arXiv subjects

Ralf Banisch

Publications and source records attributed to Ralf Banisch.

11 recordsLinked to original sources

arXiv2019

Network Measures of Mixing

Transport and mixing processes in fluid flows can be studied directly from Lagrangian trajectory data, such as obtained from particle tracking experiments. Recent work in this context highlights the application of graph-based approaches, where trajectories serve as nodes and some similarity or distance measure between them is employed to build a (possibly weighted) network, which is then analyzed using spectral methods. Here, we consider the simplest case of an unweighted, undirected network and analytically relate local network measures such as node degree or clustering coefficient to flow structures. In particular, we use these local measures to divide the family of trajectories into groups of similar dynamical behavior via manifold learning methods.

math.DS↗

arXiv2018

Data-driven Computation of Molecular Reaction Coordinates

The identification of meaningful reaction coordinates plays a key role in the study of complex molecular systems whose essential dynamics is characterized by rare or slow transition events. In a recent publication, precise defining characteristics of such reaction coordinates were identified and linked to the existence of a so-called transition manifold. This theory gives rise to a novel numerical method for the pointwise computation of reaction coordinates that relies on short parallel MD simulations only, but yields accurate approximation of the long time behavior of the system under consideration. This article presents an extension of the method towards practical applicability in computational chemistry. It links the newly defined reaction coordinates to concepts from transition path theory and Markov state model building. The main result is an alternative computational scheme that allows for a global computation of reaction coordinates based on commonly available types of simulation data, such as single long molecular trajectories, or the push-forward of arbitrary canonically-distributed point clouds. It is based on a Galerkin approximation of the transition manifold reaction coordinates, that can be tuned to individual requirements by the choice of the Galerkin ansatz functions. Moreover, we propose a ready-to-implement variant of the new scheme, that computes data-fitted, mesh-free ansatz functions directly from the available simulation data. The efficacy of the new method is demonstrated on a small protein system.

physics.comp-ph↗

arXiv2018

MIST: A Simple and Efficient Molecular Dynamics Abstraction Library for Integrator Development

We present MIST, the Molecular Integration Simulation Toolkit, a lightweight and efficient software library written in C++ which provides an abstract in- terface to common molecular dynamics codes, enabling rapid and portable development of new integration schemes for molecular dynamics. The initial release provides plug-in interfaces to NAMD-Lite, GROMACS and Amber, and includes several standard integration schemes, a constraint solver, tem- perature control using Langevin Dynamics, and two tempering schemes. We describe the architecture and functionality of the library and the C and For- tran APIs which can be used to interface additional MD codes to MIST. We show, for a range of test systems, that MIST introduces negligible overheads for serial, shared-memory parallel, and GPU-accelerated cases, except for Amber where the native integrators run directly on the GPU itself. As a demonstration of the capabilities of MIST, we describe a simulated tempering simulation used to study the free energy landscape of Alanine-12 in both vacuum and detailed solvent conditions.

physics.comp-ph↗

arXiv2018

On Hyperparameter Search in Cluster Ensembles

Quality assessments of models in unsupervised learning and clustering verification in particular have been a long-standing problem in the machine learning research. The lack of robust and universally applicable cluster validity scores often makes the algorithm selection and hyperparameter evaluation a tough guess. In this paper, we show that cluster ensemble aggregation techniques such as consensus clustering may be used to evaluate clusterings and their hyperparameter configurations. We use normalized mutual information to compare individual objects of a clustering ensemble to the constructed consensus of the whole ensemble and show, that the resulting score can serve as an overall quality measure for clustering problems. This method is capable of highlighting the standout clustering and hyperparameter configuration in the ensemble even in the case of a distorted consensus. We apply this very general framework to various data sets and give possible directions for future research.

stat.ML↗

arXiv2017

Diffusion maps tailored to arbitrary non-degenerate Ito processes

We present two generalizations of the popular diffusion maps algorithm. The first generalization replaces the drift term in diffusion maps, which is the gradient of the sampling density, with the gradient of an arbitrary density of interest which is known up to a normalization constant. The second generalization allows for a diffusion map type approximation of the forward and backward generators of general Ito diffusions with given drift and diffusion coefficients. We use the local kernels introduced by Berry and Sauer, but allow for arbitrary sampling densities. We provide numerical illustrations to demonstrate that this opens up many new applications for diffusion maps as a tool to organize point cloud data, including biased or corrupted samples, dimension reduction for dynamical systems, detection of almost invariant regions in flow fields, and importance sampling.

