## Fachbereich Mathematik

### Filtern

#### Erscheinungsjahr

#### Dokumenttyp

- Dissertation (225) (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)
- 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)
- 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)
- 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)
- 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)
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#### Fachbereich / Organisatorische Einheit

- Fachbereich Mathematik (225)
- Fraunhofer (ITWM) (2)

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.

The overall goal of the work is to simulate rarefied flows inside geometries with moving boundaries. The behavior of a rarefied flow is characterized through the Knudsen number \(Kn\), which can be very small (\(Kn < 0.01\) continuum flow) or larger (\(Kn > 1\) molecular flow). The transition region (\(0.01 < Kn < 1\)) is referred to as the transition flow regime.
Continuum flows are mainly simulated by using commercial CFD methods, which are used to solve the Euler equations. In the case of molecular flows one uses statistical methods, such as the Direct Simulation Monte Carlo (DSMC) method. In the transition region Euler equations are not adequate to model gas flows. Because of the rapid increase of particle collisions the DSMC method tends to fail, as well
Therefore, we develop a deterministic method, which is suitable to simulate problems of rarefied gases for any Knudsen number and is appropriate to simulate flows inside geometries with moving boundaries. Thus, the method we use is the Finite Pointset Method (FPM), which is a mesh-free numerical method developed at the ITWM Kaiserslautern and is mainly used to solve fluid dynamical problems.
More precisely, we develop a method in the FPM framework to solve the BGK model equation, which is a simplification of the Boltzmann equation. This equation is mainly used to describe rarefied flows.
The FPM based method is implemented for one and two dimensional physical and velocity space and different ranges of the Knudsen number. Numerical examples are shown for problems with moving boundaries. It is seen, that our method is superior to regular grid methods with respect to the implementation of boundary conditions. Furthermore, our results are comparable to reference solutions gained through CFD- and DSMC methods, respectevly.

In this work we study and investigate the minimum width annulus problem (MWAP), the circle center location or circle location problem (CLP) and the point center location or point location problem (PLP) on Rectilinear and Chebyshev planes as well as in networks. The relations between the problems have served as a basis for finding of elegant solution, algorithms for both new and well known problems. So, MWAP was formulated and investigated in Rectilinear space. In contrast to Euclidean metric, MWAP and PLP have at least one common optimal point. Therefore, MWAP on Rectilinear plane was solved in linear time with the help of PLP. Hence, the solution sequence was PLP-->MWAP. It was shown, that MWAP and CLP are equivalent. Thus, CLP can be also solved in linear time. The obtained results were analysed and transfered to Chebyshev metric. After that, the notions of circle, sphere and annulus in networks were introduced. It should be noted that the notion of a circle in a network is different from the notion of a cycle. An O(mn) time algorithm for solution of MWAP was constructed and implemented. The algorithm is based on the fact that the middle point of an edge represents an optimal solution of a local minimum width annulus on this edge. The resulting complexity is better than the complexity O(mn+n^2logn) in unweighted case of the fastest known algorithm for minimizing of the range function, which is mathematically equivalent to MWAP. MWAP in unweighted undirected networks was extended to the MWAP on subsets and to the restricted MWAP. Resulting problems were analysed and solved. Also the p–minimum width annulus problem was formulated and explored. This problem is NP–hard. However, the p–MWAP has been solved in polynomial O(m^2n^3p) time with a natural assumption, that each minimum width annulus covers all vertexes of a network having distances to the central point of annulus less than or equal to the radius of its outer circle. In contrast to the planar case MWAP in undirected unweighted networks have appeared to be a root problem among considered problems. During investigation of properties of circles in networks it was shown that the difference between planar and network circles is significant. This leads to the nonequivalence of CLP and MWAP in the general case. However, MWAP was effectively used in solution procedures for CLP giving the sequence MWAP-->CLP. The complexity of the developed and implemented algorithm is of order O(m^2n^2). It is important to mention that CLP in networks has been formulated for the first time in this work and differs from the well–studied location of cycles in networks. We have constructed an O(mn+n^2logn) algorithm for well–known PLP. The complexity of this algorithm is not worse than the complexity of the currently best algorithms. But the concept of the solution procedure is new – we use MWAP in order to solve PLP building the opposite to the planar case solution sequence MWAP-->PLP and this method has the following advantages: First, the lower bounds LB obtained in the solution procedure are proved to be in any case better than the strongest Halpern’s lower bound. Second, the developed algorithm is so simple that it can be easily applied to complex networks manually. Third, the empirical complexity of the algorithm is equal to O(mn). MWAP was extended to and explored in directed unweighted and weighted networks. The complexity bound O(n^2) of the developed algorithm for finding of the center of a minimum width annulus in the unweighted case does not depend on the number of edges in a network, because the problems can be solved in the order PLP-->MWAP. In the weighted case computational time is of order O(mn^2).

