coincides with a gradient step of its Moreau-Yosida envelope.
Efficient algorithms are needed in hyperspectral image processing,
where several hundred intensity values measured in each spatial point
give rise to large data volumes.
\(\hspace{1mm}\)
In the \(\textbf{first part}\) of this thesis, we are concerned with
models and algorithms for hyperspectral unmixing.
As part of this thesis a hyperspectral imaging system was taken into operation
at the Fraunhofer ITWM Kaiserslautern to evaluate the developed algorithms on real data.
Motivated by missing-pixel defects common in current hyperspectral imaging systems,
we propose a
total variation regularized unmixing model for incomplete and noisy data
for the case when pure spectra are given.
We minimize the proposed model by a primal-dual algorithm based on the
proximum mapping and the Fenchel transform.
To solve the unmixing problem when only a library of pure spectra is provided,
we study a modification which includes a sparsity regularizer into model.
\(\hspace{1mm}\)
We end the first part with the convergence analysis for a multiplicative
algorithm derived by optimization transfer.
The proposed algorithm extends well-known multiplicative update rules
for minimizing the Kullback-Leibler divergence,
to solve a hyperspectral unmixing model in the case
when no prior knowledge of pure spectra is given.
\(\hspace{1mm}\)
In the \(\textbf{second part}\) of this thesis, we study the properties of Moreau-Yosida envelopes,
first for functions defined on Hadamard manifolds, which are (possibly) infinite-dimensional
Riemannian manifolds with negative curvature,
and then for functions defined on Hadamard spaces.
\(\hspace{1mm}\)
In particular we extend to infinite-dimensional Riemannian manifolds an expression
for the gradient of the Moreau-Yosida envelope in terms of the proximal mapping.
With the help of this expression we show that a sequence of functions
converges to a given limit function in the sense of Mosco
if the corresponding Moreau-Yosida envelopes converge pointwise at all scales.
\(\hspace{1mm}\)
Finally we extend this result to the more general setting of Hadamard spaces.
As the reverse implication is already known, this unites two definitions of Mosco convergence
on Hadamard spaces, which have both been used in the literature,
and whose equivalence has not yet been known.

We discuss the portfolio selection problem of an investor/portfolio manager in an arbitrage-free financial market where a money market account, coupon bonds and a stock are traded continuously. We allow for stochastic interest rates and in particular consider one and two-factor Vasicek models for the instantaneous
short rates. In both cases we consider a complete and an incomplete market setting by adding a suitable number of bonds.
The goal of an investor is to find a portfolio which maximizes expected utility
from terminal wealth under budget and present expected short-fall (PESF) risk
constraints. We analyze this portfolio optimization problem in both complete and
incomplete financial markets in three different cases: (a) when the PESF risk is
minimum, (b) when the PESF risk is between minimum and maximum and (c) without risk constraints. (a) corresponds to the portfolio insurer problem, in (b) the risk constraint is binding, i.e., it is satisfied with equality, and (c) corresponds
to the unconstrained Merton investment.
In all cases we find the optimal terminal wealth and portfolio process using the
martingale method and Malliavin calculus respectively. In particular we solve in the incomplete market settings the dual problem explicitly. We compare the
optimal terminal wealth in the cases mentioned using numerical examples. Without
risk constraints, we further compare the investment strategies for complete
and incomplete market numerically.

In this thesis we address two instances of duality in commutative algebra.
In the first part, we consider value semigroups of non irreducible singular algebraic curves
and their fractional ideals. These are submonoids of Z^n closed under minima, with a conductor and which fulfill special compatibility properties on their elements. Subsets of Z^n
fulfilling these three conditions are known in the literature as good semigroups and their ideals, and their class strictly contains the class of value semigroup ideals. We examine
good semigroups both independently and in relation with their algebraic counterpart. In the combinatoric setting, we define the concept of good system of generators, and we
show that minimal good systems of generators are unique. In relation with the algebra side, we give an intrinsic definition of canonical semigroup ideals, which yields a duality
on good semigroup ideals. We prove that this semigroup duality is compatible with the Cohen-Macaulay duality under taking values. Finally, using the duality on good semigroup ideals, we show a symmetry of the Poincaré series of good semigroups with special properties.
In the second part, we treat Macaulay’s inverse system, a one-to-one correspondence
which is a particular case of Matlis duality and an effective method to construct Artinian k-algebras with chosen socle type. Recently, Elias and Rossi gave the structure of the inverse system of positive dimensional Gorenstein k-algebras. We extend their result by establishing a one-to-one correspondence between positive dimensional level k-algebras and certain submodules of the divided power ring. We give several examples to illustrate
our result.

