Well‐scaled, a‐posteriori error estimation for model order reduction of large second‐order mechanical systems

Abstract

Model Order Reduction is used to vastly speed up simulations but it also introduces an error to the simulation results, which needs to be controlled. The a‐posteriori error estimator of Ruiner et al. for second‐order systems, which is based on the residual, has the advantage of having provable upper bounds and being usable independently of the reduction method. Nevertheless a bottleneck is found in the offline phase, making it unusable for larger models. We use the spectral theorem, power series expansions, monotonicity properties, and self‐tailored algorithms to largely speed up the offline phase by one polynomial order both in terms of computation time as well as storage complexity. All properties are proven rigorously. This eliminates the aforementioned bottleneck. Hence, the error estimator of Ruiner et al. can finally be used for large, linear, second‐order mechanical systems reduced by any model reduction method based on Petrov–Galerkin reduction. The examples show speedups of up to 28.000 and the ability to compute much larger systems with a fixed amount of memory.

A Posteriori Error Estimation for Model Order Reduction of Parametric Systems

“…, K, where ρk (µ) := ∥r k (µ)∥/∥r k (µ)∥ and Φk (µ) = ∥e du ∥ + ∥V du xdu (µ)∥ with e du := A t (µ) −1 r du (µ). Here, xdu (µ) and r du (µ) are defined in (34) and (35), respectively.…”

“…In Step 8, we have used the criterion ∥V e êdu (µ)∥ to select the parameter µ * for constructing V du for the ROM (34). Recalling the state error estimator (45) for steady parametric systems proposed in (45), it is easy to see that ∥V e êdu (µ)∥ is exactly the state error estimator for the state error ∥x du (µ)−x du (µ)∥ of the ROM (34). We use this state error estimator to iteratively construct the projection matrix V du for the ROM (34).…”

Multi‐fidelity error estimation accelerates greedy model reduction of complex dynamical systems

“…10,12,17,18 Therefore, many efforts have been made in this direction to develop computable error estimators for different problems. 12,14,17,[19][20][21][22][23][24][25][26][27][28][29] However, more attention has been paid to improve the effectivity or accuracy of the error estimator than to develop more efficient strategies to accelerate the greedy process. 26,[28][29][30][31] Recently, some techniques are proposed to improve the adaptivity of the greedy algorithm.…”

“…It is known that an efficient error estimator makes the greedy algorithm successful in producing an accurate ROM without running many iterations 10,12,17,18 . Therefore, many efforts have been made in this direction to develop computable error estimators for different problems 12,14,17,19‐29 . However, more attention has been paid to improve the effectivity or accuracy of the error estimator than to develop more efficient strategies to accelerate the greedy process 26,28‐31 .…”

Multi-fidelity error estimation accelerates greedy model reduction of complex dynamical systems

“…It is known that an efficient error estimator makes the greedy algorithm successful in producing an accurate ROM without running many iterations. Therefore, many efforts have been made in this direction to develop computable error estimators for different problems [15,16,18,19,20,21,22,23,24,25,28,33,34,35]. However, more attention has been paid to improve the effectivity or accuracy of the error estimator than to develop more efficient strategies to accelerate the greedy process [13,25,33,34,36].…”