Kaiserslautern - Fachbereich Mathematik
Refine
Year of publication
Document Type
- Doctoral Thesis (291) (remove)
Has Fulltext
- yes (291)
Keywords
- Algebraische Geometrie (6)
- Portfolio Selection (6)
- Finanzmathematik (5)
- Optimization (5)
- Stochastische dynamische Optimierung (5)
- Elastizität (4)
- Homogenisierung <Mathematik> (4)
- Navier-Stokes-Gleichung (4)
- Numerische Mathematik (4)
- Portfolio-Optimierung (4)
- portfolio optimization (4)
- Bewertung (3)
- Computeralgebra (3)
- Erwarteter Nutzen (3)
- Finite-Volumen-Methode (3)
- Gröbner-Basis (3)
- Inverses Problem (3)
- Monte-Carlo-Simulation (3)
- Mosco convergence (3)
- NURBS (3)
- Numerische Strömungssimulation (3)
- Optionspreistheorie (3)
- Portfolio Optimization (3)
- Portfoliomanagement (3)
- Risikomanagement (3)
- Transaction Costs (3)
- Tropische Geometrie (3)
- Wavelet (3)
- isogeometric analysis (3)
- optimales Investment (3)
- Asymptotic Expansion (2)
- Asymptotik (2)
- B-Spline (2)
- B-splines (2)
- Derivat <Wertpapier> (2)
- Diskrete Fourier-Transformation (2)
- Elasticity (2)
- Endliche Geometrie (2)
- Erdmagnetismus (2)
- FFT (2)
- Filtergesetz (2)
- Filtration (2)
- Finite Pointset Method (2)
- Geometric Ergodicity (2)
- Hamilton-Jacobi-Differentialgleichung (2)
- Hochskalieren (2)
- IMRT (2)
- Isogeometrische Analyse (2)
- Kreditrisiko (2)
- Langevin equation (2)
- Lebensversicherung (2)
- Level-Set-Methode (2)
- Lineare Elastizitätstheorie (2)
- Lineare partielle Differentialgleichung (2)
- Local smoothing (2)
- Mathematik (2)
- Mehrskalenanalyse (2)
- Mehrskalenmodell (2)
- Mikrostruktur (2)
- Modulraum (2)
- Multiset Multicover (2)
- Partial Differential Equations (2)
- Partielle Differentialgleichung (2)
- Poröser Stoff (2)
- Regressionsanalyse (2)
- Regularisierung (2)
- Robust Optimization (2)
- Schnitttheorie (2)
- Statistisches Modell (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)
- mesh generation (2)
- optimal investment (2)
- regression analysis (2)
- splines (2)
- "Slender-Body"-Theorie (1)
- (Joint) chance constraints (1)
- 3D image analysis (1)
- A-infinity-bimodule (1)
- A-infinity-category (1)
- A-infinity-functor (1)
- ALE-Methode (1)
- Ableitungsfreie Optimierung (1)
- Adjoint method (1)
- Advanced Encryption Standard (1)
- Agriculture Loan (1)
- Algebraic dependence of commuting elements (1)
- Algebraic geometry (1)
- Algebraic groups (1)
- Algebraische Abhängigkeit der kommutierende Elementen (1)
- Algebraischer Funktionenkörper (1)
- Analysis (1)
- Angewandte Mathematik (1)
- Annulus (1)
- Anti-diffusion (1)
- Antidiffusion (1)
- Approximationsalgorithmus (1)
- Arbitrage (1)
- Arc distance (1)
- Archimedische Kopula (1)
- Asiatische Option (1)
- Asset allocation (1)
- Asset-liability management (1)
- Asympotic Analysis (1)
- Asymptotic Analysis (1)
- Asymptotische Entwicklung (1)
- Ausfallrisiko (1)
- Automorphismengruppe (1)
- Autoregressive Hilbertian model (1)
- Balance sheet (1)
- Barriers (1)
- Basic Scheme (1)
- Basis Risk (1)
- Basket Option (1)
- Bayes-Entscheidungstheorie (1)
- Beam models (1)
- Beam orientation (1)
- Bernstein–Gelfand–Gelfand construction (1)
- Beschichtungsprozess (1)
- Beschränkte Krümmung (1)
- Betrachtung des Schlimmstmöglichen Falles (1)
- Bilanzstrukturmanagement (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)
- Cluster-Analyse (1)
- Coarse graining (1)
- Cohen-Lenstra heuristic (1)
- Combinatorial Optimization (1)
- Commodity Index (1)
- Complex Structures (1)
- Composite Materials (1)
- Computer Algebra (1)
- Computer Algebra System (1)
- Computer algebra (1)
- Computeralgebra System (1)
- Conditional Value-at-Risk (1)
- Connectivity (1)
- Consistencyanalysis (1)
- Consistent Price Processes (1)
- Constraint Generation (1)
- Construction of hypersurfaces (1)
- Convergence Rate (1)
- Copula (1)
- Coupled PDEs (1)
- Coxeter-Freudenthal-Kuhn triangulation (1)
- Crash (1)
- Crash Hedging (1)
- Crash modelling (1)
- Crashmodellierung (1)
- Credit Default Swap (1)
- Credit Risk (1)
- Curvature (1)
- Curved viscous fibers (1)
- Cycle Decomposition (1)
- DSMC (1)
- Darstellungstheorie (1)
- Das Urbild von Ideal unter einen Morphismus der Algebren (1)
- Debt Management (1)