In this thesis, we deal with the finite group of Lie type \(F_4(2^n)\). The aim is to find information on the \(l\)-decomposition numbers of \(F_4(2^n)\) on unipotent blocks for \(l\neq2\) and \(n\in \mathbb{N}\) arbitrary and on the irreducible characters of the Sylow \(2\)-subgroup of \(F_4(2^n)\).
S. M. Goodwin, T. Le, K. Magaard and A. Paolini have found a parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), a Sylow \(2\)-subgroup of \(F_4(q)\), of \(F_4(p^n)\), \(p\) a prime, for the case \(p\neq2\).
We managed to adapt their methods for the parametrization of the irreducible characters of the Sylow \(2\)-subgroup for the case \(p=2\) for the group \(F_4(q)\), \(q=p^n\). This gives a nearly complete parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), namely of all irreducible characters of \(U\) arising from so-called abelian cores.
The general strategy we have applied to obtain information about the \(l\)-decomposition numbers on unipotent blocks is to induce characters of the unipotent subgroup \(U\) of \(F_4(q)\) and Harish-Chandra induce projective characters of proper Levi subgroups of \(F_4(q)\) to obtain projective characters of \(F_4(q)\). Via Brauer reciprocity, the multiplicities of the ordinary irreducible unipotent characters in these projective characters give us information on the \(l\)-decomposition numbers of the unipotent characters of \(F_4(q)\).
Sadly, the projective characters of \(F_4(q)\) we obtained were not sufficient to give the shape of the entire decomposition matrix.

In this thesis we integrate discrete dividends into the stock model, estimate
future outstanding dividend payments and solve different portfolio optimization
problems. Therefore, we discuss three well-known stock models, including
discrete dividend payments and evolve a model, which also takes early
announcement into account.
In order to estimate the future outstanding dividend payments, we develop a
general estimation framework. First, we investigate a model-free, no-arbitrage
methodology, which is based on the put-call parity for European options. Our
approach integrates all available option market data and simultaneously calculates
the market-implied discount curve. We illustrate our method using stocks
of European blue-chip companies and show within a statistical assessment that
the estimate performs well in practice.
As American options are more common, we additionally develop a methodology,
which is based on market prices of American at-the-money options.
This method relies on a linear combination of no-arbitrage bounds of the dividends,
where the corresponding optimal weight is determined via a historical
least squares estimation using realized dividends. We demonstrate our method
using all Dow Jones Industrial Average constituents and provide a robustness
check with respect to the used discount factor. Furthermore, we backtest our
results against the method using European options and against a so called
simple estimate.
In the last part of the thesis we solve the terminal wealth portfolio optimization
problem for a dividend paying stock. In the case of the logarithmic utility
function, we show that the optimal strategy is not a constant anymore but
connected to the Merton strategy. Additionally, we solve a special optimal
consumption problem, where the investor is only allowed to consume dividends.
We show that this problem can be reduced to the before solved terminal wealth
problem.

In this thesis, we focus on the application of the Heath-Platen (HP) estimator in option
pricing. In particular, we extend the approach of the HP estimator for pricing path dependent
options under the Heston model. The theoretical background of the estimator
was first introduced by Heath and Platen [32]. The HP estimator was originally interpreted
as a control variate technique and an application for European vanilla options was
presented in [32]. For European vanilla options, the HP estimator provided a considerable
amount of variance reduction. Thus, applying the technique for path dependent options
under the Heston model is the main contribution of this thesis.
The first part of the thesis deals with the implementation of the HP estimator for pricing
one-sided knockout barrier options. The main difficulty for the implementation of the HP
estimator is located in the determination of the first hitting time of the barrier. To test the
efficiency of the HP estimator we conduct numerical tests with regard to various aspects.
We provide a comparison among the crude Monte Carlo estimation, the crude control
variate technique and the HP estimator for all types of barrier options. Furthermore, we
present the numerical results for at the money, in the money and out of the money barrier
options. As numerical results imply, the HP estimator performs superior among others
for pricing one-sided knockout barrier options under the Heston model.
Another contribution of this thesis is the application of the HP estimator in pricing bond
options under the Cox-Ingersoll-Ross (CIR) model and the Fong-Vasicek (FV) model. As
suggested in the original paper of Heath and Platen [32], the HP estimator has a wide
range of applicability for derivative pricing. Therefore, transferring the structure of the
HP estimator for pricing bond options is a promising contribution. As the approximating
Vasicek process does not seem to be as good as the deterministic volatility process in the
Heston setting, the performance of the HP estimator in the CIR model is only relatively
good. However, for the FV model the variance reduction provided by the HP estimator is
again considerable.
Finally, the numerical result concerning the weak convergence rate of the HP estimator
for pricing European vanilla options in the Heston model is presented. As supported by
numerical analysis, the HP estimator has weak convergence of order almost 1.

