**Rough volatility models**

*Stemper, Benjamin Marco* (2019)

So-called rough stochastic volatility models constitute the latest advancement in option price modeling. In contrast to popular bivariate diffusion models such as Heston, here the driving noise of volatility is modeled by a fractional Brownian motion (fBM) with scaling in the rough regime of Hurst parameter H < 1/2. A major appeal of such models lies in their ability to parsimoniously recover k...

**Coupled system of Maxwell equations and circuit equations in electro-magnetism**

*Niroomand Rad, Helia* (2019)

This thesis is devoted to modeling electro-magnetic coupling, the so-called crosstalk phenomenon, in circuit simulation, where a new modeling approach via bilateral coupling of the Maxwell equations with circuit equations is considered. Using the bilaterally coupled model allows full simulation of the crosstalk phenomenon in the evolution of time, when the disturbances generated by the excitati...

**Information flow in stochastic optimal control and a stochastic representation theorem for Meyer-measurable processes**

*Beßlich, David* (2019)

Stochastic control theory determines intervention policies optimizing the evolution of a system subject to randomness. Delicate issues arise when the considered system can jump due to both exogenous shocks and endogenous controls. Here one has to specify what the controller knows when about the exogenous shocks and how and when she can act on this information. Classical optimal control resolves...

**Error analysis and adaptive control for gas flow in networks**

*Stolwijk, Jeroen Johannes* (2019)

In this thesis, uniform estimators are derived for the different errors that a numerical solution could contain, namely modeling, discretization, iteration, data uncertainty, and rounding errors. Subsequently, the errors are adaptively controlled on a network in order to bring the total error below a prescribed tolerance while keeping the computational cost low. As example problems, the simulat...

**Homogen-universelle metrische Räume**

*Rothacker, Edgar* (1976)

Homogeneous universal metric spaces are constructed in the Jónsson class of metric spaces starting from finite metric spaces over the rationals and then completing the construction over the reals; the main theorem deals with the finite homgeneity property established by Urysohn for his famous universal metric space: we show that this can be extended to totally bounded metric spaces, thereby ext...

**Homogen-universelle verallgemeinerte metrische Räume unter besonderer Berücksichtigung nicht-archimedischer Metriken**

*Rothacker, Edgar* (1978)

Homogeneous universal metric spaces are constructed over generalized distance sets such as totally ordered abelian groups or linear orderings without any further structure; the latter yields the basis for the study of hom.-univ. ultrametric spaces in the third part of this work; finally the fourth part studies the Cauchy completeness in connection with the construction of the appropriately ge...

**The Mismatch Principle and L1-analysis compressed sensing**

*Genzel, Martin* (2019)

This thesis contributes to several mathematical aspects and problems at the interface of statistical learning theory and signal processing. Although based on a common theoretical foundation, the main results of this thesis can be divided into two major topics of independent interest. The first one deals with the estimation capacity of the generalized Lasso, i.e., least squares minimization comb...

**Message routeing and percolation in interference limited multihop networks**

*Tóbiás, András József* (2019)

This thesis consists of two main parts. In the first part, we investigate a probabilistic model for routeing of messages in relay-augmented multihop ad-hoc networks, where each transmitter sends one message to the origin. Given the (random) transmitter locations, we weight the family of random, uniformly distributed message trajectories by an exponential probability weight, favouring trajectori...

**Convergence of gradient methods on hierarchical tensor varieties**

*Kutschan, Benjamin* (2019)

Subject of the attached dissertation are the sets of tensors of bounded hierarchical rank. They are algebraic varieties. The central result is a parametrization of the tangent cones of these varieties. Using this result a Riemannian gradient method is constructed. The global convergence of this gradient method is proven using a Lojasiewicz inequality.

**Subspace concentration of geometric measures**

*Pollehn, Hannes* (2019)

In this work we study geometric measures in two different extensions of the Brunn-Minkowski theory. The first part of this thesis is concerned with problems in Lp Brunn-Minkowski theory, that is based on the concept of p-addition of convex bodies, which was first introduced by Firey for p = 1 and later considered for all real p by Lutwak et al. The interplay of the volume and other functionals ...

**Connectedness of random set attractors**

*Scheutzow, Michael ; Vorkastner, Isabell* (2018-12-28)

We examine the question whether random set attractors for continuous-time random dynamical systems on a connected state space are connected. In the deterministic case, these attractors are known to be connected. In the probabilistic setup, however, connectedness has only been shown under stronger connectedness assumptions on the state space. Under a weak continuity condition on the random dynam...