- Defaultable Options (1)
- Deformationstheorie (1)
- Degenerate Diffusion Semigroups (1)
- Delaunay (1)
- Delaunay triangulation (1)
- Delaunay triangulierung (1)
- Differential forms (1)
- Differenzenverfahren (1)
- Differenzmenge (1)
- Diffusion (1)
- Diffusion processes (1)
- Diffusionsprozess (1)
- Discriminatory power (1)
- Dispersionsrelation (1)
- Dissertation (1)
- Diversifikation (1)
- Druckkorrektur (1)
- Dünnfilmapproximation (1)
- EDF observation models (1)
- EM algorithm (1)
- Edwards Model (1)
- Effective Conductivity (1)
- Efficiency (1)
- Efficient Reliability Estimation (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)
- Essential m-dissipativity (1)
- Expected shortfall (1)
- Exponential Utility (1)
- Exponentieller Nutzen (1)
- Extrapolation (1)
- Extreme Events (1)
- Extreme value theory (1)
- FEM (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-Kopplung (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)
- Garantiezins (1)
- Garbentheorie (1)
- Gebietszerlegung (1)
- Gebietszerlegungsmethode (1)
- Gebogener viskoser Faden (1)
- Geo-referenced data (1)
- Geodesie (1)
- Geometrische Ergodizität (1)
- Gewichteter Sobolev-Raum (1)
- Gittererzeugung (1)
- Gleichgewichtsstrategien (1)
- Gradient based optimization (1)
- Granular flow (1)
- Granulat (1)
- Graph Theory (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)
- Hilbert complexes (1)
- Homogenization (1)
- Homologische Algebra (1)
- Hub Location Problem (1)
- Hydrostatischer Druck (1)
- Hyperelastizität (1)
- Hyperelliptische Kurve (1)
- Hyperflächensingularität (1)
- Hyperspektraler Sensor (1)
- Hypocoercivity (1)
- Hysterese (1)
- ITSM (1)
- Idealklassengruppe (1)
- Illiquidität (1)
- Image restoration (1)
- Immiscible lattice BGK (1)
- Immobilienaktie (1)
- Index Insurance (1)
- Inflation (1)
- Infrarotspektroskopie (1)
- Insurance (1)
- Intensität (1)
- Internationale Diversifikation (1)
- Interpolation Algorithm (1)
- Inverse Problem (1)
- Irreduzibler Charakter (1)
- Isogeometric Analysis (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)
- Least-squares Monte Carlo method (1)
- Level set methods (1)
- Lie algebras (1)
- Lie-Typ-Gruppe (1)
- Lippmann-Schwinger Equation (1)
- Lippmann-Schwinger equation (1)
- Liquidität (1)
- Locally Supported Zonal Kernels (1)
- Location (1)
- MBS (1)
- MKS (1)
- ML-estimation (1)
- Macaulay’s inverse system (1)
- Magneto-Elastic Coupling (1)
- Magnetoelastic coupling (1)
- Magnetoelasticity (1)
- Magnetostriction (1)
- Marangoni-Effekt (1)
- Market Equilibrium (1)
- Markov Chain (1)
- Markov Kette (1)
- Markov-Ketten-Monte-Carlo-Verfahren (1)
- Markov-Prozess (1)
- Marktmanipulation (1)
- Marktrisiko (1)
- Martingaloptimalitätsprinzip (1)
- Maschinelles Lernen (1)
- Mathematical Finance (1)
- Mathematics (1)
- Mathematische Modellierung (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-Conjecture (1)
- McKay-Vermutung (1)
- Mehrdimensionale Bildverarbeitung (1)
- Mehrdimensionales Variationsproblem (1)
- Mehrkriterielle Optimierung (1)
- Mehrskalen (1)
- Microstructure (1)
- Mie- and Helmholtz-Representation (1)
- Mie- und Helmholtz-Darstellung (1)
- Mikroelektronik (1)
- Mixed Connectivity (1)
- Mixed integer programming (1)
- Mixed method (1)
- Model-Dynamics (1)
- Modellbildung (1)
- Molekulardynamik (1)
- Momentum and Mas Transfer (1)
- Monte Carlo (1)
- Moreau-Yosida regularization (1)
- Morphismus (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)
- 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)
- Nilpotent elements (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)
- PDE-Constrained Optimization, Robust Design, Multi-Objective Optimization (1)
- POD (1)
- Papiermaschine (1)
- Parallel Algorithms (1)
- Paralleler Algorithmus (1)
- Partikel Methoden (1)
- Patchworking Methode (1)
- Patchworking method (1)
- Pathwise Optimality (1)
- Pedestrian FLow (1)
- Periodic Homogenization (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)
- Probust optimization (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)
- 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)
- Risikoanalyse (1)
- Risikomaße (1)
- Risikotheorie (1)
- Risk Management (1)
- Risk Measures (1)
