arXiv subjects

Erich Novak

Publications and source records attributed to Erich Novak.

At least 37 records · Page 2Linked to original sources

arXiv2018

Solvable Integration Problems and Optimal Sample Size Selection

We compute the integral of a function or the expectation of a random variable with minimal cost and use, for our new algorithm and for upper bounds of the complexity, i.i.d. samples. Under certain assumptions it is possible to select a sample size based on a variance estimation, or -- more generally -- based on an estimation of a (central absolute) $p$-moment. That way one can guarantee a small absolute error with high probability, the problem is thus called solvable. The expected cost of the method depends on the $p$-moment of the random variable, which can be arbitrarily large. In order to prove the optimality of our algorithm we also provide lower bounds. These bounds apply not only to methods based on i.i.d. samples but also to general randomized algorithms. They show that -- up to constants -- the cost of the algorithm is optimal in terms of accuracy, confidence level, and norm of the particular input random variable. Since the considered classes of random variables or integrands are very large, the worst case cost would be infinite. Nevertheless one can define adaptive stopping rules such that for each input the expected cost is finite. We contrast these positive results with examples of integration problems that are not solvable.

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arXiv2017

Reproducing Kernels of Sobolev Spaces on $\mathbb{R}^d$ and Applications to Embedding Constants and Tractability

The standard Sobolev space $W^s_2(\mathbb{R}^d)$, with arbitrary positive integers $s$ and $d$ for which $s>d/2$, has the reproducing kernel $$ K_{d,s}(x,t)=\int_{\mathbb{R}^d}\frac{\prod_{j=1}^d\cos\left(2π\,(x_j-t_j)u_j\right)} {1+\sum_{0<|α|_1\le s}\prod_{j=1}^d(2π\,u_j)^{2α_j}}\,{\rm d}u $$ for all $x,t\in\mathbb{R}^d$, where $x_j,t_j,u_j,α_j$ are components of $d$-variate $x,t,u,α$, and $|α|_1=\sum_{j=1}^dα_j$ with non-negative integers $α_j$. We obtain a more explicit form for the reproducing kernel $K_{1,s}$ and find a closed form for the kernel $K_{d, \infty}$. Knowing the form of $K_{d,s}$, we present applications on the best embedding constants between the Sobolev space $W^s_2(\mathbb{R}^d)$ and $L_\infty(\mathbb{R}^d)$, and on strong polynomial tractability of integration with an arbitrary probability density. We prove that the best embedding constants are exponentially small in $d$, whereas worst case integration errors of algorithms using $n$ function values are also exponentially small in $d$ and decay at least like $n^{-1/2}$. This yields strong polynomial tractability in the worst case setting for the absolute error criterion.

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arXiv2017

Tractability of multivariate problems for standard and linear information in the worst case setting: part II

We study QPT (quasi-polynomial tractability) in the worst case setting for linear tensor product problems defined over Hilbert spaces. We assume that the domain space is a reproducing kernel Hilbert space so that function values are well defined. We prove QPT for algorithms that use only function values under the three assumptions: 1) the minimal errors for the univariate case decay polynomially fast to zero, 2) the largest singular value for the univariate case is simple and 3) the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point. The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.

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arXiv2016

Optimal Quadrature Formulas for the Sobolev Space $H^1$

We study optimal quadrature formulas for arbitrary weighted integrals and integrands from the Sobolev space $H^1([0,1])$. We obtain general formulas for the worst case error depending on the nodes $x_j$. A particular case is the computation of Fourier coefficients, where the oscillatory weight is given by $ρ_k(x) = \exp(- 2 πi k x)$. Here we study the question whether equidistant nodes are optimal or not. We prove that this depends on $n$ and $k$: equidistant nodes are optimal if $n \ge 2.7 |k| +1 $ but might be suboptimal for small $n$. In particular, the equidistant nodes $x_j = j/ |k|$ for $j=0, 1, \dots , |k| = n+1$ are the worst possible nodes and do not give any useful information. To characterize the worst case function we use certain results from the theory of weak solutions of boundary value problems and related quadratic extremal problems.

