The fast reduced QMC matrix-vector product

Josef Dick, Adrian Ebert, Lukas Herrmann, Peter Kritzer, Marcello Longo

We study the approximation of integrals $\int_D f(\boldsymbol{x}^\top A) \mathrm{d} \mu(\boldsymbol{x})$, where $A$ is a matrix, by quasi-Monte Carlo (QMC) rules $N^{-1} \sum_{k=0}^{N-1} f(\boldsymbol{x}_k^\top A)$. We are interested in cases where the main cost arises from calculating the products $\boldsymbol{x}_k^\top A$. We design QMC rules for which the computation of $\boldsymbol{x}_k^\top A$, $k = 0, 1, \ldots, N-1$, can be done fast, and for which the error of the QMC rule is similar to the standard QMC error. We do not require that $A$ has any particular structure. For instance, this approach can be used when approximating the expected value of a function with a multivariate normal random variable with a given covariance matrix, or when approximating the expected value of the solution of a PDE with random coefficients. The speed-up of the computation time is sometimes better and sometimes worse than the fast QMC matrix-vector product from [Dick, Kuo, Le Gia, and Schwab, Fast QMC Matrix-Vector Multiplication, SIAM J. Sci. Comput. 37 (2015)]. As in that paper, our approach applies to (polynomial) lattice point sets, but also to digital nets (we are currently not aware of any approach which allows one to apply the fast method from the aforementioned paper of Dick, Kuo, Le Gia, and Schwab to digital nets). Our method does not use FFT, instead we use repeated values in the quadrature points to derive a reduction in the computation time. This arises from the reduced CBC construction of lattice rules and polynomial lattice rules. The reduced CBC construction has been shown to reduce the computation time for the CBC construction. Here we show that it can also be used to also reduce the computation time of the QMC rule.

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