Graph Algorithms for Dynamical Systems

Dellnitz, Michael; Molo, Mirko Hessel-von; Metzner, Philipp; Preis, Robert; Schütte, Christof

doi:10.1007/3-540-35657-6_23

Cited by 28 publications

(12 citation statements)

References 29 publications

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“…Therefore, there are two possible paths to move from x L to x R : A direct path, and one that passes through the central well x C . Properties of (2.1) with this potential were studied by various authors (see [41,17] and references therein).…”

Section: Examplesmentioning

confidence: 99%

“…Then appropriate equations can be formulated in these variables, and in some special cases their exact form can even be found by rigorous mathematics based on the Mori-Zwanzig projection approach [24,55]. In more complex cases where a rigorous derivation of the dynamics is mathematically intractable, many numerical approaches to solve these tasks have been suggested in the literature, such as transition path sampling, the nudged elastic band, the string method, the transfer operator approach, Perron cluster analysis and many others, see [16,17,18,19,20,26,46], and references therein. In addition, given further knowledge about the system, such as a good dividing surface between reactant and product regions, several algorithms for the efficient computation of the transition rates have been developed [44,37,53].…”

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confidence: 99%

See 1 more Smart Citation

Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems

Coifman¹,

Kevrekidis²,

Lafon³

et al. 2008

Multiscale Model. Simul.

262

336

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Abstract.The concise representation of complex high dimensional stochastic systems via a few reduced coordinates is an important problem in computational physics, chemistry and biology. In this paper we use the first few eigenfunctions of the backward Fokker-Planck diffusion operator as a coarse grained low dimensional representation for the long term evolution of a stochastic system, and show that they are optimal under a certain mean squared error criterion. We denote the mapping from physical space to these eigenfunctions as the diffusion map. While in high dimensional systems these eigenfunctions are difficult to compute numerically by conventional methods such as finite differences or finite elements, we describe a simple computational data-driven method to approximate them from a large set of simulated data. Our method is based on defining an appropriately weighted graph on the set of simulated data, and computing the first few eigenvectors and eigenvalues of the corresponding random walk matrix on this graph. Thus, our algorithm incorporates the local geometry and density at each point into a global picture that merges in a natural way data from different simulation runs. Furthermore, we describe lifting and restriction operators between the diffusion map space and the original space. These operators facilitate the description of the coarse-grained dynamics, possibly in the form of a low-dimensional effective free energy surface parameterized by the diffusion map reduction coordinates. They also enable a systematic exploration of such effective free energy surfaces through the design of additional "intelligently biased" computational experiments. We conclude by demonstrating our method on a few examples.Key words. Diffusion maps, dimensional reduction, stochastic dynamical systems, Fokker Planck operator, metastable states, normalized graph Laplacian. AMS subject classifications. 60H10, 60J60, 62M051. Introduction. Systems of stochastic differential equations (SDE's) are commonly used as models for the time evolution of many chemical, physical and biological systems of interacting particles [22,45,52]. There are two main approaches to the study of such systems. The first is by detailed Brownian Dynamics (BD) or other stochastic simulations, which follow the motion of each particle (or more generally variable) in the system and generate one or more long trajectories. The second is via analysis of the time evolution of the probability densities of these trajectories using the numerical solution of the corresponding time dependent Fokker-Planck (FP) partial differential equation.For typical high dimensional systems, both approaches suffer from severe limitations, when applied directly. The main limitation of standard BD simulations is the scale gap between the atomistic time scale of single particle motions, at which the SDE's are formulated, and the macroscopic time scales that characterize the long term evolution and equilibration of these systems. This scale gap puts severe constraints on detailed simulat...

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Section: Examplesmentioning

confidence: 99%

mentioning

confidence: 99%

Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems

Coifman¹,

Kevrekidis²,

Lafon³

et al. 2008

Multiscale Model. Simul.