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arXiv2016

Product rules are optimal for numerical integration in classical smoothness spaces

We mainly study numerical integration of real valued functions defined on the $d$-dimensional unit cube with all partial derivatives up to some finite order $r\ge1$ bounded by one. It is well known that optimal algorithms that use $n$ function values achieve the error rate $n^{-r/d}$, where the hidden constant depends on $r$ and $d$. Here we prove explicit error bounds without hidden constants and, in particular, show that the optimal order of the error is $\min \bigl\{1, d \, n^{-r/d}\bigr\}$, where now the hidden constant only depends on $r$, not on $d$. For $n=m^d$, this optimal order can be achieved by (tensor) product rules. We also provide lower bounds for integration defined over an arbitrary open domain of volume one. We briefly discuss how lower bounds for integration may be applied for other problems such as multivariate approximation and optimization.

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arXiv2016

A Universal Algorithm for Multivariate Integration

We present an algorithm for multivariate integration over cubes that is unbiased and has optimal order of convergence (in the randomized sense as well as in the worst case setting) for all Sobolev spaces $H^{r, mix}([0,1]^d)$ and $H^s([0,1]^d)$ for $s>d/2$.

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arXiv2015

Some Results on the Complexity of Numerical Integration

This is a survey (21 pages, 124 references) written for the MCQMC 2014 conference in Leuven, April 2014. We start with the seminal paper of Bakhvalov (1959) and end with new results on the curse of dimension and on the complexity of oscillatory integrals. Some small errors of earlier versions are corrected.

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arXiv2015

Tractability of Multivariate Problems for Standard and Linear Information in the Worst Case Setting: Part I

We present a lower error bound for approximating linear multivariate operators defined over Hilbert spaces in terms of the error bounds for appropriately constructed linear functionals as long as algorithms use function values. Furthermore, some of these linear functionals have the same norm as the linear operators. We then apply this error bound for linear (unweighted) tensor products. In this way we use negative tractability results known for linear functionals to conclude the same negative results for linear operators. In particular, we prove that $L_2$-multivariate approximation defined for standard Sobolev space suffers the curse of dimensionality if function values are used although the curse is not present if linear functionals are allowed.

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arXiv2015

Complexity of Oscillatory Integrals on the Real Line

We analyze univariate oscillatory integrals defined on the real line for functions from the standard Sobolev space $H^s({\mathbb{R}})$ and from the space $C^s({\mathbb{R}})$ with an arbitrary integer $s\ge1$. We find tight upper and lower bounds for the worst case error of optimal algorithms that use $n$ function values. More specifically, we study integrals of the form \[ I_k^ρ(f) = \int_{ {\mathbb{R}}} f(x) \,e^{-i\,kx} ρ(x) \, {\rm d} x\ \ \ \mbox{for}\ \ f\in H^s({\mathbb{R}})\ \ \mbox{or}\ \ f\in C^s({\mathbb{R}}) \] with $k\in {\mathbb{R}}$ and a smooth density function $ρ$ such as $ ρ(x) = \frac{1}{\sqrt{2 π}} \exp( -x^2/2) $. The optimal error bounds are $Θ((n+\max(1,|k|))^{-s})$ with the factors in the $Θ$ notation dependent only on $s$ and $ρ$.

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arXiv2014

Tractability of the approximation of high-dimensional rank one tensors

We study the approximation of high-dimensional rank one tensors using point evaluations and consider deterministic as well as randomized algorithms. We prove that for certain parameters (smoothness and norm of the $r$th derivative) this problem is intractable while for other parameters the problem is tractable and the complexity is only polynomial in the dimension for every fixed $\varepsilon>0$. For randomized algorithms we completely characterize the set of parameters that lead to easy or difficult problems, respectively. In the "difficult" case we modify the class to obtain a tractable problem: The problem gets tractable with a polynomial (in the dimension) complexity if the support of the function is not too small.

