Freie Universität Berlin
Repository: Freie Universität Berlin (FU), Math Department (fu_mi_publications)Not a member yet
2251 research outputs found
Sort by
Truncated Nonsmooth Newton Multigrid Methods for Block-Separable Minimization Problems
Get PDFThe Truncated Nonsmooth Newton Multigrid method is a robust and efficient solution method for a wide range of block-separable convex minimization problems, typically stemming from discretizations of nonlinear and nonsmooth partial differential equations. This paper proves global convergence of the method under weak conditions both on the objective functional and on the local inexact subproblem solvers that are part of the method. It also discusses a range of algorithmic choices that allows to customize the algorithm for many specific problems. Numerical examples are deliberately omitted, because many such examples have already been published elsewhere
Well posedness and convergence analysis of the ensemble Kalman inversion
Get PDFThe ensemble Kalman inversion is widely used in practice to estimate unknown
parameters from noisy measurement data. Its low computational costs,
straightforward implementation, and non-intrusive nature makes the method
appealing in various areas of application. We present a complete analysis
of the ensemble Kalman inversion with perturbed observations for a fixed
ensemble size when applied to linear inverse problems. The well-posedness
and convergence results are based on the continuous time scaling limits of the
method. The resulting coupled system of stochastic differential equations allows
one to derive estimates on the long-time behaviour and provides insights into
the convergence properties of the ensemble Kalman inversion. We view the
method as a derivative free optimization method for the least-squares misfit
functional, which opens up the perspective to use the method in various areas of
applications such as imaging, groundwater flow problems, biological problems
as well as in the context of the training of neural networks
Kernel methods for detecting coherent structures in dynamical data
No full textABSTRACT
We illustrate relationships between classical kernel-based dimensionality reduction techniques and eigendecompositions of empirical estimates of reproducing kernel Hilbert space operators associated with dynamical systems. In particular, we show that kernel canonical correlation analysis (CCA) can be interpreted in terms of kernel transfer operators and that it can be obtained by optimizing the variational approach for Markov processes score. As a result, we show that coherent sets of particle trajectories can be computed by kernel CCA. We demonstrate the efficiency of this approach with several examples, namely, the well-known Bickley jet, ocean drifter data, and a molecular dynamics problem with a time-dependent potential. Finally, we propose a straightforward generalization of dynamic mode decomposition called coherent mode decomposition. Our results provide a generic machine learning approach to the computation of coherent sets with an objective score that can be used for cross-validation and the comparison of different methods.
While coherent sets of particles are common in dynamical systems, they are notoriously challenging to identify. In this article, we leverage the combination of a suite of methods designed to approximate the eigenfunctions of transfer operators with kernel embeddings in order to design an algorithm for detecting coherent structures in Langrangian data. It turns out that the resulting method is a well-known technique to analyze relationships between multidimensional variables, namely, kernel canonical correlation analysis (CCA). Our algorithm successfully identifies coherent structures in several diverse examples, including oceanic currents and a molecular dynamics problem with a moving potential. Furthermore, we show that a natural extension of our algorithm leads to a coherent mode decomposition (CMD), a counterpart to dynamic mode decomposition (DMD).
