Analytical Description of Optimally Time-Dependent Modes for Reduced-Order Modeling of Transient Instabilities

Abstract

The optimally time-dependent (OTD) modes form a time-evolving orthonormal basis that captures directions in phase space associated with transient and persistent instabilities. In the original formulation, the OTD modes are described by a set of coupled evolution equations that need to be solved along the trajectory of the system. For many applications where real-time estimation of the OTD modes is important, such as control or filtering, this is an expensive task. Here, we examine the low-dimensional structure of the OTD modes. In particular, we consider the case of slow-fast systems, and prove that OTD modes rapidly converge to a slow manifold, for which we derive an asymptotic expansion. The result is a parametric description of the OTD modes in terms of the system state in phase space. The analytical approximation of the OTD modes allows for their offline computation, making the whole framework suitable for real-time applications. In addition, we examine the accuracy of the slow-manifold approximation for systems in which there is no explicit timescale separation. In this case, we show numerically that the asymptotic expansion of the OTD modes is still valid for regions of the phase space where strongly transient behavior is observed, and for which there is an implicit scale separation. We also find an analogy between the OTD modes and the Gram-Schmidt vectors (also known as orthogonal or backward Lyapunov vectors), and thereby establish new properties of the former. Several examples of low-dimensional systems are provided to illustrate the analytical formulation.

Transient linear stability of pulsating Poiseuille flow using optimally time-dependent modes

“…A further step in the understanding of the OTD framework was taken in Blanchard & Sapsis (2019 a ), who consider the link to Gram–Schmidt vectors. They show that on choosing the rotation matrix as the OTD basis vectors become identical to the continuously orthonormalised Gram–Schmidt vectors and the evolution equation for the th OTD basis vector reduces to Apart from the theoretical appeal, this choice of also has advantageous numerical properties since the evolution equations (2.9) are lower triangular (notice the difference in the limit of the sum compared with (2.4)) and can therefore be efficiently solved through forward substitution (Blanchard & Sapsis 2019 a ). The original OTD formulation is not hierarchical in the sense that adding a basis vector requires the recomputation of all vectors.…”

“…The original OTD formulation is not hierarchical in the sense that adding a basis vector requires the recomputation of all vectors. The Blanchard & Sapsis (2019 a ) formulation overcomes this issue.…”

“…In particular, it was shown that they converge exponentially fast to the most unstable eigendirections of the Cauchy–Green tensor, making them well suited for the reduced-order computation of finite-time Lyapunov exponents (Babaee et al. 2017), and that they are identical to continuously orthonormalised Gram–Schmidt vectors for a specific choice of parameters (Blanchard & Sapsis 2019 a ), which has proved to be a relevant connection for the application of OTD modes. The strength of the OTD basis vectors is their capacity to span the instantaneously most unstable subspace of user defined size in the (possibly infinite-dimensional) phase space and to capture both modal and non-modal growth within that subspace.…”

Learning the tangent space of dynamical instabilities from data

“…One way to achieve this is to continuously apply the Gram-Schmidt algorithm to the collection {v i } r i=1 , starting with v 1 and moving down. Blanchard & Sapsis 16 showed that the resulting vectors obeyu…”

Features of functional dependence of random phenomena and values in social being in conditions of its unstability (the environmental position)

“…H. Arbaby, D. Sapsis carried out modeling and analysis of systems that have a large number of degrees of freedom, possibly combined with a significant amount of uncertainty in the parameters [1]. A. Blanchard, D. Sapsis predicted transient instability and extreme events in arrogant systems [3]. M. Hadji, J. Kluger, D. Sapsis, A. Slocum substantiated that all this is related to the amount of energy expended and noted that one of the advantages of wave energy is higher predictability and the minimum number of changes [13].…”