Adaptive simplification of complex multiscale systems.

@article{Chiavazzo2011AdaptiveSO,
  title={Adaptive simplification of complex multiscale systems.},
  author={Eliodoro Chiavazzo and Ilya Karlin},
  journal={Physical review. E, Statistical, nonlinear, and soft matter physics},
  year={2011},
  volume={83 3 Pt 2},
  pages={
          036706
        }
}
  • E. Chiavazzo, I. Karlin
  • Published 2011
  • Mathematics, Physics, Medicine
  • Physical review. E, Statistical, nonlinear, and soft matter physics
A fully adaptive methodology is developed for reducing the complexity of large dissipative systems. This represents a significant step toward extracting essential physical knowledge from complex systems, by addressing the challenging problem of a minimal number of variables needed to exactly capture the system dynamics. Accurate reduced description is achieved, by construction of a hierarchy of slow invariant manifolds, with an embarrassingly simple implementation in any dimension. The method… Expand

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References

SHOWING 1-10 OF 73 REFERENCES
Approximation of slow attracting manifolds in chemical kinetics by trajectory-based optimization approaches.
TLDR
This work presents a framework of trajectory-based optimization for model reduction in chemical kinetics and a general class of reduction criteria characterizing the relaxation of chemical forces along reaction trajectories and provides results for the computational approximation of slow attracting low-dimensional manifolds in terms of families of optimal trajectories for a six-component hydrogen combustion mechanism. Expand
Model reduction and coarse-graining approaches for multiscale phenomena
TLDR
The generic nature and the power of the pertinent conceptual, analytical and computational frameworks helps eliminate some of the traditional language barriers, which often unnecessarily impede scientific progress and the interaction of researchers between disciplines such as physics, chemistry, biology, applied mathematics and engineering. Expand
Invariant manifolds for dissipative systems
A method is given for a study of a nonlinear evolution equation for finding “slow” invariant manifolds. The method is studied for the evolution problem u˙=Au +Au,u /u,uu, u0=u0, where A is a linear,Expand
The Role of Thermodynamics in Model Reduction when using Invariant Grids
In the present work, we develop in detail the process leading to reduction of models in chemical kinetics when using the Method of Invariant Grids (MIG). To this end, reduced models (invariant grids)Expand
Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space
A general procedure for simplifying chemical kinetics is developed, based on the dynamical systems approach. In contrast to conventional reduced mechanisms no information is required concerning whichExpand
Geometric investigation of low-dimensional manifolds in systems approaching equilibrium
Many systems approach equilibrium slowly along surfaces of dimension smaller than the original dimensionality. Such systems include coupled chemical kinetics and master equations. In the past theExpand
Invariant Manifolds for Physical and Chemical Kinetics
Introduction.- The Source of Examples.- Invariance Equation in the Differential Form.- Film Extension of the Dynamics: Slowness as Stability.- Entropy, Quasi-Equilibrium and Projector Field.- NewtonExpand
Invariant grids for reaction kinetics
Abstract In this paper, we construct low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). MIM is based onExpand
Method of invariant manifold for chemical kinetics
In this paper, we review the construction of low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). The MIMExpand
Computing minimal entropy production trajectories: an approach to model reduction in chemical kinetics.
  • D. Lebiedz
  • Physics, Medicine
  • The Journal of chemical physics
  • 2004
TLDR
A global approach to model reduction based on the concept of minimal entropy production and its numerical implementation is presented, which aims to enable accurate numerical simulations of even high dimensional and spatially extended reaction systems. Expand
...
1
2
3
4
5
...