Stiff Oscillatory Systems, Delta Jumps and White Noise

Cano, Begoña; Stuart, Andrew M.; Süli, Endre; Warren, Jared

doi:10.1007/s10208001002

Cited by 12 publications

(24 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following theorem extends a result from [2] for t ∈ [0, π) to the arbitrary intervals considered above. The techniques of proof are based on ideas in [12,13].…”

Section: Discussionsupporting

confidence: 59%

Extracting macroscopic stochastic dynamics: Model problems

Huisinga¹,

Schütte²,

Stuart³

2002

Comm Pure Appl Math

View full text Add to dashboard Cite

The purpose of this work is to shed light on an algorithm designed to extract effective macroscopic models from detailed microscopic simulations. The particular algorithm we study is a recently developed transfer operator approach due to Schütte et al. [20]. The investigations involve the formulation, and subsequent numerical study, of a class of model problems. The model problems are ordinary differential equations constructed to have the property that, when projected onto a low-dimensional subspace, the dynamics is approximately that of a stochastic differential equation exhibiting a finite-state-space Markov chain structure. The numerical studies show that the transfer operator approach can accurately extract finite-state Markov chain behavior embedded within high-dimensional ordinary differential equations. In so doing the studies lend considerable weight to existing applications of the algorithm to the complex systems arising in applications such as molecular dynamics. The algorithm is predicated on the assumption of Markovian input data; further numerical studies probe the role of memory effects. Although preliminary, these studies of memory indicate interesting avenues for further development of the transfer operator methodology.

show abstract

“…The following theorem extends a result from [2] for t ∈ [0, π) to the arbitrary intervals considered above. The techniques of proof are based on ideas in [12,13].…”

Section: Discussionsupporting

confidence: 59%

Extracting macroscopic stochastic dynamics: Model problems

Huisinga¹,

Schütte²,

Stuart³

2002

Comm Pure Appl Math

View full text Add to dashboard Cite

show abstract

“…x converges strongly to X as N → ∞ and error estimates can be found [CSSW01]. However the convergence is only on a finite time-interval, because of the periodicity inherent in the construction.…”

Section: Skew-product Systemsmentioning

confidence: 99%

“…In general this approach will fail because of numerical instabilities or resonances between the time-step frequency and fast unresolved-scales (see [AR99] for example). However, there are situations where unresolved simulations correctly reproduce macroscopic behaviour, and one example is for models similar to the Hamiltonian heat bath model of Example 7.3 [SW99, CSSW01,HK02a]. One approach to coarse timestepping is introduced in [TQK00], with applications to problems of the type described in Section 8 covered, for example, in [MMK02,MMPK02].…”

Section: Coarse Time-steppingmentioning

confidence: 99%

Extracting macroscopic dynamics: model problems and algorithms

2004

View full text Add to dashboard Cite

In many applications, the primary objective of numerical simulation of time-evolving systems is the prediction of macroscopic, or coarse-grained, quantities. A representative example is the prediction of biomolecular conformations from molecular dynamics. In recent years a number of new algorithmic approaches have been introduced to extract effective, lower-dimensional, models for the macroscopic dynamics; the starting point is the full, detailed, evolution equations. In many cases the effective low-dimensional dynamics may be stochastic, even when the original starting point is deterministic.This review surveys a number of these new approaches to the problem of extracting effective dynamics, highlighting similarities and differences between them. The importance of model problems for the evaluation of these new approaches is stressed, and a number of model problems are described. When the macroscopic dynamics is stochastic, these model problems are either obtained through a clear separation of time-scales, leading to a stochastic effect of the fast dynamics on the slow dynamics, or by considering high dimensional ordinary differential equations which, when projected onto a low dimensional subspace, exhibit stochastic behaviour through the presence of a broad frequency spectrum. Models whose stochastic microscopic behaviour leads to deterministic macroscopic dynamics are also introduced.The algorithms we overview include SVD-based methods for nonlinear problems, model reduction for linear control systems, optimal prediction techniques, asymptotics-based mode elimination, coarse timestepping methods and transfer-operator based methodologies.

show abstract

“…where the new average, · n 0 denotes an average with respect to the conditioned Gaussian measure, defined just as in definition 9 but with H 0 replacing H. The "(H 0 -part)" term would be the effective Hamiltonian if H 1 were zero, and it contributes linear terms to the equations of motion that are easily evaluated by the regression formula (11). The other term in 24 is equal to a power series in To first order in k 4 , we need to evaluate where " constant " denotes terms that are independent of q 1 q n and p 1 p n (and therefore do not affect equations of motion).…”

Section: More General Modelsmentioning

confidence: 99%