A popular model for the locations of fibres or grains in composite materials
is the inhomogeneous Poisson process in dimension 3. Its local intensity function
may be estimated non-parametrically by local smoothing, e.g. by kernel
estimates. They crucially depend on the choice of bandwidths as tuning parameters
controlling the smoothness of the resulting function estimate. In this
thesis, we propose a fast algorithm for learning suitable global and local bandwidths
from the data. It is well-known, that intensity estimation is closely
related to probability density estimation. As a by-product of our study, we
show that the difference is asymptotically negligible regarding the choice of
good bandwidths, and, hence, we focus on density estimation.
There are quite a number of data-driven bandwidth selection methods for
kernel density estimates. cross-validation is a popular one and frequently proposed
to estimate the optimal bandwidth. However, if the sample size is very
large, it becomes computational expensive. In material science, in particular,
it is very common to have several thousand up to several million points.
Another type of bandwidth selection is a solve-the-equation plug-in approach
which involves replacing the unknown quantities in the asymptotically optimal
bandwidth formula by their estimates.
In this thesis, we develop such an iterative fast plug-in algorithm for estimating
the optimal global and local bandwidth for density and intensity estimation with a focus on 2- and 3-dimensional data. It is based on a detailed
asymptotics of the estimators of the intensity function and of its second
derivatives and integrals of second derivatives which appear in the formulae
for asymptotically optimal bandwidths. These asymptotics are utilised to determine
the exact number of iteration steps and some tuning parameters. For
both global and local case, fewer than 10 iterations suffice. Simulation studies
show that the estimated intensity by local bandwidth can better indicate
the variation of local intensity than that by global bandwidth. Finally, the
algorithm is applied to two real data sets from test bodies of fibre-reinforced
high-performance concrete, clearly showing some inhomogeneity of the fibre
intensity.

Using valuation theory we associate to a one-dimensional equidimensional semilocal Cohen-Macaulay ring \(R\) its semigroup of values, and to a fractional ideal of \(R\) we associate its value semigroup ideal. For a class of curve singularities (here called admissible rings) including algebroid curves the semigroups of values, respectively the value semigroup ideals, satisfy combinatorial properties defining good semigroups, respectively good semigroup ideals. Notably, the class of good semigroups strictly contains the class of value semigroups of admissible rings. On good semigroups we establish combinatorial versions of algebraic concepts on admissible rings which are compatible with their prototypes under taking values. Primarily we examine duality and quasihomogeneity.
We give a definition for canonical semigroup ideals of good semigroups which characterizes canonical fractional ideals of an admissible ring in terms of their value semigroup ideals. Moreover, a canonical semigroup ideal induces a duality on the set of good semigroup ideals of a good semigroup. This duality is compatible with the Cohen-Macaulay duality on fractional ideals under taking values.
The properties of the semigroup of values of a quasihomogeneous curve singularity lead to a notion of quasihomogeneity on good semigroups which is compatible with its algebraic prototype. We give a combinatorial criterion which allows to construct from a quasihomogeneous semigroup \(S\) a quasihomogeneous curve singularity having \(S\) as semigroup of values.
As an application we use the semigroup of values to compute endomorphism rings of maximal ideals of algebroid curves. This yields an explicit description of the intermediate rings in an algorithmic normalization of plane central arrangements of smooth curves based on a criterion by Grauert and Remmert. Applying this result to hyperplane arrangements we determine the number of steps needed to compute the normalization of a the arrangement in terms of its Möbius function.

