Calculating Macroscopic Consequences of Microscopic Interactions

20 Aug 2007 21:31

[A placeholder while I organize my thoughts]

Statistical mechanics (especially, perhaps, non-equilibrium statistical mechanics), cellular automata (especially their continuum limits), evolutionary theory, economics, sociology, simulation modeling, large deviations of stochastic processes, emergence,

David Cai, Louis Tao and David W. McLaughlin, "An embedded network approach for scale-up of fluctuation-driven systems with preservation of spike information", Proceedings of the National Academy of Sciencces (USA) 101 (2004): 14288--14293 [Interesting, but I really needed to read their prior paper on their coarse-graining method (below) to evaluate this.]
Dror Givon, Raz Kupferman and Andrew Stuart, "Extracting macroscopic dynamics: model problems and algorithms", Nonlinearity 17 (2004): R55--R127 [PDF preprint]
Olof Gonerup and Martin Nilsson Jacobi, "A method for inferring hierarchical dynamics in stochastic processes", nlin.AO/0703034
Martin Nilsson Jacobi, "Quotient Manifold Projections and Hierarchical Dynamics in Smooth Dynamical Systems" [Lie-algebraic techniques for determining when a low-dimensional projection of a high-dimensional dynamical system will itself be a self-contained dynamical system. I heard Martin talk about it, and it's very cool stuff.]
Thomas Schelling, Micromotives and Macrobehavior

Masano Aoki
- New Approaches to Macroeconomic Modeling: Evolutionary Stochastic Dynamics, Multiple Equilibria, and Externalities as Field Effects
- Modeling Aggregate Behavior and Fluctuations in Economics: Stochastic Views of Interacting Agents
Karen Ball, Tom Kurtz, Lea Popovic and Greg Rempala, "Asymptotic analysis of multiscale approximations to reaction networks", math.PR/0508015
David Cai, Louis Tao, Michael Shelley and David McLaughlin, "An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex", Proceedings of the National Academy of Sciences (USA) 101 (2004): 7757--7762
R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, B. Nadler, F. Warner and S. W. Zucker, "Geometric diffusions as a tool for harmonic analysis and structure definition of data" ["We use diffusion semigroups to generate multiscale geometries in order to organize and represent complex structures. We show that appropriately selected eigenfunctions or scaling functions of Markov matrices, which describe local transitions, lead to macroscopic descriptions at different scales. The process of iterating or diffusing the Markov matrix is seen as a generalization of some aspects of the Newtonian paradigm, in which local infinitesimal transitions of a system lead to global macroscopic descriptions by integration. We provide a unified view of ideas from data analysis, machine learning, and numerical analysis."]
1. "Diffusion maps", Proceedings of the National Academy of Sciences 102 (2005): 7426--7431
2. "Multiscale methods", Proceedings of the National Academy of Sciences 102 (2005): 7432--7437
Andreas Degenhard and Javier Rodriguez-Laguna, "Towards the Evaluation of the Relevant Degrees of Freedom in Nonlinear Partial Differential Equations," cond-mat/0106156
Radek Erban, Ioannis G. Kevrekidis and Hans G. Othmer, "An equation-free computational approach for extracting population-level behavior from individual-based models of biological dispersal", physics/0505179
G. Flores Hidalgo and Y. W. Milla, "Dressed (Renormalized) Coordinates in a Nonlinear System", physics/0410238
Navot Israeli and Nigel Goldenfeld, "Coarse-graining of cellular automata, emergence, and the predictability of complex systems", nlin.CG/0508033
Shalev Itzkovitz, Reuven Levitt, Nadav Kashtan, Ron Milo, Michael Itzkovitz and Uri Alon, "Coarse-Graining and Self-Dissimilarity of Complex Networks", q-bio.MN/0405011
Daniel Korenblum and David Shalloway, "Macrostate data clustering", Physical Review E 67 (2003): 056704
Andrew J. Majda, Rafail V. Abramov and Marcus J. Grote, Information Theory and Stochastic for Multiscale Nonlinear Systems [Sounds interesting, to judge from the blurb. PDF draft?]
Brian A. Maurer, "Statistical mechanics of complex ecological aggregates", Ecological Complexity 2 (2005): 71--85
Igor Mezic, "Spectral Properties of Dynamical Systems, Model Reduction and Decompositions", Nonlinear Dynamics 41 (2005): 309--325
Sung Joon Moon and Ioannis G. Kevrekidis, "An equation-free approach to coupled oscillator dynamics: the Kuramoto model example", nlin.AO/0502016 [An approach to predicting the behavior of macroscopic, coarse-grained variables, which is equation-free in the sense that it "circumvent[s] the derivation of explicit dynamical equations (approximately) governing [their] evolution", substituting rather "short burts of appropriately initialized simulations of the 'fine-scale', detailed" model. This sounds like an interesting idea, applicable e.g. to agent-based models, but I need to read the paper to see if/how it actually works.]
Sung Joon Moon, R Ghanem, I. G. Kevrekidis, "An equation-free approach to coupled oscillator dynamics", nlin.AO/0509022
Sung Joon Moon, B. Nabet, Naomi E. Leonard, Simon A. Levin, and I. G. Kevrekidis, "Heterogeneous animal group models and their group-level alignment dynamics; an equation-free approach", q-bio.QM/0606021
Boaz Nadler, Stephane Lafon, Ronald R. Coifman and Ioannis G. Kevrekidis
- "Diffusion maps, spectral clustering and reaction coordinates of dynamical systems", math.NA/0503445
- "Diffusion Maps, Spectral Clustering and Eigenfunctions of Fokker-Planck operators", math.NA/0506090
Liang Qiao, Radek Erban, C. T. Kelley and Ioannis G. Kevrekidis, "Spatially Distributed Stochastic Systems: equation-free and equation-assisted preconditioned computation", q-bio.QM/0606006
A. J. Roberts, "Resolve the multitude of microscale interactions to model stochastic partial differential equations", math.DS/0506533
Wolfgang Stadje, "The evolution of aggregated Markov chains", Statistics and Probability Letters 74 (2005): 303--311 ["Given a stationary two-sided Markov chain ... with finite state space ... and a partition ... we consider the aggregated sequence defined by [applying the partition], which is also stationary but in general not Markovian. We present a tractable way to determine the transition probabilities of [the aggregated process], either given a finite part of its past or given its infinite past. These probabilities are linked to the Radon-Nikodym derivative of [the density of an exponentially-decaying sum of aggregated values, conditional on the unaggregated process] with respect to [the unconditional distribution of the exponentially-decaying sum]".]
Panagiotis Stinis, "A comparative study of two stochastic mode reduction methods", math.NA/0509028
Ji-Feng Yang, "Renormalization group equations as 'decoupling' theorems", hep-th/0507024