arXiv subjects

Alexander K. Hartmann

Publications and source records attributed to Alexander K. Hartmann.

At least 73 records · Page 4Linked to original sources

arXiv2016

Practical Introduction to Clustering Data

Data clustering is an approach to seek for structure in sets of complex data, i.e., sets of "objects". The main objective is to identify groups of objects which are similar to each other, e.g., for classification. Here, an introduction to clustering is given and three basic approaches are introduced: the k-means algorithm, neighbour-based clustering, and an agglomerative clustering method. For all cases, C source code examples are given, allowing for an easy implementation.

physics.data-an↗

arXiv2016

Aging in the three-dimensional Random Field Ising Model

We studied the nonequilibrium aging behavior of the Random Field Ising Model in three dimensions for various values of the disorder strength. This allowed us to investigate how the aging behavior changes across the ferromagnetic-paramagnetic phase transition. We investigated a large system size of $N=256^3$ spins and up to $10^8$ Monte Carlo sweeps. To reach these necessary long simulation times we employed an implementation running on Intel Xeon Phi coprocessors, reaching single spin flip times as short as 6 ps. We measured typical correlation functions in space and time to extract a growing length scale and corresponding exponents.

cond-mat.stat-mech↗

arXiv2015

Phase Transitions of Traveling Salesperson Problems solved with Linear Programming and Cutting Planes

The Traveling Salesperson problem asks for the shortest cyclic tour visiting a set of cities given their pairwise distances and belongs to the NP-hard complexity class, which means that with all known algorithms in the worst case instances are not solveable in polynomial time, i.e., the problem is hard. Though that does not mean, that there are not subsets of the problem which are easy to solve. To examine numerically transitions from an easy to a hard phase, a random ensemble of cities in the Euclidean plane given a parameter σ, which governs the hardness, is introduced. Here, a linear programming approach together with suitable cutting planes is applied. Such algorithms operate outside the space of feasible solutions and are often used in practical application but rarely studied in physics so far. We observe several transitions. To characterize these transitions, scaling assumptions from continuous phase transitions are applied

cond-mat.dis-nn↗

arXiv2015

Score distributions of gapped multiple sequence alignments down to the low-probability tail

Assessing the significance of alignment scores of optimally aligned DNA or amino acid sequences can be achieved via the knowledge of the score distribution of random sequences. But this requires obtaining the distribution in the biologically relevant high-scoring region, where the probabilities are exponentially small. For gapless local alignments of infinitely long sequences this distribution is known analytically to follow a Gumbel distribution. Distributions for gapped local alignments and global alignments of finite lengths can only be obtained numerically. To obtain result for the small-probability region, specific statistical mechanics-based rare-event algorithms can be applied. In previous studies, this was achieved for pairwise alignments. They showed that, contrary to results from previous simple sampling studies, strong deviations from the Gumbel distribution occur in case of finite sequence lengths. Here we extend the studies to the for practical applications in Molecular Biology much more relevant case of multiple sequence alignments with gaps. We study the distributions of scores over a large range of the support, reaching probabilities as small as 10^-160, for global and local (sum-of-pair scores) multiple alignments. We find that even after suitable rescaling, eliminating the sequence-length dependence, the distributions for multiple alignment differ from the pairwise alignment case. Furthermore, we also show that the previously discussed Gaussian correction to the Gumbel distribution needs to be refined, also for the case of pairwise alignments.

q-bio.QM↗

arXiv2015

Fragmentation properties of two-dimensional Proximity Graphs considering random failures and targeted attacks

The pivotal quality of proximity graphs is connectivity, i.e. all nodes in the graph are connected to one another either directly or via intermediate nodes. These types of graphs are robust, i.e., they are able to function well even if they are subject to limited removal of elementary building blocks, as it may occur for random failures or targeted attacks. Here, we study how the structure of these graphs is affected when nodes get removed successively until an extensive fraction is removed such that the graphs fragment. We study different types of proximity graphs for various node removal strategies. We use different types of observables to monitor the fragmentation process, simple ones like number and sizes of connected components, and more complex ones like the hop diameter and the backup capacity, which is needed to make a network N-1 resilient. The actual fragmentation turns out to be described by a second order phase transition. Using finite-size scaling analyses we numerically assess the threshold fraction of removed nodes, which is characteristic for the particular graph type and node deletion scheme, that suffices to decompose the underlying graphs.

