Efficient multivariate approximation on the cube

We 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.

Complete Approximations by Multivariate Generalized Gauss-Weierstrass Singular Integrals

This research and survey article deals exclusively with the study of the approximation of generalized multivariate Gauss-Weierstrass singular integrals to the identity-unit operator. Here we study quantitatively most of their approximation properties. The multivariate generalized Gauss-Weierstrass operators are not in general positive linear operators. In particular we study the rate of convergence of these operators to the unit operator, as well as the related simultaneous approximation. These are given via Jackson type inequalities and by the use of multivariate high order modulus of smoothness of the high order partial derivatives of the involved function. Also we study the global smoothness preservation properties of these operators. These multivariate inequalities are nearly sharp and in one case the inequality is attained, that is sharp. Furthermore we give asymptotic expansions of Voronovskaya type for the error of multivariate approximation. The above properties are studied with respect to Lpnorm, 1 ≤ p ≤ ∞.

Modeling Geometric Varieties with Given Differential Characteristics and Its Application

The article describes an approach to modeling geometric manifolds with specified differential properties, which is based on the use of geometric interpolants of a multidimensional space. A geometric interpolant is understood as a geometric object of a multidimensional space passing through predetermined points in advance, the coordinates of which correspond to the initial experimental and statistical information. Principles for determining geometric interpolants and an example of an analytical description of a 3-parameter geometric interpolant belonging to a 4-dimensional space in the form of a geometric scheme and a computational algorithm based on a sequence of point equations are given. The main direction of practical use of geometric interpolants is geometric modeling of multifactor processes and phenomena, but they can also be an effective tool for multivariate approximation. Based on this, the article presents a general approach to modeling geometric manifolds with given differential properties and its application in the form of a numerical solution of differential equations by approximating it using geometric interpolants of a multidimensional space. To implement an approach to modeling geometric manifolds with specified differential properties, it is proposed to use a computational algorithm consisting of 10 points. The advantages of using geometric interpolants for the numerical solution of differential equations are highlighted.

Multivariate approximation of functions on irregular domains by weighted least-squares methods

We propose and analyse numerical algorithms based on weighted least squares for the approximation of a bounded real-valued function on a general bounded domain $\varOmega \subset \mathbb{R}^d$. Given any $n$-dimensional approximation space $V_n \subset L^2(\varOmega )$, the analysis in Cohen and Migliorati (2017, Optimal weighted least-squares methods. SMAI J. Comput. Math., 3, 181–203) shows the existence of stable and optimally converging weighted least-squares estimators, using a number of function evaluations $m$ of the order $n \ln n$. When an $L^2(\varOmega )$-orthonormal basis of $V_n$ is available in analytic form, such estimators can be constructed using the algorithms described in Cohen and Migliorati (2017, Optimal weighted least-squares methods. SMAI J. Comput. Math., 3, 181–203, Section 5). If the basis also has product form, then these algorithms have computational complexity linear in $d$ and $m$. In this paper we show that when $\varOmega $ is an irregular domain such that the analytic form of an $L^2(\varOmega )$-orthonormal basis is not available, stable and quasi-optimally weighted least-squares estimators can still be constructed from $V_n$, again with $m$ of the order $n \ln n$, but using a suitable surrogate basis of $V_n$ orthonormal in a discrete sense. The computational cost for the calculation of the surrogate basis depends on the Christoffel function of $\varOmega $ and $V_n$. Numerical results validating our analysis are presented.