Limited‐memory adaptive snapshot selection for proper orthogonal decomposition

Abstract

Summary Reduced order models are useful for accelerating simulations in many‐query contexts, such as optimization, uncertainty quantification, and sensitivity analysis. However, offline training of reduced order models (ROMs) can have prohibitively expensive memory and floating‐point operation costs in high‐performance computing applications, where memory per core is limited. To overcome this limitation for proper orthogonal decomposition, we propose a novel adaptive selection method for snapshots in time that limits offline training costs by selecting snapshots according an error control mechanism similar to that found in adaptive time‐stepping ordinary differential equation solvers. The error estimator used in this work is related to theory bounding the approximation error in time of proper orthogonal decomposition‐based ROMs, and memory usage is minimized by computing the singular value decomposition using a single‐pass incremental algorithm. Results for a viscous Burgers' test problem demonstrate convergence in the limit as the algorithm error tolerances go to zero; in this limit, the full‐order model is recovered to within discretization error. A parallel version of the resulting method can be used on supercomputers to generate proper orthogonal decomposition‐based ROMs, or as a subroutine within hyperreduction algorithms that require taking snapshots in time, or within greedy algorithms for sampling parameter space. Copyright © 2016 John Wiley & Sons, Ltd.

Fixed‐precision randomized low‐rank approximation methods for nonlinear model order reduction of large systems

“…Most attention in the MOR community has so far focused on SVD methods based on Arnoldi or Lanczos iterations, as proposed in the work of Chaturantabut and Sorensen and on incremental SVD algorithms. () These are also suitable for reducing the asymptotic complexity to . For an overview, we refer to the work of Bach et al…”

“…Different remedies have been proposed in literature, including the use of iterative, or incremental() SVD methods, which can achieve an asymptotic complexity of . More recently, the use of randomized low‐rank approximation methods from numerical linear algebra was proposed in this context.…”

Application of Incremental Proper Orthogonal Decomposition for the Reduction of Very Large Transient Flow Field Data

“…This algorithm is called Incremental Singular Value Decomposition (ISVD) or Incremental Proper Orthogonal Decomposition (IPOD). (6)- (7) This algorithm computes the singular vectors of datasets incrementally when new data is available. Please note that left singular vectors basically have the same meaning as POD modes as mentioned above.…”

“…Secondly, all the saved data needs to be loaded into the main memory (RAM) at the same time to compute the POD modes, which results in very huge RAM requirements. In contrast to that, Incremental Proper Orthogonal Decomposition (6)- (7) (IPOD, or Incremental Singular Value Decomposition) updates modes incrementally when new snapshot data is available. Therefore, it is not necessary to save all the snapshots of the transient flow field to disk during a CFD simulation.…”

A POD projection method for large-scale algebraic Riccati equations

“…The POD approach may generally require more computational effort and storage compared to other similar approaches; however, we emphasize that our primary goal is not offline computation time, but to move toward an efficient, highly accurate, completely data-based algorithm. We do note that it is highly likely that the computational efficiency and storage requirements of the proposed POD projection algorithm can be greatly improved using an incremental/adaptive POD algorithm; see, e.g., [54,53] and the references therein. We intend to explore this in the near future.…”