Blended reduced subspace algorithms for uncertainty quantification of quadratic systems with a stable mean state

Themistoklis P. Sapsis, Andrew J. Majda

Research output: Contribution to journalArticle

Abstract

Order-reduction schemes have been used successfully for the analysis and simplification of high-dimensional systems exhibiting low-dimensional dynamics. In this work, we first focus on presenting generic limitations of order-reduction techniques in systems with stable mean state that exhibit irreducible high-dimensional features such as non-normal dynamics, wide energy spectra, or strong energy cascades between modes. The reduced-order framework that we consider to illustrate these limitations is the dynamically orthogonal (DO) field equations. This framework is applied to a series of examples with stable mean state, including a linear non-normal system, and a nonlinear triad system in various dynamical configurations. After illustrating the weaknesses and generic limitations of order reduction, we develop a novel, two-way coupled, blended approach based on the quasilinear Gaussian (QG) closure and the DO field equations. The new method (QG-DO) overcomes the limitations of its two ingredients and achieves exceptional performance in the examples described previously as well as in other configurations with strongly transient character without using any tuned or adjustable parameters.

Original languageEnglish (US)
Pages (from-to)61-76
Number of pages16
JournalPhysica D: Nonlinear Phenomena
Volume258
DOIs
StatePublished - Sep 1 2013

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Keywords

  • Blended stochastic methods
  • Dynamical orthogonality
  • Gaussian closure
  • Limitations of reduced-order models
  • Statistical modeling of nonlinear energy fluxes
  • Uncertainty quantification

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

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