262

336

View full text Add to dashboard Cite

show abstract

“…The most common formulation of the graph partitioning problem for an undirected graph G = (V, E) asks for a division of V into k pairwise disjoint subsets (partitions ) of size at most |V |/k such that the edge-cut, i.e., the total number of edges having their incident nodes in dierent subsets, is minimized. Among others, its applications include dynamical systems [8], VLSI circuit layout [11], and image segmentation [35]. We mainly consider its use for balancing the load in numerical simulations (e. g., uid dynamics), which have become a classical application for parallel computers.…”

Section: Introductionmentioning

confidence: 99%

A new diffusion-based multilevel algorithm for computing graph partitions of very high quality

Meyerhenke

Monien

Sauerwald

2008

2008 IEEE International Symposium on Parallel and Distributed Processing

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Abstract. Graph partitioning requires the division of a graph's vertex set into k equally sized subsets s. t. some objective function is optimized. High-quality partitions are important for many applications, whose objective functions are often N P-hard to optimize. Most state-of-the-art graph partitioning libraries use a variant of the Kernighan-Lin (KL) heuristic within a multilevel framework. While these libraries are very fast, their solutions do not always meet all user requirements. Moreover, due to its sequential nature, KL is not easy to parallelize. Its use as a load balancer in parallel numerical applications therefore requires complicated adaptations. That is why we developed previously an inherently parallel algorithm, called Bubble-FOS/C (Meyerhenke et al., IPDPS'06), which optimizes partition shapes by a diusive mechanism. However, it is too slow for practical use, despite its high solution quality.In this paper, besides proving that Bubble-FOS/C converges towards a local optimum of a potential function, we develop a much faster method for the improvement of partitionings. This faster method called TruncCons is based on a dierent diusive process, which is restricted to local areas of the graph and also contains a high degree of parallelism. By coupling TruncCons with Bubble-FOS/C in a multilevel framework based on two dierent hierarchy construction methods, we obtain our new graph partitioning heuristic DibaP. Compared to Bubble-FOS/C, DibaP shows a considerable acceleration, while retaining the positive properties of the slower algorithm. Experiments with popular benchmark graphs show that DibaP computes consistently better results than the state-of-the-art libraries METIS and JOSTLE. Moreover, with our new algorithm, we have improved the best known edge-cut values for a signicant number of partitionings of six widely used benchmark graphs.

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“…30,31 Beyond this application as a diagnostic tool, there have been first attempts towards establishing prognostic methods for anticipating the emergence of El Niño based on changes in the network properties. 32,33 An alternative to defining links by statistical dependencies for establishing a complex network representation of a given flow system is to let the network links directly represent the amount of material transported among spatial locations, [34][35][36][37][38][39] thus defining a Lagrangian flow network. If nodes represent regions of an abstract phase space, then the links characterize the evolution of the corresponding dynamical system from one (coarse-grained) state to another.…”

mentioning

confidence: 99%

“…If nodes represent regions of an abstract phase space, then the links characterize the evolution of the corresponding dynamical system from one (coarse-grained) state to another. [34][35][36]40 In the context of Lagrangian approaches to fluid dynamics, a large body of work has been devoted to identifying almostinvariant or coherent sets in fluid flows from the associated transfer matrix, the so-called set-oriented approach to transport, by means of spectral methods. 36,[40][41][42][43][44][45][46] Identifying the transfer matrix with the adjacency matrix of a directed and weighted network allows in addition the use of modern community detection methods 37,47,48 or path-finding algorithms.…”

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confidence: 99%

Introduction to Focus Issue: Complex network perspectives on flow systems

Donner

Hernández-Garcı́a

Ser‐Giacomi

2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

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During the last few years, complex network approaches have demonstrated their great potentials as versatile tools for exploring the structural as well as dynamical properties of dynamical systems from a variety of different fields. Among others, recent successful examples include (i) functional (correlation) network approaches to infer hidden statistical interrelationships between macroscopic regions of the human brain or the Earth's climate system, (ii) Lagrangian flow networks allowing to trace dynamically relevant fluid-flow structures in atmosphere, ocean or, more general, the phase space of complex systems, and (iii) time series networks unveiling fundamental organization principles of dynamical systems. In this spirit, complex network approaches have proven useful for data-driven learning of dynamical processes (like those acting within and between sub-components of the Earth's climate system) that are hidden to other analysis techniques. This Focus Issue presents a collection of contributions addressing the description of flows and associated transport processes from the network point of view and its relationship to other approaches which deal with fluid transport and mixing and/or use complex network techniques.

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Graph Algorithms for Dynamical Systems

Cited by 28 publications

References 29 publications

Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems

Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems

A new diffusion-based multilevel algorithm for computing graph partitions of very high quality

Introduction to Focus Issue: Complex network perspectives on flow systems

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