Image restoration and enhancement methods that respect important features such as edges play a fundamental role in digital image processing. In the last decades a large
variety of methods have been proposed. Nevertheless, the correct restoration and
preservation of, e.g., sharp corners, crossings or texture in images is still a challenge, in particular in the presence of severe distortions. Moreover, in the context of image denoising many methods are designed for the removal of additive Gaussian noise and their adaptation for other types of noise occurring in practice requires usually additional efforts.
The aim of this thesis is to contribute to these topics and to develop and analyze new
methods for restoring images corrupted by different types of noise:
First, we present variational models and diffusion methods which are particularly well
suited for the restoration of sharp corners and X junctions in images corrupted by
strong additive Gaussian noise. For their deduction we present and analyze different
tensor based methods for locally estimating orientations in images and show how to
successfully incorporate the obtained information in the denoising process. The advantageous
properties of the obtained methods are shown theoretically as well as by
numerical experiments. Moreover, the potential of the proposed methods is demonstrated
for applications beyond image denoising.
Afterwards, we focus on variational methods for the restoration of images corrupted
by Poisson and multiplicative Gamma noise. Here, different methods from the literature
are compared and the surprising equivalence between a standard model for
the removal of Poisson noise and a recently introduced approach for multiplicative
Gamma noise is proven. Since this Poisson model has not been considered for multiplicative
Gamma noise before, we investigate its properties further for more general
regularizers including also nonlocal ones. Moreover, an efficient algorithm for solving
the involved minimization problems is proposed, which can also handle an additional
linear transformation of the data. The good performance of this algorithm is demonstrated
experimentally and different examples with images corrupted by Poisson and
multiplicative Gamma noise are presented.
In the final part of this thesis new nonlocal filters for images corrupted by multiplicative
noise are presented. These filters are deduced in a weighted maximum likelihood
estimation framework and for the definition of the involved weights a new similarity measure for the comparison of data corrupted by multiplicative noise is applied. The
advantageous properties of the new measure are demonstrated theoretically and by
numerical examples. Besides, denoising results for images corrupted by multiplicative
Gamma and Rayleigh noise show the very good performance of the new filters.

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.

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.

Nowadays one of the major objectives in geosciences is the determination of the gravitational field of our planet, the Earth. A precise knowledge of this quantity is not just interesting on its own but it is indeed a key point for a vast number of applications. The important question is how to obtain a good model for the gravitational field on a global scale. The only applicable solution - both in costs and data coverage - is the usage of satellite data. We concentrate on highly precise measurements which will be obtained by GOCE (Gravity Field and Steady State Ocean Circulation Explorer, launch expected 2006). This satellite has a gradiometer onboard which returns the second derivatives of the gravitational potential. Mathematically seen we have to deal with several obstacles. The first one is that the noise in the different components of these second derivatives differs over several orders of magnitude, i.e. a straightforward solution of this outer boundary value problem will not work properly. Furthermore we are not interested in the data at satellite height but we want to know the field at the Earth's surface, thus we need a regularization (downward-continuation) of the data. These two problems are tackled in the thesis and are now described briefly. Split Operators: We have to solve an outer boundary value problem at the height of the satellite track. Classically one can handle first order side conditions which are not tangential to the surface and second derivatives pointing in the radial direction employing integral and pseudo differential equation methods. We present a different approach: We classify all first and purely second order operators which fulfill that a harmonic function stays harmonic under their application. This task is done by using modern algebraic methods for solving systems of partial differential equations symbolically. Now we can look at the problem with oblique side conditions as if we had ordinary i.e. non-derived side conditions. The only additional work which has to be done is an inversion of the differential operator, i.e. integration. In particular we are capable to deal with derivatives which are tangential to the boundary. Auto-Regularization: The second obstacle is finding a proper regularization procedure. This is complicated by the fact that we are facing stochastic rather than deterministic noise. The main question is how to find an optimal regularization parameter which is impossible without any additional knowledge. However we could show that with a very limited number of additional information, which are obtainable also in practice, we can regularize in an asymptotically optimal way. In particular we showed that the knowledge of two input data sets allows an order optimal regularization procedure even under the hard conditions of Gaussian white noise and an exponentially ill-posed problem. A last but rather simple task is combining data from different derivatives which can be done by a weighted least squares approach using the information we obtained out of the regularization procedure. A practical application to the downward-continuation problem for simulated gravitational data is shown.

