Browsing by Author "Meyer, Fabian"

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Open Access
Error control for statistical solutions of hyperbolic systems of conservation laws
(2021) Giesselmann, Jan; Meyer, Fabian; Rohde, Christian
Statistical solutions have recently been introduced as an alternative solution framework for hyperbolic systems of conservation laws. In this work, we derive a novel a posteriori error estimate in the Wasserstein distance between dissipative statistical solutions and numerical approximations obtained from the Runge-Kutta Discontinuous Galerkin method in one spatial dimension, which rely on so-called regularized empirical measures. The error estimator can be split into deterministic parts which correspond to spatio-temporal approximation errors and a stochastic part which reflects the stochastic error. We provide numerical experiments which examine the scaling properties of the residuals and verify their splitting.
Open Access
FtsZ induces membrane deformations via torsional stress upon GTP hydrolysis
(2021) Ramirez-Diaz, Diego A.; Merino-Salomón, Adrián; Meyer, Fabian; Heymann, Michael; Rivas, Germán; Bramkamp, Marc; Schwille, Petra
FtsZ is a key component in bacterial cell division, being the primary protein of the presumably contractile Z ring. In vivo and in vitro, it shows two distinctive features that could so far, however, not be mechanistically linked: self-organization into directionally treadmilling vortices on solid supported membranes, and shape deformation of flexible liposomes. In cells, circumferential treadmilling of FtsZ was shown to recruit septum-building enzymes, but an active force production remains elusive. To gain mechanistic understanding of FtsZ dependent membrane deformations and constriction, we design an in vitro assay based on soft lipid tubes pulled from FtsZ decorated giant lipid vesicles (GUVs) by optical tweezers. FtsZ filaments actively transform these tubes into spring-like structures, where GTPase activity promotes spring compression. Operating the optical tweezers in lateral vibration mode and assigning spring constants to FtsZ coated tubes, the directional forces that FtsZ-YFP-mts rings exert upon GTP hydrolysis can be estimated to be in the pN range. They are sufficient to induce membrane budding with constricting necks on both, giant vesicles and E.coli cells devoid of their cell walls. We hypothesize that these forces result from torsional stress in a GTPase activity dependent manner.
Open Access
A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws
(2020) Giesselmann, Jan; Meyer, Fabian; Rohde, Christian
This article considers one-dimensional random systems of hyperbolic conservation laws. Existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws, which involve random initial data and random flux functions, are established. Based on these results an a posteriori error analysis for a numerical approximation of the random entropy solution is presented. For the stochastic discretization, a non-intrusive approach, namely the Stochastic Collocation method is used. The spatio-temporal discretization relies on the Runge-Kutta Discontinuous Galerkin method. The a posteriori estimator is derived using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. The scaling properties of the residuals are investigated and the efficiency of the proposed adaptive algorithms is illustrated in various numerical examples.
Open Access
Quantification of uncertainties in compressible flows
(2019) Meyer, Fabian; Rohde, Christian (Prof. Dr.)
Due to rising computing capacities, including and accounting for uncertain (model) parameters in numerical simulations is becoming more and more popular. Uncertainty Quantification (UQ) addresses this issue and provides a variety of different mathematical methods to quantify the influence of uncertain input parameters on numerical solutions and derived quantities of interest. This thesis is concerned with the development and improvement of different UQ methods for numerical simulations of compressible flow problems, described by random conservation laws like the compressible Euler or Navier-Stokes equations. We distinguish between polynomial-based (non-statistical) UQ methods and sampling-based (statistical) UQ methods. The first part of this thesis investigates non-statistical UQ methods, in particular the Stochastic Galerkin (SG), Non-Intrusive Spectral Projection (NISP) and Stochastic Collocation (SC) method. While SG is a frequently used method for UQ of random partial differential equations, the classical SG approach is not ensured to preserve hyperbolicity of the underlying random hyperbolic conservation law. To this end we develop a hyperbolicity-preserving numerical scheme, which uses a slope limiter to retain admissible solutions of the SG system, while providing high-order approximations in physical and random space. The modified numerical scheme is applied to different challenging numerical examples for which the classical SG approach fails. An important aspect when considering space-time-stochastic numerical schemes is to quantify the errors that arise from numerical discretization. In this thesis we derive a novel a posteriori error analysis framework for numerical discretizations of random hyperbolic systems of conservation laws, which rely on the Runge-Kutta Discontinuous Galerkin method in combination with polynomial-based UQ methods. Our estimates are based on the relative entropy framework of Dafermos and DiPerna and allow us to quantify the entire space-time-stochastic discretization error. Moreover, due to a splitting of the residual we are able distinguish between spatio-temporal and stochastic errors. Based on the a posteriori error estimates we design novel residual-based, space-stochastic adaptive numerical schemes. We confirm our theoretical findings by various numerical experiments. The last part of this thesis is concerned with statistical UQ methods, especially Monte Carlo (MC) type methods. We extend the Multilevel Monte Carlo (MLMC) method to what we call hp-MLMC method. Instead of considering a hierarchy of spatially refined meshes, we allow for meshes which are arbitrarily hp-refined. The classical complexity analysis of MLMC is extended to the hp-MLMC method. Moreover, to increase the robustness and efficiency of an iterative version of hp-MLMC, we construct a confidence interval for the optimal number of samples per level. To demonstrate the efficiency of the hp-MLMC method combined with the novel sample estimator we apply our method to two different compressible flow problems described by the random Navier-Stokes equations. In particular, we consider an important problem from computational acoustics that exhibits physical phenomena with high sensitivity with respect to the problem parameters and which poses a challenging problem for UQ.