physics.soc-ph↗

arXiv2015

Non-equilibrium evolution of window overlaps in spin glasses

We investigate numerically the time dependence of "window" overlaps in a three-dimensional Ising spin glass below its transition temperature after a rapid quench. Using an efficient GPU implementation, we are able to study large systems up to lateral length $L=128$ and up to long times of $t=10^8$ sweeps. We find that the data scales according to the ratio of the window size $W$ to the non-equilibrium coherence length $ξ(t)$. We also show a substantial change in behavior if the system is run for long enough that it globally equilibrates, i.e. $ξ(t) \approx L/2$, where $L$ is the lattice size. This indicates that the local behavior of a spin glass depends on the spin configurations (and presumably also the bonds) far away. We compare with similar simulations for the Ising ferromagnet. Based on these results, we speculate on a connection between the non-equilibrium dynamics discussed here and averages computed theoretically using the "metastate".

cond-mat.stat-mech↗

arXiv2015

Large-deviation properties of resilience of power grids

We study the distributions of the resilience of power flow models against transmission line failures via a so-called backup capacity. We consider three ensembles of random networks and in addition, the topology of the British transmission power grid. The three ensembles are Erdős-Rényi random graphs, Erdős-Rényi random graphs with a fixed number of links, and spatial networks where the nodes are embedded in a two dimensional plane. We investigate numerically the probability density functions (pdfs) down to the tails to gain insight in very resilient and very vulnerable networks. This is achieved via large-deviation techniques which allow us to study very rare values which occur with probability densities below $10^{-160}$. We find that the right tail of the pdfs towards larger backup capacities follows an exponential with a strong curvature. This is confirmed by the rate function which approaches a limiting curve for increasing network sizes. Very resilient networks are basically characterized by a small diameter and a large power sign ratio. In addition, networks can be made typically more resilient by adding more links.

physics.soc-ph↗

arXiv2015

Convex Hulls of Random Walks: Large-Deviation Properties

We study the convex hull of the set of points visited by a two-dimensional random walker of T discrete time steps. Two natural observables that characterize the convex hull in two dimensions are its perimeter L and area A. While the mean perimeter and the mean area have been studied before, analytically and numerically, and exact results are known for large T (Brownian motion limit), little is known about the full distributions P(A) and P(L). In this paper, we provide numerical results for these distributions. We use a sophisticated large-deviation approach that allows us to study the distributions over a larger range of the support, where the probabilities P(A) and P(L) are as small as 10^{-300}. We analyze (open) random walks as well as (closed) Brownian bridges on the two-dimensional discrete grid as well as in the two-dimensional plane. The resulting distributions exhibit, for large T, a universal scaling behavior (independent of the details of the jump distributions) as a function of A/T and L/\sqrt{T}, respectively. We are also able to obtain the rate function, describing rare events at the tails of these distributions, via a numerical extrapolation scheme and find a linear and square dependence as a function of the rescaled perimeter and the rescaled area, respectively.

cond-mat.stat-mech↗

arXiv2014

Ageing at the Spin-Glass/Ferromagnet Transition: Monte Carlo Simulation using GPUs

We study the the non-equilibrium ageing behaviour of the +/-J Edwards-Anderson model in three dimensions for samples of size up to N=128^3 and for up to 10^8 Monte Carlo sweeps. In particular we are interested in the change of the ageing when crossing from the spin-glass phase to the ferromagnetic phase. The necessary long simulation times are reached by employing a CUDA-based GPU implementation, which allows for single-spin flip times as small as 8ps. We measure typical spin glass correlation functions in space and time to determine the growing length scale and extract the constituting exponents. We observe a clear signature of the disorder-driven equilibrium transition in the non-equilibrium behavior.

