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Audio Ontologies for Intangible Cultural Heritage
Cultural heritage portals often contain intangible objects digitized as audio files. This paper presents and discusses the adaptation of existing audio ontologies intended for non-cultural heritage applications. The resulting alignment of the German Digital Library-Europeana Data Model (DDB-EDM) with Music Ontology (MO) and Audio Commons Ontology (ACO) is presented.
Non-local and local temporal cavity soliton interaction in delay models of mode-locked lasers
Interaction equations governing slow time evolution of the coordinates and phases of two interacting temporal cavity solitons in a delay differential equation model of a nonlinear mirror mode-locked laser are derived and analyzed. It is shown that non-local pulse interaction due to gain depletion and recovery can lead either to a development of harmonic mode-locking regime, or to a formation of closely packed incoherent soliton bound state with weakly oscillating intersoliton time separation. Local interaction via electric field tails can result in an anti-phase or in-phase stationary or breathing harmonic mode-locking regime.
EDP-convergence for a linear reaction-diffusion system with fast reversible reaction
We perform a fast-reaction limit for a linear reaction-diffusion system consisting of two diffusion equations coupled by a linear reaction. We understand the linear reaction-diffusion system as a gradient flow of the free energy in the space of probability measures equipped with a geometric structure, which contains the Wasserstein metric for the diffusion part and cosh-type functions for the reaction part. The fast-reaction limit is done on the level of the gradient structure by proving EDP-convergence with tilting. The limit gradient system induces a diffusion system with Lagrange multipliers on the linear slow-manifold. Moreover, the limit gradient system can be equivalently described by a coarse-grained gradient system, which induces a diffusion equation with a mixed diffusion constant for the coarse-grained slow variable.
Energy estimates and model order reduction for stochastic bilinear systems
In this paper, we investigate a large-scale stochastic system with bilinear drift and linear diffusion term. Such high dimensional systems appear for example when discretizing a stochastic partial differential equations in space. We study a particular model order reduction technique called balanced truncation (BT) to reduce the order of spatially-discretized systems and hence reduce computational complexity. We introduce suitable Gramians to the system and prove energy estimates that can be used to identify states which contribute only very little to the system dynamics. When BT is applied the reduced system is obtained by removing these states from the original system. The main contribution of this paper is an L2-error bound for BT for stochastic bilinear systems. This result is new even for deterministic bilinear equations. In order to achieve it, we develop a new technique which is not available in the literature so far.
Genealogical properties of spatial models in Population Genetics
At the interface between Phylo- and Population Genetics, and recently heavily inspired by Epidemonology, the discipline of Phylogeography comprises modelling techniques from classical theoretical biology and combines them with a spatial (2D or 3D) aspect, with the purpose of utilizing geographical information in the analysis to understand the evolutionary history of a biological system or aspects of virology such as directionality and seasonality in pandemic outbreaks [1, 2, 3, 4]. An prime example of this are datasets that take into account the sampling locations of its components (geo-referenced genomic data). In this project, we have focused on the model called "spatial Lambda-Fleming-Viot process" ( V [5, 6]) and analzed its statistical properties forward in time as well as in the ancestral (dual) process, with results that may be used for parameter inference. Of particlar interest was the spatial variance, denoted , a parameter controlling the speed at which genetic information is spread across space and therefore an analog of the reproduction number (R0) used in epidemonology e.g. to assess the infectiousness of differing viral strains. We explored the relation of this parameter to the time to coalescence between lineage pairs in this model and described methods of estimating it from sampled data under different circumstances. We have furthermore investigated similarities and differences between this model and classical models in Population Genetics, particularly Birth-Death processes, which are heavily used for all kinds of biological inference problems, but do not by themselves feature a spatial component. We compared the Vto a variant of the Birth-Death process where the location of a live individual changes over the course of its lifetime according to a Brownian motion. This process is not as easily viewed backward in time as the V, but the genalogical process is accessible by Markov-Chain Monte Carlosimulation, as the likelihoods of ancestral positions and branch lengths are easily calculated, making this model easily applicable to data. Our analysis highlights the analogy between the two processes forward in time as well as backward in time; on the other hand, we also observed a divergent behavior of the two models when no prior on the phylogenetic time scale was assumed. Lastly, this project has given rise to a study of combinatorial properties of tree shapes relevant to the V, the Birth-Death and other biological processes. In particular, we were able to identify the combinatorial class genealogical trees generated from these processes belong to and verify a conjecture regarding their enumeration. Preliminary versions of software tools for the aforementioned inference have also been provided.
