Adaptive POD‐based low‐dimensional modeling supported by residual estimates

Abstract

SUMMARYAn adaptive low-dimensional model is considered to simulate time-dependent dynamics in nonlinear dissipative systems governed by PDEs. The method combines an inexpensive POD-based Galerkin system with short runs of a standard numerical solver that provides the snapshots necessary to first construct and then update the POD modes. Switching between the numerical solver and the Galerkin system is decided 'on the fly' by monitoring (i) a truncation error estimate and (ii) a residual estimate. The latter estimate is used to control the mode truncation instability and highly improves former adaptive strategies that detected this instability by monitoring consistency with a second instrumental Galerkin system based on a larger number of POD modes. The most computationally expensive run of the numerical solver occurs at the outset, when the whole set of POD modes is calculated. This step is improved by using mode libraries, which may either be generic or result from former applications of the method. The outcome is a flexible, robust, computationally inexpensive procedure that adapts itself to the local dynamics by using the faster Galerkin system for the majority of the time and few, on demand, short runs of a numerical solver. The method is illustrated considering the complex Ginzburg-Landau equation in one and two space dimensions.

Online Adaptive Local-Global Model Reduction for Flows in Heterogeneous Porous Media

“…In this case, two reduced systems must be solved at a time, thus incurring higher computational cost. In [43], a residual based indicator was used in nonlinear dissipative systems. Here, we propose to use local error indicators based on the multiscale basis, which is cheaper to be computed.…”

Accelerating oil reservoir simulations using POD on the fly

“…In addition to an updating method to recalculate the POD modes using snapshots calculated in short runs of the FM, some ingredients have been added to the method to further increase the computational efficiency, including the use of a limited number of mesh points to project the governing equations onto the POD modes and monitoring both the approximation and possible mode truncation instabilities using appropriate estimates. Comparing the whole CPU cost (the snapshots computation using the FM included) of the ROM with its counterpart using the FM alone, the strategy gives acceleration factors as large as 15 and 350 for the complex Ginzburg-Landau equation (a paradigm of very demanding dynamics and transitions) in one and two space dimensions, respec-82 S. LE CLAINCHE, F. VARAS AND J. M. VEGA tively [25]. These ROMs are essentially different from existing adaptive preprocessed ROMs, in which adaptation is used for different ends, such as iteratively enriching the snapshots set and thus improving accuracy in multiparameter steady simulations [26] or curing the mode truncation instability [27,28] while using fixed POD modes.…”

Adaptive Reduced Order Modeling of Aeroelastic Flows Based on Proper Orthogonal Decomposition

“…An additional important property of the POD modes is that, if the snapshot matrix is approximated as S N making use of the first N POD modes, the relative root mean However, due to the fact that the covariance matrix requires the multiplication by the inner product of the snapshots, the POD modes associated with singular values such that @D jj A=@D II A < IH V are not valid (Rapún et al, 2015) with the standard double precision binary floating point format.…”

“…The new methodology, introduced here in the context of unsteady aerodynamics and computational aeroelasticity, known as POD on the fly and previously presented in (Alonso et al, 2009(Alonso et al, , 2010Rapún et al, 2015;Terragni and Vega, 2014;Terragni et al, 2011), overcomes these drawbacks.…”

“…A different perspective has been tested for simpler equations such as the Complex Ginzburg-Landau equation (Rapún et al, 2015), a paradigm of pattern forming systems that exhibits quite complex dynamic behaviors. The method consists in an adaptive strategy in which POD modes are calculated on demand as the simulation proceeds; this strategy has been proven very robust in both calculating particular solutions and constructing bifurcation diagrams (Terragni and Vega, 2014) that require a large number of particular time-dependent runs.…”