Fundamental limitations of ad hoc linear and quadratic multi-level regression models for physical systems

Andrew J. Majda, Yuan Yuan

Research output: Contribution to journalArticle

Abstract

A central issue in contemporary applied mathematics is the development of simpler dynamical models for a reduced subset of variables in complex high dimensional dynamical systems with many spatio-temporal scales. Recently, ad hoc quadratic multi-level regression models have been proposed to provide suitable reduced nonlinear models directly from data. The main results developed here are rigorous theorems demonstrating the non-physical finite time blow-up and large time instability in statistical solutions of general scalar multi-level quadratic regression models with corresponding unphysical features of the invariant measure. Surprising intrinsic model errors due to discrete sampling errors are also shown to occur rigorously even for linear multi-level regression dynamic models. all of these theoretical results are corroborated by numerical experiments with simple models. Single level nonlinear regression strategies with physical cubic damping are shown to have significant skill on the same test problems.

Original languageEnglish (US)
Pages (from-to)1333-1363
Number of pages31
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume17
Issue number4
DOIs
StatePublished - Jun 2012

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Multilevel Models
Regression Model
Discrete Sampling
Statistical Solutions
Finite Time Blow-up
Model Error
Nonlinear Regression
Reduced Model
Dynamical Model
Applied mathematics
Invariant Measure
Test Problems
Nonlinear Model
Dynamic Model
Damping
High-dimensional
Dynamical system
Numerical Experiment
Scalar
Subset

Keywords

  • Instability
  • Model error
  • Nonlinear regression models
  • Unphysical blow up

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

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