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We introduce a new method to approximate integrals $\int_{\mathbb{R}^d}f(\boldsymbol{x}) \, \mathrm{d} \boldsymbol{x}$ which simply scales latticerules from the unit cube $[0,1]^d$ to properly sized boxes on $\mathbb{R}^d$,hereby achieving higher-order convergence that matches the smoothness of theintegrand function $f$ in a certain Sobolev space of dominating mixedsmoothness. Our method only assumes that we can evaluate the integrand function$f$ and does not assume a particular density nor the ability to sample from it.In particular, for the theoretical analysis we show a new result that themethod of adding Bernoulli polynomials to a function to make it "periodic" on abox without changing its integral value over the box, is equivalent to anorthogonal projection from a well chosen Sobolev space of dominating mixedsmoothness to an associated periodic Sobolev space of the same dominating mixedsmoothness, which we call a Korobov space. We note that the Bernoullipolynomial method is often not used because of its excessive computationalcomplexity and also here we only make use of it in our theoretical analysis. Weshow that our new method of applying scaled lattice rules to increasing boxescan be interpreted as orthogonal projections with decreasing projection error.Such a method would not work on the unit cube since then the committed errorcaused by non-periodicity of the integrand would be constant, but forintegration on the Euclidean space we can use the certain decay towards zerowhen the boxes grow. Hence we can bound the truncation error as well as theprojection error and show higher-order convergence in applying scaled latticerules for integration on Euclidean space. We illustrate our theoreticalanalysis by numerical experiments which confirm our findings.

mathematics of computation

Scaled lattice rules for integration on $\mathbb{R}^d$ achieving  higher-order convergence with error analysis in terms of orthogonal  projections onto periodic spaces