An Accelerated Greedy Missing Point Estimation Procedure

Abstract

Abstract. Model reduction via Galerkin projection fails to provide considerable computational savings if applied to general nonlinear systems. This is because the reduced representation of the state vector appears as an argument to the nonlinear function, whose evaluation remains as costly as for the full model. Masked projection approaches, such as the missing point estimation and the (discrete) empirical interpolation method, alleviate this effect by evaluating only a small subset of the components of a given nonlinear term; however, the selection of the evaluated components is a combinatorial problem and is computationally intractable even for systems of small size. This has been addressed through greedy point selection algorithms, which minimize an error indicator by sequentially looping over all components. While doable, this is suboptimal and still costly. This paper introduces an approach to accelerate and improve the greedy search. The method is based on the observation that the greedy algorithm requires solving a sequence of symmetric rank-one modifications to an eigenvalue problem. For doing so, we develop fast approximations that sort the set of candidate vectors that induce the rank-one modifications, without requiring solution of the modified eigenvalue problem. Based on theoretical insights into symmetric rank-one eigenvalue modifications, we derive a variation of the greedy method that is faster than the standard approach and yields better results for the cases studied. The proposed approach is illustrated by numerical experiments, where we observe a speed-up by two orders of magnitude when compared to the standard greedy method while arriving at a better quality reduced model.

Nonlinear Unsteady Reduced Order Models based on Computational Fluid Dynamics for Gust Loads Predictions

“…The selection of the vector components (i.e., the definition of the mask matrix P) is done iteratively by applying an accelerated greedy missing point estimation procedure which improves the exhaustive greedy point selection algorithm by exploiting a rank-one SVD update strategy. 32 The original greedy algorithm minimizes an error indicator by sequentially looping over all the entries of the unsteady residual vector resulting in a very high computational cost of complexity O(N r 3 ). The error indicator is related to the inverse of the minimum singular value σ min (P T s+1 U r ) associated to the POD modes U r projected onto the iteratively populated mask matrix P s+1 ∈ R N ×(s+1) , being s ≥ r the number of already selected indices.…”

Breaking the Kolmogorov Barrier in Model Reduction of Fluid Flows

“…We note that the approach described here is called tensorial ROM, since the model predetermined terms (especially those corresponding to nonlinear terms) are computed in advance as tensors during an offline stage, while the online solution of Equation (18) scales with R 3 . Alternatively, the model terms can be calculated online while minimizing the cost of computing the nonlinear terms through approximating approaches like discrete empirical interpolation method (DEIM) [92,93], gappy POD [60,94], and missing point estimation (MPE) [95,96]. An overview of these techniques can be found in [97,98].…”

The Discrete Empirical Interpolation Method: Canonical Structure and Formulation in Weighted Inner Product Spaces

“…As already mentioned in §1.2, the PBDW framework [43] allows selecting s ≥ r approximation points, and we will proceed with the general case of rectangular S T U r . The oversampling has been successfully used in the related context of missing point estimation, see [3] [49], [70], [71]. We adopt the following notation.…”

“…As can be seen above, when the DEIM operator is generalized to the setting s = r only the projection property or the interpolation property is retained but not both simultaneously. For related developments, see [43], [71], [13].…”