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- ItemMathematical modeling and numerical simulations of diode lasers with micro-integrated external resonators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Radziunas, MindaugasThis report summarizes our scientific activities within the project MANUMIEL (BMBF Program “Förderung der Wissenschaftlich-Technologischen Zusammenarbeit (WTZ) mit der Republik Moldau”, FKZ 01DK13020A). Namely, we discuss modeling of external cavity diode lasers, numerical simulations and analysis of these devices using the software package LDSL-tool, as well as the development of this software.
- ItemCalibration methods for gas turbine performance models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Borchardt, Jürgen; Mathé, Peter; Printsypar, GalinaThe WIAS software package BOP is used to simulate gas turbine models. In order to make accurate predictions the underlying models need to be calibrated. This study compares different strategies of model calibration. These are the deterministic optimization tools as nonlinear least squares (MSO) and the sparsity promoting variant LASSO, but also the probabilistic (Bayesian) calibration. The latter allows for the quantification of the inherent uncertainty, and it gives rise to a surrogate uncertainty measure in the MSO tool. The implementation details are accompanied with a numerical case study, which highlights the advantages and drawbacks of each of the proposed calibration methods.
- ItemType II singular perturbation approximation for linear systems with Lévy noise(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Redmann, MartinWhen solving linear stochastic partial differential equations numerically, usually a high order spatial discretisation is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretised systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is singular perturbation approximation (SPA), a method which has been extensively studied for deterministic systems. As so-called type I SPA it has already been extended to stochastic equations. We provide an alternative generalisation of the deterministic setting to linear systems with Lévy noise which is called type II SPA. It turns out that the ROM from applying type II SPA has better properties than the one of using type I SPA. In this paper, we provide new energy interpretations for stochastic reachability Gramians, show the preservation of mean square stability in the ROM by type II SPA and prove two different error bounds for type II SPA when applied to Lévy driven systems.
- ItemGlobal-in-time existence for liquid mixtures subject to a generalised incompressibility constraint(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Druet, Pierre-ÉtienneWe 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.
- ItemNon-local and local temporal cavity soliton interaction in delay models of mode-locked lasers(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Vladimirov, Andrei G.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.
- ItemTowards time-limited H2-optimal model order reduction(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Goyal, Pawan; Redmann, MartinIn order to solve partial differential equations numerically and accurately, a high order spatial discretization is usually needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretized systems and hence reduce computational complexity. A particular class of MOR techniques are H2-optimal methods such as the iterative rational Krylov subspace algorithm (IRKA) and related schemes. However, these methods are used to obtain good approximations on a infinite time-horizon. Thus, in this work, our main goal is to discuss MOR schemes for time-limited linear systems. For this, we propose an alternative time-limited H2-norm and show its connection with the time-limited Gramians. We then provide first-order optimality conditions for an optimal reduced order model (ROM) with respect to the time-limited H2-norm. Based on these optimality conditions, we propose an iterative scheme which upon convergences aims at satisfying these conditions. Then, we analyze how far away the obtained ROM is from satisfying the optimality conditions.We test the efficiency of the proposed iterative scheme using various numerical examples and illustrate that the newly proposed iterative method can lead to a better reduced-order compared to unrestricted IRKA in the time interval of interest.
- ItemExponential decay of covariances for the supercritical membrane model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Bolthausen, Erwin; Cipriani, Alessandra; Kurt, NoemiWe consider the membrane model, that is the centered Gaussian field on Zd whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a delta-pinning condition, giving a reward of strength " for the field to be 0 at any site of the lattice. In this paper we prove that in dimensions d ≥ 5 covariances of the pinned field decay at least exponentially, as opposed to the field without pinning, where the decay is polynomial. The proof is based on estimates for certain discrete weighted norms, a percolation argument and on a Bernoulli domination result.
- ItemCoarse-graining and reconstruction for Markov matrices(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Stephan, ArturWe 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.
- ItemMass transport in multicomponent compressible fluids: Local and global well-posedness in classes of strong solutions for general class-one models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Bothe, Dieter; Druet, Pierre-ÉtienneWe 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.
- ItemTime-periodic boundary layer solutions to singularly perturbed parabolic problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Omelchenko, Oleh; Recke, Lutz; Butuzov, Valentin; Nefedov, NikolayIn this paper, we present a result of implicit function theorem type, which was designed for applications to singularly perturbed problems. This result is based on fixed point iterations for contractive mappings, in particular, no monotonicity or sign preservation properties are needed. Then we apply our abstract result to time-periodic boundary layer solutions (which are allowed to be non-monotone with respect to the space variable) in semilinear parabolic problems with two independent singular perturbation parameters. We prove existence and local uniqueness of those solutions, and estimate their distance to certain approximate solutions.