Dimension Reduction for Systems with Slow Relaxation

We develop reduced, stochastic models for high dimensional, dissipative dynamical systems that relax very slowly to equilibrium and can encode long term memory. We present a variety of empirical and first principles approaches for model reduction, and build a mathematical framework for analyzing the reduced models. We introduce the notions of universal and asymptotic filters to characterize ‘optimal’ model reductions for sloppy linear models. We illustrate our methods by applying them to the practically important problem of modeling evaporation in oil spills.

[1]  Gilbert T. Walker,et al.  On Periodicity in Series of Related Terms , 1931 .

[2]  A J Chorin,et al.  Optimal prediction and the Mori-Zwanzig representation of irreversible processes. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[3]  Stéphane Lafon,et al.  Diffusion maps , 2006 .

[4]  Alan V. Oppenheim,et al.  Discrete-time signal processing (2nd ed.) , 1999 .

[5]  J. Restrepo,et al.  Modelling the mass exchange dynamics of oceanic surface and subsurface oil , 2018, Ocean Modelling.

[6]  G. U. Yule,et al.  The Foundations of Econometric Analysis: On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers ( Philosophical Transactions of the Royal Society of London , A, vol. 226, 1927, pp. 267–73) , 1995 .

[7]  V. Barnett,et al.  Applied Linear Statistical Models , 1975 .

[8]  I. Mezić,et al.  Applied Koopmanism. , 2012, Chaos.

[9]  R. Zwanzig.  Nonequilibrium statistical mechanics , 2001, Physics Subject Headings (PhySH).

[10]  A. Majda,et al.  Nonlinear Laplacian spectral analysis for time series with intermittency and low-frequency variability , 2012, Proceedings of the National Academy of Sciences.

[11]  J. Fineberg,et al.  Slip-stick and the evolution of frictional strength , 2010, Nature.

[12]  Aging in glassy systems: new experiments, simple models, and open questions , 1999, cond-mat/9910387.

[13]  K. S. Brown,et al.  Sloppy-model universality class and the Vandermonde matrix. , 2006, Physical review letters.

[14]  Andrew J. Majda,et al.  Data-driven prediction strategies for low-frequency patterns of North Pacific climate variability , 2015, Climate Dynamics.

[15]  Mervin F. Fingas,et al.  A literature review of the physics and predictive modelling of oil spill evaporation , 1995 .

[16]  K. S. Brown,et al.  Statistical mechanical approaches to models with many poorly known parameters. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Alexandre J. Chorin,et al.  Data-based stochastic model reduction for the Kuramoto--Sivashinsky equation , 2015, 1509.09279.

[18]  Robert Zwanzig,et al.  Problems in nonlinear transport theory , 1980 .

[19]  R. Vautard,et al.  Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series , 1989 .

[20]  A. Amir,et al.  On relaxations and aging of various glasses , 2011, Proceedings of the National Academy of Sciences.

[21]  Michael Ghil,et al.  Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation , 2011, Proceedings of the National Academy of Sciences.

[22]  M. Spaulding.  A state-of-the-art review of oil spill trajectory and fate modeling* , 1988 .

[23]  D Venturi,et al.  Convolutionless Nakajima–Zwanzig equations for stochastic analysis in nonlinear dynamical systems , 2014, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[24]  Crumpling a thin sheet. , 2001, Physical review letters.

[25]  Leo P. Kadanoff,et al.  Discrete Charges on a Two Dimensional Conductor , 2004 .

[26]  Eric Darve,et al.  Computing generalized Langevin equations and generalized Fokker–Planck equations , 2009, Proceedings of the National Academy of Sciences.

[27]  M. Fingas,et al.  Modeling evaporation using models that are not boundary-layer regulated. , 2004, Journal of hazardous materials.

[28]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1972 .

[29]  G. Karniadakis,et al.  Mori-Zwanzig Approach to Uncertainty Quantification , 2015 .

[30]  F. Takens.  Detecting strange attractors in turbulence , 1981 .

[31]  D. Mackay,et al.  Evaporation rates of liquid hydrocarbon spills on land and water , 1973 .

[32]  M. Fingas,et al.  Modeling Oil and Petroleum Evaporation , 2013 .

[33]  R. Zwanzig.  Nonlinear generalized Langevin equations , 1973 .