In this dissertation, we discuss how to price American-style options. Our aim is to study and improve the regression-based Monte Carlo methods. In order to have good benchmarks to compare with them, we also study the tree methods.
In the second chapter, we investigate the tree methods specifically. We do research firstly within the Black-Scholes model and then within the Heston model. In the Black-Scholes model, based on Müller's work, we illustrate how to price one dimensional and multidimensional American options, American Asian options, American lookback options, American barrier options and so on. In the Heston model, based on Sayer's research, we implement his algorithm to price one dimensional American options. In this way, we have good benchmarks of various American-style options and put them all in the appendix.
In the third chapter, we focus on the regression-based Monte Carlo methods theoretically and numerically. Firstly, we introduce two variations, the so called "Tsitsiklis-Roy method" and the "Longstaff-Schwartz method". Secondly, we illustrate the approximation of American option by its Bermudan counterpart. Thirdly we explain the source of low bias and high bias. Fourthly we compare these two methods using in-the-money paths and all paths. Fifthly, we examine the effect using different number and form of basis functions. Finally, we study the Andersen-Broadie method and present the lower and upper bounds.
In the fourth chapter, we study two machine learning techniques to improve the regression part of the Monte Carlo methods: Gaussian kernel method and kernel-based support vector machine. In order to choose a proper smooth parameter, we compare fixed bandwidth, global optimum and suboptimum from a finite set. We also point out that scaling the training data to [0,1] can avoid numerical difficulty. When out-of-sample paths of stock prices are simulated, the kernel method is robust and even performs better in several cases than the Tsitsiklis-Roy method and the Longstaff-Schwartz method. The support vector machine can keep on improving the kernel method and needs less representations of old stock prices during prediction of option continuation value for a new stock price.
In the fifth chapter, we switch to the hardware (FGPA) implementation of the Longstaff-Schwartz method and propose novel reversion formulas for the stock price and volatility within the Black-Scholes and Heston models. The test for this formula within the Black-Scholes model shows that the storage of data is reduced and also the corresponding energy consumption.