- Risk Sharing (1)
- Robust smoothing (1)
- Rohstoffhandel (1)
- Rohstoffindex (1)
- Räumliche Statistik (1)
- SWARM (1)
- Sandwiching algorithm (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)
- Semi-infinite optimization (1)
- Sequenzieller Algorithmus (1)
- Serre functor (1)
- Shallow Water Equations (1)
- Shape optimization (1)
- Simulation (1)
- Singular <Programm> (1)
- Singularity theory (1)
- Singularität (1)
- Singularitätentheorie (1)
- Slender body theory (1)
- Sobolev spaces (1)
- Sobolev-Raum (1)
- Solvency II (1)
- Solvency-II-Richtlinie (1)
- Spannungs-Dehn (1)
- Spatial Statistics (1)
- Spectral Method (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)
- Statistics (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)
- Stochastisches Modell (1)
- Stokes-Gleichung (1)
- Stop- und Spieloperator (1)
- Stornierung (1)
- Stoßdämpfer (1)
- Strahlentherapie (1)
- Strahlungstransport (1)
- Structural Reliability (1)
- Strukturiertes Finanzprodukt (1)
- Strukturoptimierung (1)
- Strömungsdynamik (1)
- Strömungsmechanik (1)
- Subset Simulationen (1)
- Success Run (1)
- Survival Analysis (1)
- Systemidentifikation (1)
- Sägezahneffekt (1)
- Tail Dependence Koeffizient (1)
- Temporal Variational Autoencoders (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-Scale Convergence (1)
- Two-phase flow (1)
- Unreinheitsfunktion (1)
- Untermannigfaltigkeit (1)
- Upwind-Verfahren (1)
- Usage modeling (1)
- Utility (1)
- Value at Risk (1)
- Value at risk (1)
- Value-at-Risk (1)
- Variational autoencoders (1)
- Variationsrechnung (1)
- Vectorfield approximation (1)
- Vektorfeldapproximation (1)
- Vektorkugelfunktionen (1)
- Verschwindungsatz (1)
- Versicherung (1)
- Viskoelastische Flüssigkeiten (1)
- Viskose Transportschemata (1)
- Volatilität (1)
- Volatilitätsarbitrage (1)
- Vorkonditionierer (1)
- Vorwärts-Rückwärts-Stochastische-Differentialgleichung (1)
- Water reservoir management (1)
- Wave Based Method (1)
- Wavelet-Theorie (1)
- Wavelet-Theory (1)
- Weißes Rauschen (1)
- White Noise (1)
- Wirbelabtrennung (1)
- Wirbelströmung (1)
- Wissenschaftliches Rechnen (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)
- adaptive algorithm (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)
- arbitrary Lagrangian-Eulerian methods (ALE) (1)
- archimedean copula (1)
- asian option (1)
- asymptotic-preserving (1)
- auto-pruning (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)
- characteristic polynomial (1)
- closure approximation (1)
- clustering (1)
- clustering methods (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)
- determinant (1)
- diffusion models (1)
- discrepancy (1)
- diversification (1)
- domain parametrization (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)
- fibre lay-down dynamics (1)
- filtration (1)
- financial mathematics (1)
- finite difference schemes (1)
- finite element method (1)
- finite groups of Lie type (1)
- finite spin group (1)
- first hitting time (1)
- float glass (1)
- flood risk (1)
- fluid structure (1)
- fluid structure interaction (1)
- fluid-structure interaction (FSI) (1)
- forward-shooting grid (1)
- free surface (1)
- freie Oberfläche (1)
- gebietszerlegung (1)
- generic character table (1)
- gitter (1)
- glioblastoma (1)
- good semigroup (1)
- graph p-Laplacian (1)
- gravitation (1)
- group action (1)
- groups of Lie type (1)
- großer Investor (1)
- haptotaxis (1)
- hedging (1)
- heuristic (1)
- hierarchical matrix (1)
- hyperbolic systems (1)
- hyperelliptic function field (1)
- hyperelliptische Funktionenkörper (1)
- hyperspectal unmixing (1)
- hypocoercivity (1)
- idealclass group (1)
- image analysis (1)
- image denoising (1)
- impulse control (1)
- impurity functions (1)
- incompressible elasticity (1)
- infinite-dimensional analysis (1)
- infinite-dimensional manifold (1)
- inflation-linked product (1)
- integer programming (1)
- integral constitutive equations (1)
- intensity (1)
- inverse optimization (1)
- inverse problem (1)
- isogeometric analysis (IGA) (1)
- jump-diffusion process (1)
- kernel (1)
- kinetic equations (1)
- large investor (1)
- large scale integer programming (1)
- lattice Boltzmann (1)
- level K-algebras (1)