**Low rank tensor decompositions for high dimensional data approximation, recovery and prediction**

*Wolf, Alexander Sebastian Johannes Wolf* (2019)

In this thesis, we examine different approaches for efficient high dimensional data acquisition and reconstruction using low rank tensor decomposition techniques. High dimensional here refers to the order of the ambient tensor space in which the data is contained. Examples of such data include tomographic videos, solutions to parametric differential equations and quantum states of many particle...

**Stability analysis in the inverse Robin transmission problem**

*Meftahi, Houcine* (2016)

In this paper, we consider the conductivity problem with piecewise‐constant conductivity and Robin‐type boundary condition on the interface of discontinuity. When the quantity of interest is the jump of the conductivity, we perform a local stability estimate for a parameterized non‐monotone family of domains. We give also a quantitative stability result of local optimal solution with respect to...

**Spectral properties of the random conductance model**

*Flegel, Franziska* (2019)

Charge and exciton transport in disordered media plays an essential role in modern technologies. Classical examples are amorphous and organic semiconductors where the disorder can give rise to localized electron states. These localized electrons effectively behave like discrete particles hopping between discrete sites in an inhomogeneous environment. A popular model for such a hopping process ...

**A stable, polynomial-time algorithm for the eigenpair problem**

*Armentano, Diego ; Beltrán, Carlos ; Bürgisser, Peter ; Cucker, Felipe ; Shub, Michael* (2018)

We describe algorithms for computing eigenpairs (eigenvalue-eigenvector pairs) of a complex n×n matrix A. These algorithms are numerically stable, strongly accurate, and theoretically efficient (i.e., polynomial-time). We do not believe they outperform in practice the algorithms currently used for this computational problem. The merit of our paper is to give a positive answer to a long-standing...

**The Loewner equation and Lipschitz graphs**

*Rohde, Steffen ; Tran, Huy ; Zinsmeister, Michel* (2018)

The proofs of continuity of Loewner traces in the stochastic and in the deterministic settings employ different techniques. In the former setting of the Schramm–Loewner evolution SLE, Hölder continuity of the conformal maps is shown by estimating the derivatives, whereas the latter setting uses the theory of quasiconformal maps. In this note, we adopt the former method to the deterministic sett...

**The hyperbolic geometry of Markov’s theorem on Diophantine approximation and quadratic forms**

*Springborn, Boris* (2017)

Markov’s theorem classifies the worst irrational numbers with respect to rational approximation and the indefinite binary quadratic forms whose values for integer arguments stay farthest away from zero. The main purpose of this paper is to present a new proof of Markov’s theorem using hyperbolic geometry. The main ingredients are a dictionary to translate between hyperbolic geometry and algebra...

**Moments of quantum Lévy areas using sticky shuffle Hopf algebras**

*Hudson, Robin ; Schauz, Uwe ; Wu, Yue* (2018)

We study a family of quantum analogs of Lévy's stochastic area for planar Brownian motion depending on a variance parameter σ ≥ 1 which deform to the classical Lévy area as σ → ∞. They are defined as second rank iterated stochastic integrals against the components of planar Brownian motion, which are one-dimensional Brownian motions satisfying Heisenberg-type commutation relations. Such iterate...

**A coupling approach to Doob’s theorem**

*Kulik, Alexei ; Scheutzow, Michael* (2015)

We provide a coupling proof of Doob’s theorem which says that the transition probabilities of a regular Markov process which has an invariant probability measure μ converge to μ in the total variation distance. In addition we show that non-singularity (rather than equivalence) of the transition probabilities suffices to ensure convergence of the transition probabilities for μ-almost all initial...

**Planar graphs and face areas: Area-Universality**

*Kleist, Linda* (2018)

In this work, we study planar graphs with prescribed face areas. This field is inspired by cartograms. A 'cartogram' is a distorted map where the size of the regions are proportional to some statistical parameter such as the population, the total birth, the gross national product, or some other special property. As a mathematical abstraction we are interested in straight-line drawings of plane...

- FG Differentialgeometrie und Visualisierung
- FG Diskrete Mathematik
- FG Diskrete Mathematik / Geometrie
- FG Finanz- und Versicherungsmathematik
- FG Finanzmathematik
- FG Geometrie und Integrable Systeme
- FG Geometrie und Visualisierung
- FG Kombinatorische Optimierung und Graphenalgorithmen
- FG Mathematische Stochastik / Stochastische Prozesse in den Neurowissenschaften
- FG Mathematische Stochastik und Anwendungen in statistischer Physik und Biologie
- FG Modellierung, Simulation und Optimierung in Natur- und Ingenieurwissenschaften
- FG Numerische Analysis partieller Differentialgleichungen
- FG Numerische Lineare Algebra
- FG Numerische Mathematik
- FG Stochastische Analysis