This thesis, whose subject is located in the field of algorithmic commutative algebra and algebraic geometry, consists of three parts.
The first part is devoted to parallelization, a technique which allows us to take advantage of the computational power of modern multicore processors. First, we present parallel algorithms for the normalization of a reduced affine algebra A over a perfect field. Starting from the algorithm of Greuel, Laplagne, and Seelisch, we propose two approaches. For the local-to-global approach, we stratify the singular locus Sing(A) of A, compute the normalization locally at each stratum and finally reconstruct the normalization of A from the local results. For the second approach, we apply modular methods to both the global and the local-to-global normalization algorithm.
Second, we propose a parallel version of the algorithm of Gianni, Trager, and Zacharias for primary decomposition. For the parallelization of this algorithm, we use modular methods for the computationally hardest steps, such as for the computation of the associated prime ideals in the zero-dimensional case and for the standard bases computations. We then apply an innovative fast method to verify that the result is indeed a primary decomposition of the input ideal. This allows us to skip the verification step at each of the intermediate modular computations.
The proposed parallel algorithms are implemented in the open-source computer algebra system SINGULAR. The implementation is based on SINGULAR's new parallel framework which has been developed as part of this thesis and which is specifically designed for applications in mathematical research.
In the second part, we propose new algorithms for the computation of syzygies, based on an in-depth analysis of Schreyer's algorithm. Here, the main ideas are that we may leave out so-called "lower order terms" which do not contribute to the result of the algorithm, that we do not need to order the terms of certain module elements which occur at intermediate steps, and that some partial results can be cached and reused.
Finally, the third part deals with the algorithmic classification of singularities over the real numbers. First, we present a real version of the Splitting Lemma and, based on the classification theorems of Arnold, algorithms for the classification of the simple real singularities. In addition to the algorithms, we also provide insights into how real and complex singularities are related geometrically. Second, we explicitly describe the structure of the equivalence classes of the unimodal real singularities of corank 2. We prove that the equivalences are given by automorphisms of a certain shape. Based on this theorem, we explain in detail how the structure of the equivalence classes can be computed using SINGULAR and present the results in concise form. The probably most surprising outcome is that the real singularity type \(J_{10}^-\) is actually redundant.

In the first part of the thesis we develop the theory of standard bases in free modules over (localized) polynomial rings. Given that linear equations are solvable in the coefficients of the polynomials, we introduce an algorithm to compute standard bases with respect to arbitrary (module) monomial orderings. Moreover, we take special care to principal ideal rings, allowing zero divisors. For these rings we design modified algorithms which are new and much faster than the general ones. These algorithms were motivated by current limitations in formal verification of microelectronic System-on-Chip designs. We show that our novel approach using computational algebra is able to overcome these limitations in important classes of applications coming from industrial challenges.
The second part is based on research in collaboration with Jason Morton, Bernd Sturmfels and Anne Shiu. We devise a general method to describe and compute a certain class of rank tests motivated by statistics. The class of rank tests may loosely be described as being based on computing the number of linear extensions to given partial orders. In order to apply these tests to actual data we developed two algorithms and used our implementations to apply the methodology to gene expression data created at the Stowers Institute for Medical Research. The dataset is concerned with the development of the vertebra. Our rankings proved valuable to the biologists.

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.

This thesis builds a bridge between singularity theory and computer algebra. To an isolated hypersurface singularity one can associate a regular meromorphic connection, the Gauß-Manin connection, containing a lattice, the Brieskorn lattice. The leading terms of the Brieskorn lattice with respect to the weight and V-filtration of the Gauß-Manin connection define the spectral pairs. They correspond to the Hodge numbers of the mixed Hodge structure on the cohomology of the Milnor fibre and belong to the finest known invariants of isolated hypersurface singularities. The differential structure of the Brieskorn lattice can be described by two complex endomorphisms A0 and A1 containing even more information than the spectral pairs. In this thesis, an algorithmic approach to the Brieskorn lattice in the Gauß-Manin connection is presented. It leads to algorithms to compute the complex monodromy, the spectral pairs, and the differential structure of the Brieskorn lattice. These algorithms are implemented in the computer algebra system Singular.

In the first part of this thesis we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus is the study of moduli spaces, where the underlying combinatorics of the varieties involved allow a much more efficient way of computing certain tropical cycles. The algorithms discussed here have been implemented in an extension for polymake, a software for polyhedral computations.
In the second part we apply the algorithmic toolkit developed in the first part to the study of tropical double Hurwitz cycles. Hurwitz cycles are a higher-dimensional generalization of Hurwitz numbers, which count covers of \(\mathbb{P}^1\) by smooth curves of a given genus with a certain fixed ramification behaviour. Double Hurwitz numbers provide a strong connection between various mathematical disciplines, including algebraic geometry, representation theory and combinatorics. The tropical cycles have a rather complex combinatorial nature, so it is very difficult to study them purely "by hand". Being able to compute examples has been very helpful
in coming up with theoretical results. Our main result states that all marked and unmarked Hurwitz cycles are connected in codimension one and that for a generic choice of simple ramification points the marked cycle is a multiple of an irreducible cycle. In addition we provide computational examples to show that this is the strongest possible statement.

