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We study numerical integration over bounded regions in $\mathbb{R}^s, s\ge1$with respect to some probability measure. We replace random sampling withquasi-Monte Carlo methods, where the underlying point set is derived fromdeterministic constructions that aim to fill the space more evenly than randompoints. Such quasi-Monte Carlo point sets are ordinarily designed for theuniform measure, and the theory only works for product measures when acoordinate-wise transformation is applied. Going beyond this setting, we firstconsider the case where the target density is a mixture distribution where eachterm in the mixture comes from a product distribution. Next we consider targetdensities which can be approximated with such mixture distributions. We requirethe approximation to be a sum of coordinate-wise products and the approximationto be positive everywhere (so that they can be re-scaled to probability densityfunctions). We use tensor product hat function approximations for this purposehere, since a hat function approximation of a positive function is itselfpositive.  We also study more complex algorithms, where we first approximate the targetdensity with a general Gaussian mixture distribution and approximate themixtures with an adaptive hat function approximation on rotated intervals. TheGaussian mixture approximation allows us to locate the essential parts of thetarget density, whereas the adaptive hat function approximation allows us toapproximate the finer structure of the target density. We prove convergencerates for each of the integration techniques based on quasi-Monte Carlosampling for integrands with bounded partial mixed derivatives. The employedalgorithms are based on digital $(t,s)$-sequences over the finite field$\mathbb{F}_2$ and an inversion method. Numerical examples illustrate theperformance of the algorithms for some target densities and integrands.

Quasi-Monte Carlo methods for mixture distributions and approximated distributions via piecewise linear interpolation