Coarse-graining and reconstruction for Markov matrices
We present a coarse-graining (or model order reduction) procedure for stochastic matrices by clustering. The method is consistent with the natural structure of Markov theory, preserving positivity and mass, and does not rely on any tools from Hilbert space theory. The reconstruction is provided by a generalized Penrose-Moore inverse of the coarse-graining operator incorporating the inhomogeneous invariant measure of the Markov matrix. As we show, the method provides coarse-graining and reconstruction also on the level of tensor spaces, which is consistent with the notion of an incidence matrix and quotient graphs, and, moreover, allows to coarse-grain and reconstruct fluxes. Furthermore, we investigate the connection with functional inequalities and Poincaré-type constants.
Low-dimensional approximations of high-dimensional asset price models
We consider high-dimensional asset price models that are reduced in their dimension in order to reduce the complexity of the problem or the effect of the curse of dimensionality in the context of option pricing. We apply model order reduction (MOR) to obtain a reduced system. MOR has been previously studied for asymptotically stable controlled stochastic systems with zero initial conditions. However, stochastic differential equations modeling price processes are uncontrolled, have non-zero initial states and are often unstable. Therefore, we extend MOR schemes and combine ideas of techniques known for deterministic systems. This leads to a method providing a good pathwise approximation. After explaining the reduction procedure, the error of the approximation is analyzed and the performance of the algorithm is shown conducting several numerical experiments. Within the numerics section, the benefit of the algorithm in the context of option pricing is pointed out.
Global-in-time existence for liquid mixtures subject to a generalised incompressibility constraint
We consider a system of partial differential equations describing diffusive and convective mass transport in a fluid mixture of N > 1 chemical species. A weighted sum of the partial mass densities of the chemical species is assumed to be constant, which expresses the incompressibility of the fluid, while accounting for different reference sizes of the involved molecules. This condition is different from the usual assumption of a constant total mass density, and it leads in particular to a non-solenoidal velocity field in the Navier-Stokes equations. In turn, the pressure gradient occurs in the diffusion fluxes, so that the PDE-system of mass transport equations and momentum balance is fully coupled. Another striking feature of such incompressible mixtures is the algebraic formula connecting the pressure and the densities, which can be exploited to prove a pressure bound in L1. In this paper, we consider incompressible initial states with bounded energy and show the global existence of weak solutions with defect measure.
Mass transport in multicomponent compressible fluids: Local and global well-posedness in classes of strong solutions for general class-one models
We consider a system of partial differential equations describing mass transport in a multicomponent isothermal compressible fluid. The diffusion fluxes obey the Fick-Onsager or Maxwell- Stefan closure approach. Mechanical forces result into one single convective mixture velocity, the barycentric one, which obeys the Navier-Stokes equations. The thermodynamic pressure is defined by the Gibbs-Duhem equation. Chemical potentials and pressure are derived from a thermodynamic potential, the Helmholtz free energy, with a bulk density allowed to be a general convex function of the mass densities of the constituents. The resulting PDEs are of mixed parabolic-hyperbolic type. We prove two theoretical results concerning the well-posedness of the model in classes of strong solutions: 1. The solution always exists and is unique for short-times and 2. If the initial data are sufficiently near to an equilibrium solution, the well-posedness is valid on arbitrary large, but finite time intervals. Both results rely on a contraction principle valid for systems of mixed type that behave like the compressible Navier- Stokes equations. The linearised parabolic part of the operator possesses the self map property with respect to some closed ball in the state space, while being contractive in a lower order norm only. In this paper, we implement these ideas by means of precise a priori estimates in spaces of exact regularity.
Hydrodynamics at the moving contact line
By removing the smearing effect of capillary waves in molecular dynamics simulations we are able to provide a microscopic picture of the region around the moving contact line (MCL) at an unprecedented resolution. On this basis, we show that the continuum character of the velocity field is unaffected by molecular layering down to below the molecular scale. The solution of the continuum Stokes problem with MCL and Navier-slip matches very well the molecular dynamics data and is consistent with a slip-length of 42 Å and small contact line dissipation. This is consistent with observations of the local force balance near the liquid-solid interface.