cond-mat.stat-mech↗

arXiv2014

Exact ground states of one-dimensional long-range random-field Ising magnets

We investigate the one-dimensional long-range random-field Ising magnet with Gaussian distribution of the random fields. In this model, a ferromagnetic bond between two spins is placed with a probability $p \sim r^{-1-σ}$, where $r$ is the distance between these spins and $σ$ is a parameter to control the effective dimension of the model. Exact ground states at zero temperature are calculated for system sizes up to $L = 2^{19}$ via graph theoretical algorithms for four different values of $σ\in \{0.25,0.4,0.5,1.0\}$ while varying the strength $h$ of the random fields. For each of these values several independent physical observables are calculated, i.e., magnetization, Binder parameter, susceptibility and a specific-heat-like quantity. The ferromagnet-paramagnet transitions at critical values $h_c(σ)$ as well as the corresponding critical exponents are obtained. The results agree well with theory and interestingly we find for $σ= 1/2$ the data is compatible with a critical random-field strength $h_c > 0$.

cond-mat.dis-nn↗

arXiv2014

Large-deviation properties of resilience of transportation networks

Distributions of the resilience of transport networks are studied numerically, in particular the large-deviation tails. Thus, not only typical quantities like average or variance but the distributions over the (almost) full support can be studied. For a proof of principle, a simple transport model based on the edge-betweenness and three abstract yet widely studied random network ensembles are considered here: Erdoes-Renyi random networks with finite connectivity, small world networks and spatial networks embedded in a two-dimensional plane. Using specific numerical large-deviation techniques, probability densities as small as 10^(-80) are obtained here. This allows one to study typical but also the most and the least resilient networks. The resulting distributions fulfill the mathematical large-deviation principle, i.e., can be well described by rate functions in the thermodynamic limit. The analysis of the limiting rate function reveals that the resilience follows an exponential distribution almost everywhere. An analysis of the structure of the network shows that the most-resilient networks can be obtained, as a rule of thumb, by minimizing the diameter of a network. Also, trivially, by including more links a network can typically be made more resilient. On the other hand, the least-resilient networks are very rare and characterized by one (or few) small core(s) to which all other nodes are connected. In total, the spatial network ensemble turns out to be most suitable for obtaining and studying resilience of real mostly finite-dimensional networks. Studying this ensemble in combination with the presented large-deviation approach for more realistic, in particular dynamic transport networks appears to be very promising.

physics.soc-ph↗

arXiv2013

Spin glasses: the game

This document presents the rules of a tactical two-player board game which is inspired by spin glasses. The aim is, while placing bonds and spins, to achieve a majority of the spins facing the chosen direction of each player. The game has been already successfully used in university teaching but should be accessible to players from age 10 and up. (Note that there is a non-physics version of the rules available at www.compphys.uni-oldenburg.de which formulates the game in terms of a comptetion between party hosts.) Material is included such that a cheap version of the board game can be made using a color printer, scissors and glue.

cond-mat.dis-nn↗

arXiv2013

Diluted antiferromagnets in a field seem to be in a different universality class than the random-field Ising model

We perform large-scale Monte Carlo simulations using the Machta-Newman-Chayes algorithms to study the critical behavior of both the diluted antiferromagnet in a field with 30% dilution and the random-field Ising model with Gaussian random fields for different field strengths. Analytical calculations by Cardy [Phys. Rev. B 29, 505 (1984)] predict that both models map onto each other and share the same universality class in the limit of vanishing fields. However, a detailed finite-size scaling analysis of both the Binder cumulant and the two-point finite-size correlation length suggests that even in the limit of small fields, where the mapping is expected to work, both models are not in the same universality class. Therefore, care should be taken when interpreting (experimental) data for diluted antiferromagnets in a field using the random-field Ising model. Based on our numerical data, we present analytical expressions for the phase boundaries of both models.

cond-mat.dis-nn↗

arXiv2013

Sampling fractional Brownian motion in presence of absorption: a Markov Chain method