- level set method (1)
- life insurance (1)
- limit theorems (1)
- linear code (1)
- linear systems (1)
- local-global conjectures (1)
- localizing basis (1)
- longevity bonds (1)
- loss analysis (1)
- low-rank approximation (1)
- machine learning (1)
- macro derivative (1)
- market crash (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)
- mesh deformation (1)
- micromechanics (1)
- minimal polynomial (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)
- multi-scale model (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)
- naive diversification (1)
- network flows (1)
- network synthesis (1)
- netzgenerierung (1)
- nicht-newtonsche Strömungen (1)
- nichtlineare Druckkorrektor (1)
- nichtlineare Modellreduktion (1)
- nichtlineare Netzwerke (1)
- non square linear system solving (1)
- non-desarguesian plane (1)
- non-newtonian flow (1)
- nonconvex optimization (1)
- nonlinear circuits (1)
- nonlinear diffusion filtering (1)
- nonlinear elasticity (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)
- number fields (1)
- numerical irreducible decomposition (1)
- numerical methods (1)
- numerics (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)
- posterior collapse (1)
- potential (1)
- preconditioners (1)
- pressure correction (1)
- primal-dual algorithm (1)
- probability distribution (1)
- projective surfaces (1)
- proximation (1)
- proxy modeling (1)
- quadrinomial tree (1)
- quasi-Monte Carlo (1)
- quasi-variational inequalities (1)
- quasihomogeneity (1)
- quasiregular group (1)
- quasireguläre Gruppe (1)
- radiation therapy (1)
- radiotherapy (1)
- rare disasters (1)
- rate of convergence (1)
- raum-zeitliche Analyse (1)
- real quadratic number fields (1)
- reconstructions (1)
- redundant constraint (1)
- reflectionless boundary condition (1)
- reflexionslose Randbedingung (1)
- regime-shift model (1)
- regularization methods (1)
- rheology (1)
- risk analysis (1)
- risk measures (1)
- risk reduction (1)
- sampling (1)
- sawtooth effect (1)
- scalar and vectorial wavelets (1)
- scaled boundary isogeometric analysis (1)
- scaled boundary parametrizations (1)
- second class group (1)
- seismic tomography (1)
- semigroup of values (1)
- semisprays (1)
- sheaf theory (1)
- similarity measures (1)
- singularities (1)
- sparse interpolation of multivariate rational functions (1)
- sparse multivariate polynomial interpolation (1)
- sparsity (1)
- spherical approximation (1)
- sputtering process (1)
- star-shaped domain (1)
- stochastic arbitrage (1)
- stochastic coefficient (1)
- stochastic optimal control (1)
- stochastic processes (1)
- stochastische Arbitrage (1)
- stop- and play-operator (1)
- stratifolds (1)
- subgradient (1)
- superposed fluids (1)
- surface measures (1)
- surrender options (1)
- surrogate algorithm (1)
- syzygies (1)
- tail dependence coefficient (1)
- tax (1)
- tensions (1)
- time delays (1)
- topological asymptotic expansion (1)
- toric geometry (1)
- torische Geometrie (1)
- total variation (1)
- total variation spatial regularization (1)
- translation invariant spaces (1)
- translinear circuits (1)
- translineare Schaltungen (1)
- transmission conditions (1)
- tropical geometry (1)
- unbeschränktes Potential (1)
- unbounded potential (1)
- unimodular certification (1)
- unimodularity (1)
- value semigroup (1)
- valuing contracts (1)
- variable selection (1)
- variational methods (1)
- variational model (1)
- vector bundles (1)
- vector spherical harmonics (1)
- vectorial wavelets (1)
- vertical velocity (1)
- vertikale Geschwindigkeiten (1)
- viscoelastic fluids (1)
- volatility arbitrage (1)
- vortex seperation (1)
- well-posedness (1)
- worst-case (1)
- worst-case scenario (1)
- Äquisingularität (1)
- Überflutung (1)
- Überflutungsrisiko (1)
- Übergangsbedingungen (1)
Faculty / Organisational entity
Structure and Construction of Instanton Bundles on P3
Abstract
The main theme of this thesis is about Graph Coloring Applications and Defining Sets in Graph Theory.
As in the case of block designs, finding defining sets seems to be difficult problem, and there is not a general conclusion. Hence we confine us here to some special types of graphs like bipartite graphs, complete graphs, etc.
