Two-Step Greedy Algorithm for Reduced Order Quadratures

Abstract

We present an algorithm to generate application-specific, global reduced order quadratures (ROQ) for multiple fast evaluations of weighted inner products between parameterized functions. If a reduced basis (RB) or any other projection-based model reduction technique is applied, the dimensionality of integrands is reduced dramatically; however, the cost of approximating the integrands by projection still scales as the size of the original problem. In contrast, using discrete empirical interpolation (DEIM) points as ROQ nodes leads to a computational cost which depends linearly on the dimension of the reduced space. Generation of a reduced basis via a greedy procedure requires a training set, which for products of functions can be very large. Since this direct approach can be impractical in many applications, we propose instead a two-step greedy targeted towards approximation of such products. We present numerical experiments demonstrating the accuracy and the efficiency of the two-step approach. The presented ROQ are expected to display very fast convergence whenever there is regularity with respect to parameter variation. We find that for the particular application here considered, one driven by gravitational wave physics, the two-step approach speeds up the offline computations to build the ROQ by more than two orders of magnitude. Furthermore, the resulting ROQ rule is found to converge exponentially with the number of nodes, and a factor of ∼ 50 savings, without loss of accuracy, is observed in evaluations of inner products when ROQ are used as a downsampling strategy for equidistant samples using the trapezoidal rule. While the primary focus of this paper is on quadrature rules for inner products of parameterized functions, our method can be easily adapted to integrations of single parameterized functions, and some examples of this type are considered.

Towards beating the curse of dimensionality for gravitational waves using reduced basis

“…The exploration of the 8D parameter space will require further development and/or implementation of technical but critical aspects, including the efficient and adaptive sampling techniques for large training spaces (see, e.g., [38]) and rapid evaluation of high accuracy quadratures for parametrized problems [39]. For the problem at hand, a splitting of dimensions as discussed in Sec.…”

Fast and accurate inference on gravitational waves from precessing compact binaries

“…[12,18,19] which have been implemented in C þ þ and parallelized with message passing interface [21,22]. First, on this training set, we apply a greedy algorithm (see algorithm 1 of Ref.…”

“…Our implementation of the empirical interpolation method uses the modification suggested by Ref. [18] which reduces the overall cost from OðN 4 Þ to OðN 3 Þ (see algorithm 2 of Ref. [19]).…”

“…To motivate our approach, notice that a tensor product of the linear basis is sufficient to describeh i¼1 is built following the same iterative enrichment procedure used for the linear basis. A more direct (but more costly) two-step approach to treating these product terms is given in [18].…”

“…(11). Since the ROQ's error, j log LðdjΛÞ − log LðdjΛÞ ROQ j, is controlled by the empirical interpolant's error [18], we are especially interested in quantifying the latter error for any possible gravitational wave model evaluation. Given an approximationâ ≈ a, which could stand for either the linear or quadratic parts, we report the error as the square of the unweighted (S n ¼ 1) norm ofâ − a, which is related to the unweighted overlap, ðâ; aÞ, by ∥â − a∥ 2 ¼ ðâ − a;â − aÞ;…”