I. INTRODUCTIO
Feedback control theory & Model order reduction for stochastic equations
No full textAbstract:
We analyze structure-preserving model order reduction methods for
Ornstein-Uhlenbeck processes and linear SPDEs with multiplicative noise based on
balanced truncation with non-zero initial data. We then marry these model order
reduction methods with stochastic optimal control theory and prove error bounds
for a class of linear quadratic regulator problems. We discuss the application of our
approach to enhanced sampling methods from non-equilibrium statistical mechanics
Time scales and exponential trends to equilibrium: Gaussian model problems
Get PDFWe review results on the exponential convergence of multi- dimensional Ornstein-Uhlenbeck processes and discuss related notions of characteristic timescales with concrete model systems. We focus, on the one hand, on exit time distributions and provide ecplicit expressions for the exponential rate of the distribution in the small noise limit. On the other hand, we consider relaxation timescales of the process to its equi- librium measured in terms of relative entropy and discuss the connection with exit probabilities. Along these lines, we study examples which il- lustrate specific properties of the relaxation and discuss the possibility of deriving a simulation-based, empirical definition of slow and fast de- grees of freedom which builds upon a partitioning of the relative entropy functional in conjuction with the observed relaxation behaviour
The mechanism of RNA base fraying: Molecular dynamics simulations analyzed with core-set Markov state models
Get PDFThe process of RNA base fraying (i.e. the transient opening of the termini of a helix) is involved in many aspects of RNA dynamics. We here use molecular dynamics simulations and Markov state models to characterize the kinetics of RNA fraying and its sequence and direction dependence. In particular, we first introduce a method for determining biomolecular dynamics employing core-set Markov state models constructed using an advanced clustering technique. The method is validated on previously reported simulations. We then use the method to analyze extensive trajectories for four different RNA model duplexes. Results obtained using D. E. Shaw research and AMBER force fields are compared and discussed in detail, and show a non-trivial interplay between the stability of intermediate states and the overall fraying kinetics
Part Load Control for a Shockless Explosion Combustion Cycle
No full textSince a significant increase in the efficiency of conventional gas turbines is unlikely due to various reasons, new concepts are needed. One option is to redesign the thermodynamic process itself. Replacing the constant pressure combustion with constant volume combustion (CVC) offers such an increase in efficiency. A promising new process that approximates constant volume combustion is the so-called shockless explosion combustion (SEC). SEC utilizes a homogeneous auto-ignition inside a combustion tube to avoid gas expansion during combustion. An acoustic interaction within the tube is exploited to ensure a self-sustained cyclic operation. For this, chemical and acoustic time-scales have to match. As this is impossible under ambient pressure conditions, for which SEC has been tested experimentally, this study focuses on simulations that mimic the situation of elevated pressure to design a controller. Herein, a control system is introduced within the numerical simulation of SEC that is capable of driving the process to different operating points. It expands on an iterative learning control from recent publications, which adjusts ignition time over the length of the tube. The control system proposed here can be used to realize a part load operation within the observed simulation
Jarzysnki equality, fluctuation theorems and variance reduction: Mathematical analysis and numerical algorithms
Get PDFIn this paper, we study Jarzynski's equality and fluctuation theorems for diffusion processes. While some of the results considered in the current work are known in the (mainly physics) literature, we review and generalize these nonequilibrium theorems using mathematical arguments, therefore enabling further investigations in the mathematical community. On the numerical side, variance reduction approaches such as importance sampling method are studied in order to compute free energy differences based on Jarzynski's equality
Rassismus in Deutschland. Eine macht-reflexive, biographietheoretische und diskursanalytische Studie
No full textAnna-Christin Ransiek untersucht die Wirkweisen von Rassismus in Deutschland. Sie zeigt auf, wie Rassismus in Deutschland biographisch und gesellschaftlich bearbeitet und interaktiv ausgehandelt wird. Dazu werden vier Typen des biographischen Umgangs mit Rassismus vorgestellt: das selbstgewählte Auffallen, die Distanzierung, die Aufrechterhaltung von Autonomie und die Interventionen. Ihre Studie macht zudem die gegenwärtige gesellschaftliche Auseinandersetzung mit Rassismus vor dem Hintergrund von Kolonialismus und Nationalsozialismus sichtbar. Es werden zwei wirkmächtige Diskursstränge präsentiert, vor denen die Biographen und Biographinnen ihre Erfahrungen aufschichten: Rassismus als Randphänomen und Rassismus als gesamtgesellschaftliches Phänomen. Außerdem wird ein Zugang vorgeschlagen, um die Forscherinnen- und Forscherperspektive machtkritisch zu beleuchten
A limiter‐based well‐balanced discontinuous Galerkin method for shallow‐water flows with wetting and drying: Triangular grids
No full textA novel wetting and drying treatment for second‐order Runge‐Kutta discontinuous Galerkin methods solving the nonlinear shallow‐water equations is proposed. It is developed for general conforming two‐dimensional triangular meshes and utilizes a slope limiting strategy to accurately model inundation. The method features a nondestructive limiter, which concurrently meets the requirements for linear stability and wetting and drying. It further combines existing approaches for positivity preservation and well balancing with an innovative velocity‐based limiting of the momentum. This limiting controls spurious velocities in the vicinity of the wet/dry interface. It leads to a computationally stable and robust scheme, even on unstructured grids, and allows for large time steps in combination with explicit time integrators. The scheme comprises only one free parameter, to which it is not sensitive in terms of stability. A number of numerical test cases, ranging from analytical tests to near‐realistic laboratory benchmarks, demonstrate the performance of the method for inundation applications. In particular, superlinear convergence, mass conservation, well balancedness, and stability are verified