In this work, four new concepts of defining sets are introduced:
• Defining sets for perfect (maximum) matchings
• Defining sets for independent sets
• Defining sets for edge colorings
• Defining set for maximal (maximum) clique
Furthermore, some algorithms to find and construct the defining sets are introduced. A review on some known kinds of defining sets in graph theory is also incorporated, in chapter 2 the basic definitions and some relevant notations used in this work are introduced.
chapter 3 discusses the maximum and perfect matchings and a new concept for a defining set for perfect matching.
Different kinds of graph colorings and their applications are the subject of chapter 4.
Chapter 5 deals with defining sets in graph coloring. New results are discussed along with already existing research results, an algorithm is introduced, which enables to determine a defining set of a graph coloring.
In chapter 6, cliques are discussed. An algorithm for the determination of cliques using their defining sets. Several examples are included.
The study of families of curves with prescribed singularities has a long tradition. Its foundations were laid by Plücker, Severi, Segre, and Zariski at the beginning of the 20th century. Leading to interesting results with applications in singularity theory and in the topology of complex algebraic curves and surfaces it has attained the continuous attraction of algebraic geometers since then. Throughout this thesis we examine the varieties V(D,S1,...,Sr) of irreducible reduced curves in a fixed linear system |D| on a smooth projective surface S over the complex numbers having precisely r singular points of types S1,...,Sr. We are mainly interested in the following three questions: 1) Is V(D,S1,...,Sr) non-empty? 2) Is V(D,S1,...,Sr) T-smooth, that is smooth of the expected dimension? 3) Is V(D,S1,...Sr) irreducible? We would like to answer the questions in such a way that we present numerical conditions depending on invariants of the divisor D and of the singularity types S1,...,Sr, which ensure a positive answer. The main conditions which we derive will be of the type inv(S1)+...+inv(Sr) < aD^2+bD.K+c, where inv is some invariant of singularity types, a, b and c are some constants, and K is some fixed divisor. The case that S is the projective plane has been very well studied by many authors, and on other surfaces some results for curves with nodes and cusps have been derived in the past. We, however, consider arbitrary singularity types, and the results which we derive apply to large classes of surfaces, including surfaces in projective three-space, K3-surfaces, products of curves and geometrically ruled surfaces.
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.
Matrix Compression Methods for the Numerical Solution of Radiative Transfer in Scattering Media
(2002)
Radiative transfer in scattering media is usually described by the radiative transfer equation, an integro-differential equation which describes the propagation of the radiative intensity along a ray. The high dimensionality of the equation leads to a very large number of unknowns when discretizing the equation. This is the major difficulty in its numerical solution. In case of isotropic scattering and diffuse boundaries, the radiative transfer equation can be reformulated into a system of integral equations of the second kind, where the position is the only independent variable. By employing the so-called momentum equation, we derive an integral equation, which is also valid in case of linear anisotropic scattering. This equation is very similar to the equation for the isotropic case: no additional unknowns are introduced and the integral operators involved have very similar mapping properties. The discretization of an integral operator leads to a full matrix. Therefore, due to the large dimension of the matrix in practical applcation, it is not feasible to assemble and store the entire matrix. The so-called matrix compression methods circumvent the assembly of the matrix. Instead, the matrix-vector multiplications needed by iterative solvers are performed only approximately, thus, reducing, the computational complexity tremendously. The kernels of the integral equation describing the radiative transfer are very similar to the kernels of the integral equations occuring in the boundary element method. Therefore, with only slight modifications, the matrix compression methods, developed for the latter are readily applicable to the former. As apposed to the boundary element method, the integral kernels for radiative transfer in absorbing and scattering media involve an exponential decay term. We examine how this decay influences the efficiency of the matrix compression methods. Further, a comparison with the discrete ordinate method shows that discretizing the integral equation may lead to reductions in CPU time and to an improved accuracy especially in case of small absorption and scattering coefficients or if local sources are present.
Different aspects of geomagnetic field modelling from satellite data are examined in the framework of modern multiscale approximation. The thesis is mostly concerned with wavelet techniques, i.e. multiscale methods based on certain classes of kernel functions which are able to realize a multiscale analysis of the funtion (data) space under consideration. It is thus possible to break up complicated functions like the geomagnetic field, electric current densities or geopotentials into different pieces and study these pieces separately. Based on a general approach to scalar and vectorial multiscale methods, topics include multiscale denoising, crustal field approximation and downward continuation, wavelet-parametrizations of the magnetic field in Mie-representation as well as multiscale-methods for the analysis of time-dependent spherical vector fields. For each subject the necessary theoretical framework is established and numerical applications examine and illustrate the practical aspects.
