Journal ArticleDOI
Equation-free: The computer-aided analysis of complex multiscale systems
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Over the last few years with several collaborators, a mathematically inspired, computational enabling technology is developed and validated that allows the modeler to perform macroscopic tasks acting on the microscopic models directly, and can lead to experimental protocols for the equation-free exploration of complex system dynamics.Abstract:
The best available descriptions of systems often come at a fine level (atomistic, stochastic, microscopic, agent based), whereas the questions asked and the tasks required by the modeler (prediction, parametric analysis, optimization, and control) are at a much coarser, macroscopic level. Traditional modeling approaches start by deriving macroscopic evolution equations from microscopic models, and then bringing an arsenal of computational tools to bear on these macroscopic descriptions. Over the last few years with several collaborators, we have developed and validated a mathematically inspired, computational enabling technology that allows the modeler to perform macroscopic tasks acting on the microscopic models directly. We call this the “equation-free” approach, since it circumvents the step of obtaining accurate macroscopic descriptions. The backbone of this approach is the design of computational “experiments”. In traditional numerical analysis, the main code “pings“ a subroutine containing the model, and uses the returned information (time derivatives, etc.) to perform computer-assisted analysis. In our approach the same main code “pings“ a subroutine that runs an ensemble of appropriately initialized computational experiments from which the same quantities are estimated. Traditional continuum numerical algorithms can, thus, be viewed as protocols for experimental design (where “experiment“ means a computational experiment set up, and performed with a model at a different level of description). Ultimately, what makes it all possible is the ability to initialize computational experiments at will. Short bursts of appropriately initialized computational experimentation -through matrix-free numerical analysis, and systems theory tools like estimationbridge microscopic simulation with macroscopic modeling. If enough control authority exists to initialize laboratory experiments “at will” this computational enabling technology can lead to experimental protocols for the equation-free exploration of complex system dynamics.read more
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Book ChapterDOI
Modeling Momentum and Mass Transport in Cellular Biological Media: From the Molecular to the Tissue Scale
TL;DR: This chapter presents a comprehensive review of the theoretical modeling of momentum and mass transport in cellular biological media and presents the conceptual framework and the mathematical foundation for various theoretical approaches, including single-scale–single-phase models, the theory of interacting continua, upscaling methods and multiscale equation-free approaches.
Journal ArticleDOI
An equation-free approach to nonlinear control: coarse feedback linearization with pole-placement
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Journal ArticleDOI
Exploration of effective potential landscapes using coarse reverse integration
TL;DR: In these "equation-free" computational illustrations, estimation techniques are applied to the results of short bursts of inner simulation to obtain the unavailable quantities required for reverse ring integration, which naturally leads to approximations of the effective landscape.
Journal ArticleDOI
Ergodic SDEs on submanifolds and related numerical sampling schemes
TL;DR: In this paper, the authors constructed a family of ergodic diffusion processes on the level set of a smooth function whose invariant measures coincide with the given one, and showed that the corresponding SDEs of these processes have relatively simple forms, and, moreover, a consistent numerical scheme which samples the conditional measure asymptotically.
References
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Numerical Integration of the Cartesian Equations of Motion of a System with Constraints: Molecular Dynamics of n-Alkanes
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Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
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A Reflection on Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
J. Guckenheimer,P. J. Holmes +1 more
TL;DR: In this paper, the authors introduce differential equations and dynamical systems, including hyperbolic sets, Sympolic Dynamics, and Strange Attractors, and global bifurcations.
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