Composite materials are used in many modern tools and engineering applications and
consist of two or more materials that are intermixed. Features like inclusions in a matrix
material are often very small compared to the overall structure. Volume elements that
are characteristic for the microstructure can be simulated and their elastic properties are
then used as a homogeneous material on the macroscopic scale.
Moulinec and Suquet [2] solve the so-called Lippmann-Schwinger equation, a reformulation of the equations of elasticity in periodic homogenization, using truncated
trigonometric polynomials on a tensor product grid as ansatz functions.
In this thesis, we generalize their approach to anisotropic lattices and extend it to
anisotropic translation invariant spaces. We discretize the partial differential equation
on these spaces and prove the convergence rate. The speed of convergence depends on
the smoothness of the coefficients and the regularity of the ansatz space. The spaces of
translates unify the ansatz of Moulinec and Suquet with de la Vallée Poussin means and
periodic Box splines, including the constant finite element discretization of Brisard and
Dormieux [1].
For finely resolved images, sampling on a coarser lattice reduces the computational
effort. We introduce mixing rules as the means to transfer fine-grid information to the
smaller lattice.
Finally, we show the effect of the anisotropic pattern, the space of translates, and the
convergence of the method, and mixing rules on two- and three-dimensional examples.
References
[1] S. Brisard and L. Dormieux. “FFT-based methods for the mechanics of composites:
A general variational framework”. In: Computational Materials Science 49.3 (2010),
pp. 663–671. doi: 10.1016/j.commatsci.2010.06.009.
[2] H. Moulinec and P. Suquet. “A numerical method for computing the overall response
of nonlinear composites with complex microstructure”. In: Computer Methods in
Applied Mechanics and Engineering 157.1-2 (1998), pp. 69–94. doi: 10.1016/s00457825(97)00218-1.

Multiphase materials combine properties of several materials, which makes them interesting for high-performing components. This thesis considers a certain set of multiphase materials, namely silicon-carbide (SiC) particle-reinforced aluminium (Al) metal matrix composites and their modelling based on stochastic geometry models.
Stochastic modelling can be used for the generation of virtual material samples: Once we have fitted a model to the material statistics, we can obtain independent three-dimensional “samples” of the material under investigation without the need of any actual imaging. Additionally, by changing the model parameters, we can easily simulate a new material composition.
The materials under investigation have a rather complicated microstructure, as the system of SiC particles has many degrees of freedom: Size, shape, orientation and spatial distribution. Based on FIB-SEM images, that yield three-dimensional image data, we extract the SiC particle structure using methods of image analysis. Then we model the SiC particles by anisotropically rescaled cells of a random Laguerre tessellation that was fitted to the shapes of isotropically rescaled particles. We fit a log-normal distribution for the volume distribution of the SiC particles. Additionally, we propose models for the Al grain structure and the Aluminium-Copper (\({Al}_2{Cu}\)) precipitations occurring on the grain boundaries and on SiC-Al phase boundaries.
Finally, we show how we can estimate the parameters of the volume-distribution based on two-dimensional SEM images. This estimation is applied to two samples with different mean SiC particle diameters and to a random section through the model. The stereological estimations are within acceptable agreement with the parameters estimated from three-dimensional image data
as well as with the parameters of the model.

Certain brain tumours are very hard to treat with radiotherapy due to their irregular shape caused by the infiltrative nature of the tumour cells. To enhance the estimation of the tumour extent one may use a mathematical model. As the brain structure plays an important role for the cell migration, it has to be included in such a model. This is done via diffusion-MRI data. We set up a multiscale model class accounting among others for integrin-mediated movement of cancer cells in the brain tissue, and the integrin-mediated proliferation. Moreover, we model a novel chemotherapy in combination with standard radiotherapy.
Thereby, we start on the cellular scale in order to describe migration. Then we deduce mean-field equations on the mesoscopic (cell density) scale on which we also incorporate cell proliferation. To reduce the phase space of the mesoscopic equation, we use parabolic scaling and deduce an effective description in the form of a reaction-convection-diffusion equation on the macroscopic spatio-temporal scale. On this scale we perform three dimensional numerical simulations for the tumour cell density, thereby incorporating real diffusion tensor imaging data. To this aim, we present programmes for the data processing taking the raw medical data and processing it to the form to be included in the numerical simulation. Thanks to the reduction of the phase space, the numerical simulations are fast enough to enable application in clinical practice.