We study fractional Brownian motion (fBm) characterized by the Hurst exponent H. Using a Monte Carlo sampling technique, we are able to numerically generate fBm processes with an absorbing boundary at the origin at discrete times for a large number of 10^7 time steps even for small values like H=1/4. The results are compatible with previous analytical results that the distribution of (rescaled) endpoints y follow a power law P(y) y^ϕwith ϕ=(1-H)/H, even for small values of H. Furthermore, for the case H=0.5 we also study analytically the finite-length corrections to the first order, namely a plateau of P(y) for y->0 which decreases with increasing process length. These corrections are compatible with the numerical results.

cond-mat.stat-mech↗

arXiv2013

Generalized black-box large deviation simulations: High-precision work distributions for extreme non-equilibrium processes in large systems

The distributions of work for strongly non-equilibrium processes are studied using a very general form of a large-deviation approach, which allows one to study distributions of almost arbitrary quantities of interest for equilibrium, non-equilibrium stationary and even non-stationary processes. The method is applied to varying quickly the external field in a wide range B=3 <-> 0 for critical (T=2.269) two-dimensional Ising system of size LxL=128x128. To obtain free energy differences from the work distributions, they must be studied in ranges where the probabilities are as small as 10^{-240}, which is not possible using direct simulation approaches. By comparison with the exact free energies, which are available for this model for the zero-field case, one sees that the present approach allows one to obtain the free energy with a very high relative precision of 10^{-4}. This works well also for non-zero field, i.e., for a case where standard umbrella-sampling methods seem to be not so efficient to calculate free energies. Furthermore, for the present case it is verified that the resulting distributions of work for forward and backward process fulfill Crooks theorem with high precision. Finally, the free energy for the Ising magnet as a function of the field strength is obtained.

cond-mat.stat-mech↗

arXiv2012

Ultrametric probe of the spin-glass state in a field

We study the ultrametric structure of phase space of one-dimensional Ising spin glasses with random power-law interaction in an external random field. Although in zero field the model in both the mean-field and non-mean-field universality classes shows an ultrametric signature [Phys. Rev. Lett. 102, 037207 (2009)], when a field is applied ultrametricity seems only present in the mean-field regime. The results for the non-mean field case in an external field agree with data for spin glasses studied within the Migdal-Kadanoff approximation. Our results therefore suggest that the spin-glass state might be fragile to external fields below the upper critical dimension.

cond-mat.dis-nn↗

arXiv2012

Random number generators for massively parallel simulations on GPU

High-performance streams of (pseudo) random numbers are crucial for the efficient implementation for countless stochastic algorithms, most importantly, Monte Carlo simulations and molecular dynamics simulations with stochastic thermostats. A number of implementations of random number generators has been discussed for GPU platforms before and some generators are even included in the CUDA supporting libraries. Nevertheless, not all of these generators are well suited for highly parallel applications where each thread requires its own generator instance. For this specific situation encountered, for instance, in simulations of lattice models, most of the high-quality generators with large states such as Mersenne twister cannot be used efficiently without substantial changes. We provide a broad review of existing CUDA variants of random-number generators and present the CUDA implementation of a new massively parallel high-quality, high-performance generator with a small memory load overhead.

physics.comp-ph↗

arXiv2012

Phase transition for cutting-plane approach to vertex-cover problem

We study the vertex-cover problem which is an NP-hard optimization problem and a prototypical model exhibiting phase transitions on random graphs, e.g., Erdoes-Renyi (ER) random graphs. These phase transitions coincide with changes of the solution space structure, e.g, for the ER ensemble at connectivity c=e=2.7183 from replica symmetric to replica-symmetry broken. For the vertex-cover problem, also the typical complexity of exact branch-and-bound algorithms, which proceed by exploring the landscape of feasible configurations, change close to this phase transition from "easy" to "hard". In this work, we consider an algorithm which has a completely different strategy: The problem is mapped onto a linear programming problem augmented by a cutting-plane approach, hence the algorithm operates in a space OUTSIDE the space of feasible configurations until the final step, where a solution is found. Here we show that this type of algorithm also exhibits an "easy-hard" transition around c=e, which strongly indicates that the typical hardness of a problem is fundamental to the problem and not due to a specific representation of the problem.

cond-mat.dis-nn↗