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 present work, we investigated how to correct the questionable normality, linear and quadratic assumptions underlying existing Value-at-Risk methodologies. In order to take also into account the skewness, the heavy tailedness and the stochastic feature of the volatility of the market values of financial instruments, the constant volatility hypothesis widely used by existing Value-at-Risk appproches has also been investigated and corrected and the tails of the financial returns distributions have been handled via Generalized Pareto or Extreme Value Distributions. Artificial Neural Networks have been combined by Extreme Value Theory in order to build consistent and nonparametric Value-at-Risk measures without the need to make any of the questionable assumption specified above. For that, either autoregressive models (AR-GARCH) have been used or the direct characterization of conditional quantiles due to Bassett, Koenker [1978] and Smith [1987]. In order to build consistent and nonparametric Value-at-Risk estimates, we have proved some new results extending White Artificial Neural Network denseness results to unbounded random variables and provide a generalisation of the Bernstein inequality, which is needed to establish the consistency of our new Value-at-Risk estimates. For an accurate estimation of the quantile of the unexpected returns, Generalized Pareto and Extreme Value Distributions have been used. The new Artificial Neural Networks denseness results enable to build consistent, asymptotically normal and nonparametric estimates of conditional means and stochastic volatilities. The denseness results uses the Sobolev metric space L^m (my) for some m >= 1 and some probability measure my and which holds for a certain subclass of square integrable functions. The Fourier transform, the new extension of the Bernstein inequality for unbounded random variables from stationary alpha-mixing processes combined with the new generalization of a result of White and Wooldrige [1990] have been the main tool to establich the extension of White's neural network denseness results. To illustrate the goodness and level of accuracy of the new denseness results, we were able to demonstrate the applicability of the new Value-at-Risk approaches by means of three examples with real financial data mainly from the banking sector traded on the Frankfort Stock Exchange.
One crucial assumption of continuous financial mathematics is that the portfolio can be rebalanced continuously and that there are no transaction costs. In reality, this of course does not work. On the one hand, continuous rebalancing is impossible, on the other hand, each transaction causes costs which have to be subtracted from the wealth. Therefore, we focus on trading strategies which are based on discrete rebalancing - in random or equidistant times - and where transaction costs are considered. These strategies are considered for various utility functions and are compared with the optimal ones of continuous trading.
The immiscible lattice BGK method for solving the two-phase incompressible Navier-Stokes equations is analysed in great detail. Equivalent moment analysis and local differential geometry are applied to examine how interface motion is determined and how surface tension effects can be included such that consistency to the two-phase incompressible Navier-Stokes equations can be expected. The results obtained from theoretical analysis are verified by numerical experiments. Since the intrinsic interface tracking scheme of immiscible lattice BGK is found to produce unsatisfactory results in two-dimensional simulations several approaches to improving it are discussed but all of them turn out to yield no substantial improvement. Furthermore, the intrinsic interface tracking scheme of immiscible lattice BGK is found to be closely connected to the well-known conservative volume tracking method. This result suggests to couple the conservative volume tracking method for determining interface motion with the Navier-Stokes solver of immiscible lattice BGK. Applied to simple flow fields, this coupled method yields much better results than plain immiscible lattice BGK.
In this work we present and estimate an explanatory model with a predefined system of explanatory equations, a so called lag dependent model. We present a locally optimal, on blocked neural network based lag estimator and theorems about consistensy. We define the change points in context of lag dependent model, and present a powerfull algorithm for change point detection in high dimensional high dynamical systems. We present a special kind of bootstrap for approximating the distribution of statistics of interest in dependent processes.
Extensions of Shallow Water Equations The subject of the thesis of Michael Hilden is the simulation of floods in urban areas. In case of strong rain events, water can flow out of the overloaded sewer system onto the street and damage the connected houses. The dependable simulation of water flow out of a manhole ("manhole") and over a curb ("curb") is crucial for the assessment of the flood risks. The incompressible 3D-Navier-Stokes Equations (3D-NSE) describe the free surface flow of water accurately, but require expensive computations. Therefore, the less CPU-intensive (factor ca.1/100) Shallow Water Equations (SWE) are usually applied in hydrology. They can be derived from 3D-NSE under the assumption of a hydrostatic pressure distribution via depth-integration and are applied successfully in particular to simulations of river flow processes. The SWE-computations of the flow problems "manhole" and "curb" differ to the 3D-NSE results. Thus, SWE need to be extended appropriately to give reliable forecasts for flood risks in urban areas within reduced computational efforts. These extensions are developed based on physical considerations not considered in the classical SWE. In one extension, a vortex layer on the ground is separated from the main flow representing its new bottom. In a further extension, the hydrostatic pressure distribution is corrected by additional terms due to approximations of vertical velocities and their interaction with the flow. These extensions increase the quality of the SWE results for these flow problems up to the quality level of the NSE results within a moderate increase of the CPU efforts.
