TY - JOUR
T1 - Blended reduced subspace algorithms for uncertainty quantification of quadratic systems with a stable mean state
AU - Sapsis, Themistoklis P.
AU - Majda, Andrew J.
N1 - Funding Information:
The research of A. Majda is partially supported by NFS grant DMS-0456713 , NSF CMG grant DMS-1025468 , and ONR grants ONR-DRI N00014-10-1-0554 and N00014-11-1-0306 . T. Sapsis is supported as a postdoctoral fellow on the first and third grants.
PY - 2013/9/1
Y1 - 2013/9/1
N2 - 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.
AB - 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.
KW - Blended stochastic methods
KW - Dynamical orthogonality
KW - Gaussian closure
KW - Limitations of reduced-order models
KW - Statistical modeling of nonlinear energy fluxes
KW - Uncertainty quantification
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U2 - 10.1016/j.physd.2013.05.004
DO - 10.1016/j.physd.2013.05.004
M3 - Article
AN - SCOPUS:84879973498
SN - 0167-2789
VL - 258
SP - 61
EP - 76
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
ER -