Harbir Antil - Research

Reduced Order Quadrature with Application to Numerical Relativity

For a description of model reduction techniques we refer to: link


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 evaluating the reduced integrals 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 converges 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 also considered.


Related Publications

  • H. Antil, D. Chen, and S. E. Field.
    A Note on QR-Based Model Reduction: Algorithm, Software, and Gravitational Wave Applications.
    Submitted 2017.

    See Code

  • H. Antil, S. Field, F. Herrmann, R. H. Nochetto, and M. Tiglio.
    Two-step Greedy Algorithm for Reduced Order Quadratures.
    Journal of Scientific Computing, 57:604--637, 2013. arxiv
  • Literature

  • S. Field C. Galley, F. Herrmann, J. Hesthaven, E. Ochsner M. Tiglio.
    Reduced basis catalogs for gravitational wave templates . Physical Review Letters (2011), 106(22). link