This thesis contains the mathematical treatment of a special class of analog microelectronic circuits called translinear circuits. The goal is to provide foundations of a new coherent synthesis approach for this class of circuits. The mathematical methods of the suggested synthesis approach come from graph theory, combinatorics, and from algebraic geometry, in particular symbolic methods from computer algebra. Translinear circuits form a very special class of analog circuits, because they rely on nonlinear device models, but still allow a very structured approach to network analysis and synthesis. Thus, translinear circuits play the role of a bridge between the "unknown space" of nonlinear circuit theory and the very well exploited domain of linear circuit theory. The nonlinear equations describing the behavior of translinear circuits possess a strong algebraic structure that is nonetheless flexible enough for a wide range of nonlinear functionality. Furthermore, translinear circuits offer several technical advantages like high functional density, low supply voltage and insensitivity to temperature. This unique profile is the reason that several authors consider translinear networks as the key to systematic synthesis methods for nonlinear circuits. The thesis proposes the usage of a computer-generated catalog of translinear network topologies as a synthesis tool. The idea to compile such a catalog has grown from the observation that on the one hand, the topology of a translinear network must satisfy strong constraints which severely limit the number of "admissible" topologies, in particular for networks with few transistors, and on the other hand, the topology of a translinear network already fixes its essential behavior, at least for static networks, because the so-called translinear principle requires the continuous parameters of all transistors to be the same. Even though the admissible topologies are heavily restricted, it is a highly nontrivial task to compile such a catalog. Combinatorial techniques have been adapted to undertake this task. In a catalog of translinear network topologies, prototype network equations can be stored along with each topology. When a circuit with a specified behavior is to be designed, one can search the catalog for a network whose equations can be matched with the desired behavior. In this context, two algebraic problems arise: To set up a meaningful equation for a network in the catalog, an elimination of variables must be performed, and to test whether a prototype equation from the catalog and a specified equation of desired behavior can be "matched", a complex system of polynomial equations must be solved, where the solutions are restricted to a finite set of integers. Sophisticated algorithms from computer algebra are applied in both cases to perform the symbolic computations. All mentioned algorithms have been implemented using C++, Singular, and Mathematica, and are successfully applied to actual design problems of humidity sensor circuitry at Analog Microelectronics GmbH, Mainz. As result of the research conducted, an exhaustive catalog of all static formal translinear networks with at most eight transistors is available. The application for the humidity sensor system proves the applicability of the developed synthesis approach. The details and implementations of the algorithms are worked out only for static networks, but can easily be adopted for dynamic networks as well. While the implementation of the combinatorial algorithms is stand-alone software written "from scratch" in C++, the implementation of the algebraic algorithms, namely the symbolic treatment of the network equations and the match finding, heavily rely on the sophisticated Gröbner basis engine of Singular and thus on more than a decade of experience contained in a special-purpose computer algebra system. It should be pointed out that the thesis contains the new observation that the translinear loop equations of a translinear network are precisely represented by the toric ideal of the network's translinear digraph. Altogether, this thesis confirms and strengthenes the key role of translinear circuits as systematically designable nonlinear circuits.

Advantage of Filtering for Portfolio Optimization in Financial Markets with Partial Information
(2016)