math.DS↗

arXiv2017

Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics

We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.

math.DS↗

arXiv2017

Cycle flow based module detection in directed recurrence networks

We present a new cycle flow based method for finding fuzzy partitions of weighted directed networks coming from time series data. We show that this method overcomes essential problems of most existing clustering approaches, which tend to ignore important directional information by considering only one-step, one-directional node connections. Our method introduces a novel measure of communication between nodes using multi-step, bidirectional transitions encoded by a cycle decomposition of the probability flow. Symmetric properties of this measure enable us to construct an undirected graph that captures information flow of the original graph seen by the data and apply clustering methods designed for undirected graphs. Finally, we demonstrate our algorithm by analyzing earthquake time series data, which naturally induce (time-)directed networks. This article has been published originally in EPL, DOI: 10.1209/0295-5075/108/68008. This version differs from the published version by minor formatting details.

physics.data-an↗

arXiv2016

Understanding the geometry of transport: diffusion maps for Lagrangian trajectory data unravel coherent sets

Dynamical systems often exhibit the emergence of long-lived coherent sets, which are regions in state space that keep their geometric integrity to a high extent and thus play an important role in transport. In this article, we provide a method for extracting coherent sets from possibly sparse Lagrangian trajectory data. Our method can be seen as an extension of diffusion maps to trajectory space, and it allows us to construct "dynamical coordinates" which reveal the intrinsic low-dimensional organization of the data. The only a priori knowledge about the dynamics that we require is a locally valid notion of distance, which renders our method highly suitable for automated data analysis. We show convergence of our method to the analytic transfer operator framework of coherence in the infinite data limit, and illustrate its potential on several two- and three-dimensional examples as well as real world data.

math.DS↗

arXiv2014

Modularity of Directed Networks: Cycle Decomposition Approach

The problem of decomposing networks into modules (or clusters) has gained much attention in recent years, as it can account for a coarse-grained description of complex systems, often revealing functional subunits of these systems. A variety of module detection algorithms have been proposed, mostly oriented towards finding hard partitionings of undirected networks. Despite the increasing number of fuzzy clustering methods for directed networks, many of these approaches tend to neglect important directional information. In this paper, we present a novel random walk based approach for finding fuzzy partitions of directed, weighted networks, where edge directions play a crucial role in defining how well nodes in a module are interconnected. We will show that cycle decomposition of a random walk process connects the notion of network modules and information transport in a network, leading to a new, symmetric measure of node communication. walk process, for which we will prove that although being time-reversible it inherits all necessary information about directions and modular structure of the original network. Finally, we will use this measure to introduce a communication graph, for which we will show that although being undirected it inherits all necessary information about modular structures from the original network.

math-ph↗

arXiv2013

The Unruh-deWitt Detector and the Vacuum in the General Boundary formalism

We discuss how to formulate a condition for choosing the vacuum state of a quantum scalar field on a timelike hyperplane in the general boundary formulation (GBF) using the coupling to an Unruh-DeWitt detector. We explicitly study the response of an Unruh-DeWitt detector for evanescent modes which occur naturally in quantum field theory in the presence of the equivalent of a dielectric boundary. We find that the physically correct vacuum state has to depend on the physical situation outside of the boundaries of the spacetime region considered. Thus it cannot be determined by general principles pertaining only to a subset of spacetime.

hep-th↗

arXiv2013

Meshless discretization of LQ-type stochastic control problems

We propose a novel Galerkin discretization scheme for stochastic optimal control problems on an indefinite time horizon. The control problems are linear-quadratic in the controls, but possibly nonlinear in the state variables, and the discretization is based on the fact that problems of this kind can be transformed into linear boundary value problems by a logarithmic transformation. We show that the discretized linear problem is dual to a Markov decision problem, the precise form of which depends on the chosen Galerkin basis. We prove a strong error bound in $L^{2}$ for the general scheme and discuss two special cases: a variant of the known Markov chain approximation obtained from a basis of characteristic functions of a box discretization, and a sparse approximation that uses the basis of committor functions of metastable sets of the dynamics; the latter is particularly suited for high-dimensional systems, e.g., control problems in molecular dynamics. We illustrate the method with several numerical examples, one being the optimal control of Alanine dipeptide to its helical conformation.

math.OC↗