In modern algebraic geometry solutions of polynomial equations are studied from a qualitative point of view using highly sophisticated tools such as cohomology, \(D\)-modules and Hodge structures. The latter have been unified in Saito’s far-reaching theory of mixed Hodge modules, that has shown striking applications including vanishing theorems for cohomology. A mixed Hodge module can be seen as a special type of filtered \(D\)-module, which is an algebraic counterpart of a system of linear differential equations. We present the first algorithmic approach to Saito’s theory. To this end, we develop a Gröbner basis theory for a new class of algebras generalizing PBW-algebras.
The category of mixed Hodge modules satisfies Grothendieck’s six-functor formalism. In part these functors rely on an additional natural filtration, the so-called \(V\)-filtration. A key result of this thesis is an algorithm to compute the \(V\)-filtration in the filtered setting. We derive from this algorithm methods for the computation of (extraordinary) direct image functors under open embeddings of complements of pure codimension one subvarieties. As side results we show
how to compute vanishing and nearby cycle functors and a quasi-inverse of Kashiwara’s equivalence for mixed Hodge modules.
Describing these functors in terms of local coordinates and taking local sections, we reduce the corresponding computations to algorithms over certain bifiltered algebras. It leads us to introduce the class of so-called PBW-reduction-algebras, a generalization of the class of PBW-algebras. We establish a comprehensive Gröbner basis framework for this generalization representing the involved filtrations by weight vectors.

Numerical Godeaux surfaces are minimal surfaces of general type with the smallest possible numerical invariants. It is known that the torsion group of a numerical Godeaux surface is cyclic of order \(m\leq 5\). A full classification has been given for the cases \(m=3,4,5\) by the work of Reid and Miyaoka. In each case, the corresponding moduli space is 8-dimensional and irreducible.
There exist explicit examples of numerical Godeaux surfaces for the orders \(m=1,2\), but a complete classification for these surfaces is still missing.
In this thesis we present a construction method for numerical Godeaux surfaces which is based on homological algebra and computer algebra and which arises from an experimental approach by Schreyer. The main idea is to consider the canonical ring \(R(X)\) of a numerical Godeaux surface \(X\) as a module over some graded polynomial ring \(S\). The ring \(S\) is chosen so that \(R(X)\) is finitely generated as an \(S\)-module and a Gorenstein \(S\)-algebra of codimension 3. We prove that the canonical ring of any numerical Godeaux surface, considered as an \(S\)-module, admits a minimal free resolution whose middle map is alternating. Moreover, we show that a partial converse of this statement is true under some additional conditions.
Afterwards we use these results to construct (canonical rings of) numerical Godeaux surfaces. Hereby, we restrict our study to surfaces whose bicanonical system has no fixed component but 4 distinct base points, in the following referred to as marked numerical Godeaux surfaces.
The particular interest of this thesis lies on marked numerical Godeaux surfaces whose torsion group is trivial. For these surfaces we study the fibration of genus 4 over \(\mathbb{P}^1\) induced by the bicanonical system. Catanese and Pignatelli showed that the general fibre is non-hyperelliptic and that the number \(\tilde{h}\) of hyperelliptic fibres is bounded by 3. The two explicit constructions of numerical Godeaux surfaces with a trivial torsion group due to Barlow and Craighero-Gattazzo, respectively, satisfy \(\tilde{h} = 2\).
With the method from this thesis, we construct an 8-dimensional family of numerical Godeaux surfaces with a trivial torsion group and whose general element satisfy \(\tilde{h}=0\).
Furthermore, we establish a criterion for the existence of hyperelliptic fibres in terms of a minimal free resolution of \(R(X)\). Using this criterion, we verify experimentally the
existence of a numerical Godeaux surface with \(\tilde{h}=1\).