In a financial market we consider three types of investors trading with a finite
time horizon with access to a bank account as well as multliple stocks: the
fully informed investor, the partially informed investor whose only source of
information are the stock prices and an investor who does not use this infor-
mation. The drift is modeled either as following linear Gaussian dynamics
or as being a continuous time Markov chain with finite state space. The
optimization problem is to maximize expected utility of terminal wealth.
The case of partial information is based on the use of filtering techniques.
Conditions to ensure boundedness of the expected value of the filters are
developed, in the Markov case also for positivity. For the Markov modulated
drift, boundedness of the expected value of the filter relates strongly to port-
folio optimization: effects are studied and quantified. The derivation of an
equivalent, less dimensional market is presented next. It is a type of Mutual
Fund Theorem that is shown here.
Gains and losses eminating from the use of filtering are then discussed in
detail for different market parameters: For infrequent trading we find that
both filters need to comply with the boundedness conditions to be an advan-
tage for the investor. Losses are minimal in case the filters are advantageous.
At an increasing number of stocks, again boundedness conditions need to be
met. Losses in this case depend strongly on the added stocks. The relation
of boundedness and portfolio optimization in the Markov model leads here to
increasing losses for the investor if the boundedness condition is to hold for
all numbers of stocks. In the Markov case, the losses for different numbers
of states are negligible in case more states are assumed then were originally
present. Assuming less states leads to high losses. Again for the Markov
model, a simplification of the complex optimal trading strategy for power
utility in the partial information setting is shown to cause only minor losses.
If the market parameters are such that shortselling and borrowing constraints
are in effect, these constraints may lead to big losses depending on how much
effect the constraints have. They can though also be an advantage for the
investor in case the expected value of the filters does not meet the conditions
for boundedness.
All results are implemented and illustrated with the corresponding numerical
findings.

Diese Arbeit gehört in die algebraische Geometrie und die Darstellungstheorie und stellt eine Beziehung zwischen beiden Gebieten dar. Man beschäftigt sich mit den abgeleiteten Kategorien auf flachen Entartungen projektiver Geraden und elliptischer Kurven. Als Mittel benutzt man die Technik der Matrixprobleme. Das Hauptergebnis dieser Dissertation ist der folgende Satz: SATZ. Sei X ein Zykel projektiver Geraden. Dann gibt es drei Typen unzerlegbarer Objekte in D^-(Coh_X): - Shifts von Wolkenkratzergarben in einem regulären Punkt; - Bänder B(w,m,lambda), - Saiten S(w). Ganz analog beweist man die Zahmheit der abgeleiteten Kategorien vieler assoziativer Algebren.

This dissertation is intended to transport the theory of Serre functors into the context of A-infinity-categories. We begin with an introduction to multicategories and closed multicategories, which form a framework in which the theory of A-infinity-categories is developed. We prove that (unital) A-infinity-categories constitute a closed symmetric multicategory. We define the notion of A-infinity-bimodule similarly to Tradler and show that it is equivalent to an A-infinity-functor of two arguments which takes values in the differential graded category of complexes of k-modules, where k is a commutative ground ring. Serre A-infinity-functors are defined via A-infinity-bimodules following ideas of Kontsevich and Soibelman. We prove that a unital closed under shifts A-infinity-category over a field admits a Serre A-infinity-functor if and only if its homotopy category admits an ordinary Serre functor. The proof uses categories and Serre functors enriched in the homotopy category of complexes of k-modules. Another important ingredient is an A-infinity-version of the Yoneda Lemma.

The interest of the exploration of new hydrocarbon fields as well as deep geothermal reservoirs is permanently growing. The analysis of seismic data specific for such exploration projects is very complex and requires the deep knowledge in geology, geophysics, petrology, etc from interpreters, as well as the ability of advanced tools that are able to recover some particular properties. There again the existing wavelet techniques have a huge success in signal processing, data compression, noise reduction, etc. They enable to break complicate functions into many simple pieces at different scales and positions that makes detection and interpretation of local events significantly easier.
In this thesis mathematical methods and tools are presented which are applicable to the seismic data postprocessing in regions with non-smooth boundaries. We provide wavelet techniques that relate to the solutions of the Helmholtz equation. As application we are interested in seismic data analysis. A similar idea to construct wavelet functions from the limit and jump relations of the layer potentials was first suggested by Freeden and his Geomathematics Group.
The particular difficulty in such approaches is the formulation of limit and
jump relations for surfaces used in seismic data processing, i.e., non-smooth
surfaces in various topologies (for example, uniform and
quadratic). The essential idea is to replace the concept of parallel surfaces known for a smooth regular surface by certain appropriate substitutes for non-smooth surfaces.
By using the jump and limit relations formulated for regular surfaces, Helmholtz wavelets can be introduced that recursively approximate functions on surfaces with edges and corners. The exceptional point is that the construction of wavelets allows the efficient implementation in form of
a tree algorithm for the fast numerical computation of functions on the boundary.
In order to demonstrate the
applicability of the Helmholtz FWT, we study a seismic image obtained by the reverse time migration which is based on a finite-difference implementation. In fact, regarding the requirements of such migration algorithms in filtering and denoising the wavelet decomposition is successfully applied to this image for the attenuation of low-frequency
artifacts and noise. Essential feature is the space localization property of
Helmholtz wavelets which numerically enables to discuss the velocity field in
pointwise dependence. Moreover, the multiscale analysis leads us to reveal additional geological information from optical features.

