arXiv subjects

G. W. Wasilkowski

Publications and source records attributed to G. W. Wasilkowski.

9 recordsLinked to original sources

arXiv2020

On Quasi-Monte Carlo Methods in Weighted ANOVA Spaces

In the present paper we study quasi-Monte Carlo rules for approximating integrals over the $d$-dimensional unit cube for functions from weighted Sobolev spaces of regularity one. While the properties of these rules are well understood for anchored Sobolev spaces, this is not the case for the ANOVA spaces, which are another very important type of reference spaces for quasi-Monte Carlo rules. Using a direct approach we provide a formula for the worst case error of quasi-Monte Carlo rules for functions from weighted ANOVA spaces. As a consequence we bound the worst case error from above in terms of weighted discrepancy of the employed integration nodes. On the other hand we also obtain a general lower bound in terms of the number $n$ of used integration nodes. For the one-dimensional case our results lead to the optimal integration rule and also in the two-dimensional case we provide rules yielding optimal convergence rates.

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arXiv2019

On alternative quantization for doubly weighted approximation and integration over unbounded domains

It is known that for a $ρ$-weighted $L_q$-approximation of single variable functions $f$ with the $r$th derivatives in a $ψ$-weighted $L_p$ space, the minimal error of approximations that use $n$ samples of $f$ is proportional to $\|ω^{1/α}\|_{L_1}^α\|f^{(r)}ψ\|_{L_p}n^{-r+(1/p-1/q)_+},$ where $ω=ρ/ψ$ and $α=r-1/p+1/q.$ Moreover, the optimal sample points are determined by quantiles of $ω^{1/α}.$ In this paper, we show how the error of best approximations changes when the sample points are determined by a quantizer $κ$ other than $ω.$ Our results can be applied in situations when an alternative quantizer has to be used because $ω$ is not known exactly or is too complicated to handle computationally. The results for $q=1$ are also applicable to $ρ$-weighted integration over unbounded domains.

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arXiv2018

On efficient weighted integration via a change of variables

In this paper, we study the approximation of $d$-dimensional $\rho$-weighted integrals over unbounded domains $\mathbb{R}_+^d$ or $\mathbb{R}^d$ using a special change of variables, so that quasi-Monte Carlo (QMC) or sparse grid rules can be applied to the transformed integrands over the unit cube. We consider a class of integrands with bounded $L_p$ norm of mixed partial derivatives of first order, where $p\in[1,+\infty].$ The main results give sufficient conditions on the change of variables $\nu$ which guarantee that the transformed integrand belongs to the standard Sobolev space of functions over the unit cube with mixed smoothness of order one. These conditions depend on $\rho$ and $p$. The proposed change of variables is in general different than the standard change based on the inverse of the cumulative distribution function. We stress that the standard change of variables leads to integrands over a cube; however, those integrands have singularities which make the application of QMC and sparse grids ineffective. Our conclusions are supported by numerical experiments.

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arXiv2018

Embeddings for Infinite-Dimensional Integration and $L_2$-Approximation with Increasing Smoothness

We study integration and $L_2$-approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh, Haar, and Sobolev spaces. For the proofs we derive embedding theorems between spaces of increasing smoothness and appropriate weighted function spaces of fixed smoothness.

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arXiv2017

Truncation in Average and Worst Case Settings for Special Classes of $\infty$-Variate Functions

The paper considers truncation errors for functions of the form $f(x_1,x_2,\dots)=g(\sum_{j=1}^\infty x_j\,\xi_j)$, i.e., errors of approximating $f$ by $f_k(x_1,\dots,x_k)=g(\sum_{j=1}^k x_j\,\xi_j)$, where the numbers $\xi_j$ converge to zero sufficiently fast and $x_j$'s are i.i.d. random variables. As explained in the introduction, functions $f$ of the form above appear in a number of important applications. To have positive results for possibly large classes of such functions, the paper provides sharp bounds on truncation errors in both the average and worst case settings. In the former case, the functions $g$ are from a Hilbert space $G$ endowed with a zero mean probability measure with a given covariance kernel. In the latter case, the functions $g$ are from a reproducing kernel Hilbert space, or a space of functions satisfying a H\"older condition.

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arXiv2017

Truncation Dimension for Linear Problems on Multivariate Function Spaces

The paper considers linear problems on weighted spaces of multivariate functions of many variables. The main questions addressed are: When is it possible to approximate the solution for the original function of very many variables by the solution for the same function; however with all but the first $k$ variables set to zero, so that the corresponding error is small? What is the truncation dimension, i.e., the smallest number $k=k(\varepsilon)$ such that the corresponding error is bounded by a given error demand $\varepsilon$? Surprisingly, $k(\varepsilon)$ could be very small even for weights with a modest speed of convergence to zero.

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arXiv2016

Truncation Dimension for Function Approximation

We consider approximation of functions of $s$ variables, where $s$ is very large or infinite, that belong to weighted anchored spaces. We study when such functions can be approximated by algorithms designed for functions with only very small number ${\rm dim^{trnc}}(\varepsilon)$ of variables. Here $\varepsilon$ is the error demand and we refer to ${\rm dim^{trnc}}(\varepsilon)$ as the $\varepsilon$-truncation dimension. We show that for sufficiently fast decaying product weights and modest error demand (up to about $\varepsilon \approx 10^{-5}$) the truncation dimension is surprisingly very small.

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arXiv2015

On Equivalence of Anchored and ANOVA Spaces; Lower Bounds

We provide lower bounds for the norms of embeddings between $\boldsymbol{\gamma}$-weighted Anchored and ANOVA spaces of $s$-variate functions with mixed partial derivatives of order one bounded in $L_p$ norm ($p\in[1,\infty]$). In particular we show that the norms behave polynomially in $s$ for Finite Order Weights and Finite Diameter Weights if $p>1$, and increase faster than any polynomial in $s$ for Product Order-Dependent Weights and any $p$.

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arXiv2015

Very Low Truncation Dimension for High Dimensional Integration Under Modest Error Demand

We consider the problem of numerical integration for weighted anchored and ANOVA Sobolev spaces of $s$-variate functions. Here $s$ is large including $s=\infty$. Under the assumption of sufficiently fast decaying weights, we prove in a constructive way that such integrals can be approximated by quadratures for functions $f_k$ with only $k$ variables, where $k=k(\varepsilon)$ depends solely on the error demand $\varepsilon$ and is surprisingly small when $s$ is sufficiently large relative to $\varepsilon$. This holds, in particular, for $s=\infty$ and arbitrary $\varepsilon$ since then $k(\varepsilon)<\infty$ for all $\varepsilon$. Moreover $k(\varepsilon)$ does not depend on the function being integrated, i.e., is the same for all functions from the unit ball of the space.

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