Destructive diseases of the lung like lung cancer or fibrosis are still often lethal. Also in case of fibrosis in the liver, the only possible cure is transplantation.
In this thesis, we investigate 3D micro computed synchrotron radiation (SR\( \mu \)CT) images of capillary blood vessels in mouse lungs and livers. The specimen show so-called compensatory lung growth as well as different states of pulmonary and hepatic fibrosis.
During compensatory lung growth, after resecting part of the lung, the remaining part compensates for this loss by extending into the empty space. This process is accompanied by an active vessel growing.
In general, the human lung can not compensate for such a loss. Thus, understanding this process in mice is important to improve treatment options in case of diseases like lung cancer.
In case of fibrosis, the formation of scars within the organ's tissue forces the capillary vessels to grow to ensure blood supply.
Thus, the process of fibrosis as well as compensatory lung growth can be accessed by considering the capillary architecture.
As preparation of 2D microscopic images is faster, easier, and cheaper compared to SR\( \mu \)CT images, they currently form the basis of medical investigation. Yet, characteristics like direction and shape of objects can only properly be analyzed using 3D imaging techniques. Hence, analyzing SR\( \mu \)CT data provides valuable additional information.
For the fibrotic specimen, we apply image analysis methods well-known from material science. We measure the vessel diameter using the granulometry distribution function and describe the inter-vessel distance by the spherical contact distribution. Moreover, we estimate the directional distribution of the capillary structure. All features turn out to be useful to characterize fibrosis based on the deformation of capillary vessels.
It is already known that the most efficient mechanism of vessel growing forms small torus-shaped holes within the capillary structure, so-called intussusceptive pillars. Analyzing their location and number strongly contributes to the characterization of vessel growing. Hence, for all three applications, this is of great interest. This thesis provides the first algorithm to detect intussusceptive pillars in SR\( \mu \)CT images. After segmentation of raw image data, our algorithm works automatically and allows for a quantitative evaluation of a large amount of data.
The analysis of SR\( \mu \)CT data using our pillar algorithm as well as the granulometry, spherical contact distribution, and directional analysis extends the current state-of-the-art in medical studies. Although it is not possible to replace certain 3D features by 2D features without losing information, our results could be used to examine 2D features approximating the 3D findings reasonably well.

Magnetoelastic coupling describes the mutual dependence of the elastic and magnetic fields and can be observed in certain types of materials, among which are the so-called "magnetostrictive materials". They belong to the large class of "smart materials", which change their shape, dimensions or material properties under the influence of an external field. The mechanical strain or deformation a material experiences due to an externally applied magnetic field is referred to as magnetostriction; the reciprocal effect, i.e. the change of the magnetization of a body subjected to mechanical stress is called inverse magnetostriction. The coupling of mechanical and electromagnetic fields is particularly observed in "giant magnetostrictive materials", alloys of ferromagnetic materials that can exhibit several thousand times greater magnitudes of magnetostriction (measured as the ratio of the change in length of the material to its original length) than the common magnetostrictive materials. These materials have wide applications areas: They are used as variable-stiffness devices, as sensors and actuators in mechanical systems or as artificial muscles. Possible application fields also include robotics, vibration control, hydraulics and sonar systems.
Although the computational treatment of coupled problems has seen great advances over the last decade, the underlying problem structure is often not fully understood nor taken into account when using black box simulation codes. A thorough analysis of the properties of coupled systems is thus an important task.
The thesis focuses on the mathematical modeling and analysis of the coupling effects in magnetostrictive materials. Under the assumption of linear and reversible material behavior with no magnetic hysteresis effects, a coupled magnetoelastic problem is set up using two different approaches: the magnetic scalar potential and vector potential formulations. On the basis of a minimum energy principle, a system of partial differential equations is derived and analyzed for both approaches. While the scalar potential model involves only stationary elastic and magnetic fields, the model using the magnetic vector potential accounts for different settings such as the eddy current approximation or the full Maxwell system in the frequency domain.
The distinctive feature of this work is the analysis of the obtained coupled magnetoelastic problems with regard to their structure, strong and weak formulations, the corresponding function spaces and the existence and uniqueness of the solutions. We show that the model based on the magnetic scalar potential constitutes a coupled saddle point problem with a penalty term. The main focus in proving the unique solvability of this problem lies on the verification of an inf-sup condition in the continuous and discrete cases. Furthermore, we discuss the impact of the reformulation of the coupled constitutive equations on the structure of the coupled problem and show that in contrast to the scalar potential approach, the vector potential formulation yields a symmetric system of PDEs. The dependence of the problem structure on the chosen formulation of the constitutive equations arises from the distinction of the energy and coenergy terms in the Lagrangian of the system. While certain combinations of the elastic and magnetic variables lead to a coupled magnetoelastic energy function yielding a symmetric problem, the use of their dual variables results in a coupled coenergy function for which a mixed problem is obtained.
The presented models are supplemented with numerical simulations carried out with MATLAB for different examples including a 1D Euler-Bernoulli beam under magnetic influence and a 2D magnetostrictive plate in the state of plane stress. The simulations are based on material data of Terfenol-D, a giant magnetostrictive materials used in many industrial applications.