In this thesis, we investigate a statistical model for precipitation time series recorded at a single site. The sequence of observations consists of rainfall amounts aggregated over time periods of fixed duration. As the properties of this sequence depend strongly on the length of the observation intervals, we follow the approach of Rodriguez-Iturbe et. al. [1] and use an underlying model for rainfall intensity in continuous time. In this idealized representation, rainfall occurs in clusters of rectangular cells, and each observations is treated as the sum of cell contributions during a given time period. Unlike the previous work, we use a multivariate lognormal distribution for the temporal structure of the cells and clusters. After formulating the model, we develop a Markov-Chain Monte-Carlo algorithm for fitting it to a given data set. A particular problem we have to deal with is the need to estimate the unobserved intensity process alongside the parameter of interest. The performance of the algorithm is tested on artificial data sets generated from the model. [1] I. Rodriguez-Iturbe, D. R. Cox, and Valerie Isham. Some models for rainfall based on stochastic point processes. Proc. R. Soc. Lond. A, 410:269-288, 1987.

The dissertation is concerned with the numerical solution of Fokker-Planck equations in high dimensions arising in the study of dynamics of polymeric liquids. Traditional methods based on tensor product structure are not applicable in high dimensions for the number of nodes required to yield a fixed accuracy increases exponentially with the dimension; a phenomenon often referred to as the curse of dimension. Particle methods or finite point set methods are known to break the curse of dimension. The Monte Carlo method (MCM) applied to such problems are 1/sqrt(N) accurate, where N is the cardinality of the point set considered, independent of the dimension. Deterministic version of the Monte Carlo method called the quasi Monte Carlo method (QMC) are quite effective in integration problems and accuracy of the order of 1/N can be achieved, up to a logarithmic factor. However, such a replacement cannot be carried over to particle simulations due to the correlation among the quasi-random points. The method proposed by Lecot (C.Lecot and F.E.Khettabi, Quasi-Monte Carlo simulation of diffusion, Journal of Complexity, 15 (1999), pp.342-359) is the only known QMC approach, but it not only leads to large particle numbers but also the proven order of convergence is 1/N^(2s) in dimension s. We modify the method presented there, in such a way that the new method works with reasonable particle numbers even in high dimensions and has better order of convergence. Though the provable order of convergence is 1/sqrt(N), the results show less variance and thus the proposed method still slightly outperforms standard MCM.

A Multi-Phase Flow Model Incorporated with Population Balance Equation in a Meshfree Framework
(2011)

This study deals with the numerical solution of a meshfree coupled model of Computational Fluid Dynamics (CFD) and Population Balance Equation (PBE) for liquid-liquid extraction columns. In modeling the coupled hydrodynamics and mass transfer in liquid extraction columns one encounters multidimensional population balance equation that could not be fully resolved numerically within a reasonable time necessary for steady state or dynamic simulations. For this reason, there is an obvious need for a new liquid extraction model that captures all the essential physical phenomena and still tractable from computational point of view. This thesis discusses a new model which focuses on discretization of the external (spatial) and internal coordinates such that the computational time is drastically reduced. For the internal coordinates, the concept of the multi-primary particle method; as a special case of the Sectional Quadrature Method of Moments (SQMOM) is used to represent the droplet internal properties. This model is capable of conserving the most important integral properties of the distribution; namely: the total number, solute and volume concentrations and reduces the computational time when compared to the classical finite difference methods, which require many grid points to conserve the desired physical quantities. On the other hand, due to the discrete nature of the dispersed phase, a meshfree Lagrangian particle method is used to discretize the spatial domain (extraction column height) using the Finite Pointset Method (FPM). This method avoids the extremely difficult convective term discretization using the classical finite volume methods, which require a lot of grid points to capture the moving fronts propagating along column height.

