Renormalization Group Methods for Coarse-Graining of Evolution Equations

Degenhard, Andreas; Rodríguez-Laguna, Javier

doi:10.1007/3-540-35888-9_8

Cited by 2 publications

(4 citation statements)

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“…Let's now consider the case where the core and action regions have half-width a > 0 . As in Section 3.2, we evaluate numerical examples of eigenvalue and eigenvectors using Matlab and from these we determine the analytic forms, which we confirm by substitution into the matrix equations (9). The eigenvalues of matrix equations (9) are…”

Section: Action Regions and Core Average Over Part Of A Patchmentioning

confidence: 86%

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Accuracy of Patch Dynamics with Mesoscale Temporal Coupling for Efficient Massively Parallel Simulations

Bunder¹,

Roberts²,

Kevrekidis³

2016

SIAM J. Sci. Comput.

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Massive parallelisation has lead to a dramatic increase in available computational power. However, data transfer speeds have failed to keep pace and are the major limiting factor in the development of exascale computing. New algorithms must be developed which minimise the transfer of data. Patch dynamics is a computational macroscale modelling scheme which provides a coarse macroscale solution of a problem defined on a fine microscale by dividing the domain into many nonoverlapping, coupled patches. Patch dynamics is readily adaptable to massive parallelisation as each processor can evaluate the dynamics on one, or a few, patches. However, patch coupling conditions interpolate across the unevaluated parts of the domain

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Section: Action Regions and Core Average Over Part Of A Patchmentioning

confidence: 86%

“…From these numerical examples we determine the general analytic forms of the eigenvalues and eigenvectors and confirm by substitution into the matrix equations. The eigenvalues of the matrix equations (9) are…”

Section: Act and Sample On A Lattice Pointmentioning

confidence: 99%

Accuracy of Patch Dynamics with Mesoscale Temporal Coupling for Efficient Massively Parallel Simulations

Bunder¹,

Roberts²,

Kevrekidis³

2016

SIAM J. Sci. Comput.

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“…In fact, one of the main results of this work is the derivation of a physically sound and mathematically rigorous dimensional reduction scheme for MSMs based on the so-called Real-Space RG formalism. This approach was inspired by the work of Degenhard and Rodriguez-Laguna [20,21], who used the RG to lower the computational cost of integrating non-linear partial differential equations.…”

Section: Arxiv:160407614v1 [Cond-matstat-mech] 26 Apr 2016mentioning

confidence: 99%

“…There are many possible norms between which one can choose, a priori. We follow the choice of Ref.s[20,21].…”

mentioning

confidence: 99%

Dimensional reduction of Markov state models from renormalization group theory

Orioli¹,

Faccioli²

2016

The Journal of Chemical Physics

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Renormalization Group (RG) theory provides the theoretical framework to define Effective Theories (ETs), i.e. systematic low-resolution approximations of arbitrary microscopic models. Markov State Models (MSMs) are shown to be rigorous ETs for Molecular Dynamics (MD). Based on this fact, we use Real Space RG to vary the resolution of a MSM and define an algorithm for clustering microstates into macrostates. The result is a lower dimensional stochastic model which, by construction, provides the optimal coarse-grained Markovian representation of the system's relaxation kinetics. To illustrate and validate our theory, we analyze a number of test systems of increasing complexity, ranging from synthetic toy models to two realistic applications, built form all-atom MD simulations. The computational cost of computing the low-dimensional model remains affordable on a desktop computer even for thousands of microstates.

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Renormalization Group Methods for Coarse-Graining of Evolution Equations

Cited by 2 publications

References 50 publications

Accuracy of Patch Dynamics with Mesoscale Temporal Coupling for Efficient Massively Parallel Simulations

Accuracy of Patch Dynamics with Mesoscale Temporal Coupling for Efficient Massively Parallel Simulations

Dimensional reduction of Markov state models from renormalization group theory

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