Transporting higher-order quadrature rules - Quasi-Monte Carlo points and sparse grids for mixture distributions

Abstract Integration against, and hence sampling from, high-dimensional probability distributions is of essential importance in many application areas and has been an active research area for decades. One approach that has drawn increasing attention in recent years has been the generation of samples from a target distribution $$\mathbb {P}_{\text {tar} }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mtext>tar</mml:mtext> </mml:msub> </mml:math> using transport maps: if $$\mathbb {P}_{\text {tar} } = T_\sharp \mathbb {P}_{\text {ref} }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mtext>tar</mml:mtext> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mo>♯</mml:mo> </mml:msub> <mml:msub> <mml:mi>P</mml:mi> <mml:mtext>ref</mml:mtext> </mml:msub> </mml:mrow> </mml:math> is the pushforward of an easily-sampled probability distribution $$\mathbb {P}_{\text {ref} }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mtext>ref</mml:mtext> </mml:msub> </mml:math> under the transport map T , then the application of T to $$\mathbb {P}_{\text {ref} }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mtext>ref</mml:mtext> </mml:msub> </mml:math> -distributed samples yields $$\mathbb {P}_{\text {tar} }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mtext>tar</mml:mtext> </mml:msub> </mml:math> -distributed samples. This paper proposes the application of transport maps not just to random samples, but also to quasi-Monte Carlo points, higher-order nets, and sparse grids so that the transformed samples inherit the original convergence rates that are often better than $$N^{-1/2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> , N being the number of samples/quadrature nodes. Our main result is the derivation of an explicit transport map for the case that $$\mathbb {P}_{\text {tar} }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mtext>tar</mml:mtext> </mml:msub> </mml:math> is a mixture of simple distributions, e.g. a Gaussian mixture, in which case application of the transport map T requires the solution of an explicit ODE with closed-form right-hand side. Mixture distributions are of particular applicability and interest since many methods proceed by first approximating $$\mathbb {P}_{\text {tar} }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mtext>tar</mml:mtext> </mml:msub> </mml:math> by a mixture and then sampling from that mixture (often using importance reweighting). Hence, this paper allows for the sampling step to provide a better convergence rate than $$N^{-1/2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> for all such methods.