[34]  Michael H. Kutner.  Applied Linear Statistical Models , 1974 .

[35]  G. Pólya,et al.  Theory of functions, zeros, polynomials, determinants, number theory, geometry , 1977 .

[36]  A. Majda,et al.  Regression models with memory for the linear response of turbulent dynamical systems , 2013 .

[37]  Clint Dawson,et al.  Nearshore sticky waters , 2014 .

[38]  B. Nadler,et al.  Diffusion maps, spectral clustering and reaction coordinates of dynamical systems , 2005, math/0503445.

[39]  R. Kubo.  The fluctuation-dissipation theorem , 1966 .

[40]  A. Stuart,et al.  Extracting macroscopic dynamics: model problems and algorithms , 2004 .

[41]  Alexandre J. Chorin,et al.  Optimal prediction with memory , 2002 .

[42]  V. Baladi.  Positive transfer operators and decay of correlations , 2000 .

[43]  G. Pólya,et al.  Problems and theorems in analysis , 1983 .

[44]  John Harlim,et al.  Forecasting turbulent modes with nonparametric diffusion models: Learning from noisy data , 2015, 1501.06848.

[45]  Alexandre J. Chorin,et al.  Problem reduction, renormalization, and memory , 2005 .

[46]  R. Vautard,et al.  Singular-spectrum analysis: a toolkit for short, noisy chaotic signals , 1992 .

[47]  Kyozi Kawasaki,et al.  Simple derivations of generalized linear and nonlinear Langevin equations , 1973 .

[48]  I. Moroz,et al.  Stochastic parametrizations and model uncertainty in the Lorenz ’96 system , 2013, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[49]  H. Mori.  Transport, Collective Motion, and Brownian Motion , 1965 .

[50]  M. Mézard,et al.  Out of equilibrium dynamics in spin-glasses and other glassy systems , 1997, cond-mat/9702070.

[51]  J. Restrepo,et al.  Modelling the mass exchange dynamics of oceanic surface and subsurface oil , 2017, Ocean Modelling.

[52]  A. Crisanti,et al.  TOPICAL REVIEW: Violation of the fluctuation dissipation theorem in glassy systems: basic notions and the numerical evidence , 2002 .

[53]  Alexandre J. Chorin,et al.  Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics , 2015, Proceedings of the National Academy of Sciences.

[54]  E. Ott.  Chaos in Dynamical Systems: Contents , 1993 .

[55]  Mark K Transtrum,et al.  Geometry of nonlinear least squares with applications to sloppy models and optimization. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[56]  Michael Ghil,et al.  Data-driven non-Markovian closure models , 2014, 1411.4700.

[57]  Juan M. Restrepo,et al.  An Oil Fate Model for Shallow-Waters , 2015 .

[58]  Panos Stinis,et al.  Renormalized Mori–Zwanzig-reduced models for systems without scale separation , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[59]  J. Weber,et al.  Fluctuation Dissipation Theorem , 1956 .

[60]  Andrew J. Majda,et al.  Physics constrained nonlinear regression models for time series , 2012 .

[61]  A. Chorin,et al.  Stochastic Tools in Mathematics and Science , 2005 .

[62]  Dror Givon,et al.  Existence proof for orthogonal dynamics and the Mori-Zwanzig formalism , 2005 .

[63]  D. Sherrington.  Stochastic Processes in Physics and Chemistry , 1983 .

[64]  Philippe Flajolet,et al.  Singularity Analysis of Generating Functions , 1990, SIAM J. Discret. Math..

[65]  Wu,et al.  Scaling in the relaxation of supercooled liquids. , 1990, Physical review letters.

[66]  D Mackay,et al.  Evaporation rate of spills of hydrocarbons and petroleum mixtures. , 1984, Environmental science & technology.

[67]  Oliver Graham Sutton,et al.  Wind structure and evaporation in a turbulent atmosphere , 1934 .

[68]  K. Kawasaki.  Theoretical Methods Dealing with Slow Dynamics , 2000 .

[69]  D. Hoult.  OIL ON THE SEA. PROCEEDINGS OF A SYMPOSIUM ON SCIENTIFIC AND ENGINEERING ASPECTS OF OIL POLLUTION OF THE SEA , 1969 .