The thesis discusses discrete-time dynamic flows over a finite time horizon T. These flows take time, called travel time, to pass an arc of the network. Travel times, as well as other network attributes, such as, costs, arc and node capacities, and supply at the source node, can be constant or time-dependent. Here we review results on discrete-time dynamic flow problems (DTDNFP) with constant attributes and develop new algorithms to solve several DTDNFPs with time-dependent attributes. Several dynamic network flow problems are discussed: maximum dynamic flow, earliest arrival flow, and quickest flow problems. We generalize the hybrid capacity scaling and shortest augmenting path algorithmic of the static network flow problem to consider the time dependency of the network attributes. The result is used to solve the maximum dynamic flow problem with time-dependent travel times and capacities. We also develop a new algorithm to solve earliest arrival flow problems with the same assumptions on the network attributes. The possibility to wait (or park) at a node before departing on outgoing arc is also taken into account. We prove that the complexity of new algorithm is reduced when infinite waiting is considered. We also report the computational analysis of this algorithm. The results are then used to solve quickest flow problems. Additionally, we discuss time-dependent bicriteria shortest path problems. Here we generalize the classical shortest path problems in two ways. We consider two - in general contradicting - objective functions and introduce a time dependency of the cost which is caused by a travel time on each arc. These problems have several interesting practical applications, but have not attained much attention in the literature. Here we develop two new algorithms in which one of them requires weaker assumptions as in previous research on the subject. Numerical tests show the superiority of the new algorithms. We then apply dynamic network flow models and their associated solution algorithms to determine lower bounds of the evacuation time, evacuation routes, and maximum capacities of inhabited areas with respect to safety requirements. As a macroscopic approach, our dynamic network flow models are mainly used to produce good lower bounds for the evacuation time and do not consider any individual behavior during the emergency situation. These bounds can be used to analyze existing buildings or help in the design phase of planning a building.
The main two problems of continuous-time financial mathematics are option pricing and portfolio optimization. In this thesis, various new aspects of these major topics of financial mathematics will be discussed. In all our considerations we will assume the standard diffusion type setting for securitiy prices which is today well-know under the term "Black-Scholes model". This setting and the basic results of option pricing and portfolio optimization are surveyed in the first chapter. The next three chapters deal with generalizations of the standard portfolio problem, also know as "Merton's problem". Here, we will always use the stochastic control approach as introduced in the seminal papers by Merton (1969, 1971, 1990). One such problem is the very realistic setting of an investor who is faced with fixed monetary streams. More precisely, in addition to maximizing the utility from final wealth via choosing an investment strategy, the investor also has to fulfill certain consumption needs. Also the opposite situation, an additional income stream can now be taken into account in our portfolio optimization problem. We consider various examples and solve them on one hand via classical stochastic control methods and on the other hand by our new separation theorem. This together with some numerical examples forms Chapter 2. Chapter 3 is mainly concerned with the portfolio problem if the investor has different lending and borrowing rates. We give explicit solutions (where possible) and numerical methods to calculate the optimal strategy in the cases of log utility and HARA utility for three different modelling approaches of the dependence of the borrowing rate on the fraction of wealth financed by a credit. The further generalization of the standard Merton problem in Chapter 4 consists in considering simultaneously the possibilities for continuous and discrete consumption. In our general approach there is a possibility for assigning the different consumption times different weights which is a generalization of the usual way of making them comparable via discounting. Chapter 5 deals with the special case of pricing basket options. Here, the main problem is not path-dependence but the multi-dimensionality which makes it impossible to give usuefull analytical representations of the option price. We review the literature and compare six different numerical methods in a systematic way. Thereby we also look at the influence of various parameters such as strike, correlation, forwards or volatilities on the erformance of the different numerical methods. The problem of pricing Asian options on average spot with average strike is the topic of Chapter 6. We here apply the bivariate normal distribution to obtain an approximate option price. This method proves to be very reliable and e±cient for the valuation of different variants of Asian options on average spot with average strike.
The focus of this work has been to develop two families of wavelet solvers for the inner displacement boundary-value problem of elastostatics. Our methods are particularly suitable for the deformation analysis corresponding to geoscientifically relevant (regular) boundaries like sphere, ellipsoid or the actual Earth's surface. The first method, a spatial approach to wavelets on a regular (boundary) surface, is established for the classical (inner) displacement problem. Starting from the limit and jump relations of elastostatics we formulate scaling functions and wavelets within the framework of the Cauchy-Navier equation. Based on numerical integration rules a tree algorithm is constructed for fast wavelet computation. This method can be viewed as a first attempt to "short-wavelength modelling", i.e. high resolution of the fine structure of displacement fields. The second technique aims at a suitable wavelet approximation associated to Green's integral representation for the displacement boundary-value problem of elastostatics. The starting points are tensor product kernels defined on Cauchy-Navier vector fields. We come to scaling functions and a spectral approach to wavelets for the boundary-value problems of elastostatics associated to spherical boundaries. Again a tree algorithm which uses a numerical integration rule on bandlimited functions is established to reduce the computational effort. For numerical realization for both methods, multiscale deformation analysis is investigated for the geoscientifically relevant case of a spherical boundary using test examples. Finally, the applicability of our wavelet concepts is shown by considering the deformation analysis of a particular region of the Earth, viz. Nevada, using surface displacements provided by satellite observations. This represents the first step towards practical applications.
