Efficient multivariate approximation on the cube

Abstract

AbstractWe combine a periodization strategy for weighted $$L_{2}$$ L 2 -integrands with efficient approximation methods in order to approximate multivariate non-periodic functions on the high-dimensional cube $$\left[ -\frac{1}{2},\frac{1}{2}\right] ^{d}$$ - 1 2 , 1 2 d . Our concept allows to determine conditions on the d-variate torus-to-cube transformations $${\psi :\left[ -\frac{1}{2},\frac{1}{2}\right] ^{d}\rightarrow \left[ -\frac{1}{2},\frac{1}{2}\right] ^{d}}$$ ψ : - 1 2 , 1 2 d → - 1 2 , 1 2 d such that a non-periodic function is transformed into a smooth function in the Sobolev space $${\mathcal {H}}^{m}(\mathbb {T}^{d})$$ H m ( T d ) when applying $$\psi $$ ψ . We adapt $$L_{\infty }(\mathbb {T}^{d})$$ L ∞ ( T d ) - and $$L_{2}(\mathbb {T}^{d})$$ L 2 ( T d ) -approximation error estimates for single rank-1 lattice approximation methods and adjust algorithms for the fast evaluation and fast reconstruction of multivariate trigonometric polynomials on the torus in order to apply these methods to the non-periodic setting. We illustrate the theoretical findings by means of numerical tests in up to $$d=5$$ d = 5 dimensions.