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A Generic C++ Library for Multilevel Quasi-Monte Carlo

by R. N. Gantner

(Report number 2016-30)

Abstract
We present a high-performance framework for single- and multilevel quasi-Monte Carlo methods applicable to a large class of problems in uncertainty quantification. A typical application of interest is the approximation of expectations of quantities depending on solutions to e.g. partial differential equations with uncertain inputs, often yielding integrals over high-dimensional spaces. The goal of the software presented in this paper is to allow recently developed quasi-Monte Carlo (QMC) methods with high, dimension-independent orders of convergence to be combined with a multilevel approach and applied to large problems in a generic way, easing distributed memory parallel implementation. The well-known multilevel Monte Carlo method is also supported as a particular case, and some standard choices of distributions are included. For so-called interlaced polynomial lattice rules, a recently developed QMC method, precomputed generating vectors are required; some such vectors are provided for common cases. After a theoretical introduction, the implementation is briefly explained and a user guide is given to aid in applying the framework to a concrete problem. We give two examples: a simple model problem designed to illustrate the mathematical concepts as well as the advantages of the multilevel method, including a measured decrease in computational work of multiple orders of magnitude for engineering accuracy, and a larger problem that exploits high-performance computing to show excellent parallel scaling. Some concrete use cases and applications are mentioned, including uncertainty quantification of partial differential equation models, and the approximation of solutions of corresponding Bayesian inverse problems. This software framework easily admits modifications; custom features like schedulers and load balancers can be implemented without hassle, and the code documentation includes relevant details.

Keywords: Multilevel Monte Carlo, MLMC, multilevel, Quasi-Monte Carlo, C++, High-Performance Computing

@Techreport{G16_667,
  author = {R. N. Gantner},
  title = {A Generic C++ Library for Multilevel Quasi-Monte Carlo},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-30},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-30.pdf },
  year = {2016}
}

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