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.
The central theme in this thesis concerns the development of enhanced methods and algorithms for appraising market and credit risks and their application within the context of standard and more advanced market models. Generally, methods and algorithms for analysing market risk of complex portfolios involve detailed knowledge of option sensitivities, the so-called "Greeks". Based on an analysis of symmetries in financial market models, relations between option sensitivities are obtained, which can be used for the efficient valuation of the Greeks. Mainly, the relations are derived within the Black Scholes model, however, some relations are also valid for more general models, for instance the Heston model. Portfolios are usually influenced by lots of underlyings, so it is necessary to characterise the dependencies of these basic instruments. It is usual to describe such dependencies by correlation matrices. However, estimations of correlation matrices in practice are disturbed by statistical noise and usually have the problem of rank deficiency due to missing data. A fast algorithm is presented which performs a generalized Cholesky decomposition of a perturbed correlation matrix. In contrast to the standard Cholesky algorithm, an advantage of the generalized method is that it works for semi-positive, rank deficient matrices as well. Moreover, it gives an approximative decomposition when the input matrix is indefinite. A comparison with known algorithms with similar features is performed and it turns out, that the new algorithm can be recommended in situations where computation time is the critical issue. The determination of a profit and loss distribution by Fourier inversion of its characteristic function is a powerful tool, but it can break down when the characteristic function is not integrable. In this thesis, methods for Fourier inversion of non-integrable characteristic functions are studied. In this respect, two theorems are obtained which are based on a suitable approximation of the unknown distribution with known density and characteristic function. Further it will be shown, that straightforward Fast Fourier inversion works, when the according density lives on a bounded interval. The above techniques are of crucial importance to determine the profit and loss distribution (P&L) of large portfolios efficiently. The so-called Delta Gamma normal approach has become industrial standard for the estimation of market risk. It is shown, that the performance of the Delta Gamma normal approach can be improved substantially by application of the developed methods. The same optimization procedure also applies to the Delta Gamma Student model. A standard tool for computing the P&L distribution of a loan portfolio is the CreditRisk+ model. Basically, the CreditRisk+ distribution is a discrete distribution which can be computed from its probability generating function. For this a numerically stable method is presented and as an alternative, a new algorithm based on Fourier inversion is proposed. Finally, an extension of the CreditRisk+ model to market risk is developed, which distribution can be obtained efficiently by the presented Fourier inversion methods as well.
The thesis deals with the subgradient optimization methods which are serving to solve nonsmooth optimization problems. We are particularly concerned with solving large-scale integer programming problems using the methodology of Lagrangian relaxation and dualization. The goal is to employ the subgradient optimization techniques to solve large-scale optimization problems that originated from radiation therapy planning problem. In the thesis, different kinds of zigzagging phenomena which hamper the speed of the subgradient procedures have been investigated and identified. Moreover, we have established a new procedure which can completely eliminate the zigzagging phenomena of subgradient methods. Procedures used to construct both primal and dual solutions within the subgradient schemes have been also described. We applied the subgradient optimization methods to solve the problem of minimizing total treatment time of radiation therapy. The problem is NP-hard and thus far there exists no method for solving the problem to optimality. We present a new, efficient, and fast algorithm which combines exact and heuristic procedures to solve the problem.
The thesis is concerned with the modelling of ionospheric current systems and induced magnetic fields in a multiscale framework. Scaling functions and wavelets are used to realize a multiscale analysis of the function spaces under consideration and to establish a multiscale regularization procedure for the inversion of the considered operator equation. First of all a general multiscale concept for vectorial operator equations between two separable Hilbert spaces is developed in terms of vector kernel functions. The equivalence to the canonical tensorial ansatz is proven and the theory is transferred to the case of multiscale regularization of vectorial inverse problems. As a first application, a special multiresolution analysis of the space of square-integrable vector fields on the sphere, e.g. the Earth’s magnetic field measured on a spherical satellite’s orbit, is presented. By this, a multiscale separation of spherical vector-valued functions with respect to their sources can be established. The vector field is split up into a part induced by sources inside the sphere, a part which is due to sources outside the sphere and a part which is generated by sources on the sphere, i.e. currents crossing the sphere. The multiscale technqiue is tested on a magnetic field data set of the satellite CHAMP and it is shown that crustal field determination can be improved by previously applying our method. In order to reconstruct ionspheric current systems from magnetic field data, an inversion of the Biot-Savart’s law in terms of multiscale regularization is defined. The corresponding operator is formulated and the singular values are calculated. Based on the konwledge of the singular system a regularzation technique in terms of certain product kernels and correponding convolutions can be formed. The method is tested on different simulations and on real magnetic field data of the satellite CHAMP and the proposed satellite mission SWARM.