In the filling process of a car tank, the formation of foam plays an unwanted role, as it may prevent the tank from being completely filled or at least delay the filling. Therefore it is of interest to optimize the geometry of the tank using numerical simulation in such a way that the influence of the foam is minimized. In this dissertation, we analyze the behaviour of the foam mathematically on the mezoscopic scale, that is for single lamellae. The most important goals are on the one hand to gain a deeper understanding of the interaction of the relevant physical effects, on the other hand to obtain a model for the simulation of the decay of a lamella which can be integrated in a global foam model. In the first part of this work, we give a short introduction into the physical properties of foam and find that the Marangoni effect is the main cause for its stability. We then develop a mathematical model for the simulation of the dynamical behaviour of a lamella based on an asymptotic analysis using the special geometry of the lamella. The result is a system of nonlinear partial differential equations (PDE) of third order in two spatial and one time dimension. In the second part, we analyze this system mathematically and prove an existence and uniqueness result for a simplified case. For some special parameter domains the system can be further simplified, and in some cases explicit solutions can be derived. In the last part of the dissertation, we solve the system using a finite element approach and discuss the results in detail.

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 \(\tilde{h}=1\).

The various uses of fiber-reinforced composites, for example in the enclosures of planes, boats and cars, generates the demand for a detailed analysis of these materials. The final goal is to optimize fibrous materials by the means of “virtual material design”. New fibrous materials are virtually created as realizations of a stochastic model and evaluated with physical simulations. In that way, materials can be optimized for specific use cases, without constructing expensive prototypes or performing mechanical experiments. In order to design a practically fabricable material, the stochastic model is first adapted to an existing material and then slightly modified. The virtual reconstruction of the existing material requires a precise knowledge of the geometry of its microstructure. The first part of this thesis describes a fiber quantification method by the means of local measurements of the fiber radius and orientation. The combination of a sparse chord length transform and inertia moments leads to an efficient and precise new algorithm. It outperforms existing approaches with the possibility to treat different fiber radii within one sample, with high precision in continuous space and comparably fast computing time. This local quantification method can be directly applied on gray value images by adapting the directional distance transforms on gray values. In this work, several approaches of this kind are developed and evaluated. Further characterization of the fiber system requires a segmentation of each single fiber. Using basic morphological operators with specific structuring elements, it is possible to derive a probability for each pixel describing if the pixel belongs to a fiber core in a region without overlapping fibers. Tracking high probabilities leads to a partly reconstruction of the fiber cores in non crossing regions. These core parts are then reconnected over critical regions, if they fulfill certain conditions ensuring the affiliation to the same fiber. In the second part of this work, we develop a new stochastic model for dense systems of non overlapping fibers with a controllable level of bending. Existing approaches in the literature have at least one weakness in either achieving high volume fractions, producing non overlapping fibers, or controlling the bending or the orientation distribution. This gap can be bridged by our stochastic model, which operates in two steps. Firstly, a random walk with the multivariate von Mises-Fisher orientation distribution defines bent fibers. Secondly, a force-biased packing approach arranges them in a non overlapping configuration. Furthermore, we provide the estimation of all parameters needed for the fitting of this model to a real microstructure. Finally, we simulate the macroscopic behavior of different microstructures to derive their mechanical and thermal properties. This part is mostly supported by existing software and serves as a summary of physical simulation applied to random fiber systems. The application on a glass fiber reinforced polymer proves the quality of the reconstruction by our stochastic model, as the effective properties match for both the real microstructure and the realizations of the fitted model. This thesis includes all steps to successfully perform virtual material design on various data sets. With novel and efficient algorithms it contributes to the science of analysis and modeling of fiber reinforced materials.

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.