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arXiv2014

Complexity of Oscillatory Integration for Univariate Sobolev Spaces

We analyze univariate oscillatory integrals for the standard Sobolev spaces $H^s$ of periodic and non-periodic functions with an arbitrary integer $s\ge1$. We find matching lower and upper bounds on the minimal worst case error of algorithms that use $n$ function or derivative values. We also find sharp bounds on the information complexity which is the minimal $n$ for which the absolute or normalized error is at most $\varepsilon$. We show surprising relations between the information complexity and the oscillatory weight. We also briefly consider the case of $s=\infty$.

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arXiv2014

Computation of expectations by Markov chain Monte Carlo methods

Markov chain Monte Carlo (MCMC) methods are a very versatile and widely used tool to compute integrals and expectations. In this short survey we focus on error bounds, rules for choosing the burn in, high dimensional problems and tractability versus curse of dimension.

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arXiv2014

On Weak Tractability of the Clenshaw-Curtis Smolyak Algorithm

We consider the problem of integration of d-variate analytic functions defined on the unit cube with directional derivatives of all orders bounded by 1. We prove that the Clenshaw Curtis Smolyak algorithm leads to weak tractability of the problem. This seems to be the first positive tractability result for the Smolyak algorithm for a normalized and unweighted problem. The space of integrands is not a tensor product space and therefore we have to develop a different proof technique. We use the polynomial exactness of the algorithm as well as an explicit bound on the operator norm of the algorithm.

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arXiv2013

The Curse of Dimensionality for Numerical Integration of Smooth Functions II

We prove the curse of dimensionality in the worst case setting for numerical integration for a number of classes of smooth $d$-variate functions. Roughly speaking, we consider different bounds for the derivatives of $f \in C^k(D_d)$ and ask whether the curse of dimensionality holds for the respective classes of functions. We always assume that $D_d \subset \mathbb{R}^d$ has volume one and consider various values of $k$ including the case $k=\infty$ which corresponds to infinitely many differentiable functions. We obtain necessary and sufficient conditions, and in some cases a full characterization for the curse of dimensionality. For infinitely many differentiable functions we prove the curse if the bounds on the successive derivatives are appropriately large. The proof technique is based on a volume estimate of a neighborhood of the convex hull of $n$ points which decays exponentially fast if $n$ is small relative to $d$. For $k=\infty$, we also also study conditions for quasi-polynomial, weak and uniform weak tractability.

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arXiv2013

The Curse of Dimensionality for Numerical Integration of Smooth Functions

We prove the curse of dimensionality for multivariate integration of C^r functions: The number of needed function values to achieve an error ε is larger than c_r (1+γ)^d for ε\le ε_0, where c_r,γ>0 and d is the dimension. The proofs are based on volume estimates for r=1 together with smoothing by convolution. This allows us to obtain smooth fooling functions for r>1.

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arXiv2011

Discontinuous information in the worst case and randomized settings

We believe that discontinuous linear information is never more powerful than continuous linear information for approximating continuous operators. We prove such a result in the worst case setting. In the randomized setting we consider compact linear operators defined between Hilbert spaces. In this case, the use of discontinuous linear information in the randomized setting cannot be much more powerful than continuous linear information in the worst case setting. These results can be applied when function evaluations are used even if function values are defined only almost everywhere.

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arXiv2011

On the power of function values for the approximation problem in various settings

This is an expository paper on approximating functions from general Hilbert or Banach spaces in the worst case, average case and randomized settings with error measured in the $L_p$ sense. We define the power function as the ratio between the best rate of convergence of algorithms that use function values over the best rate of convergence of algorithms that use arbitrary linear functionals for a worst possible Hilbert or Banach space for which the problem of approximating functions is well defined. Obviously, the power function takes values at most one. If these values are one or close to one than the power of function values is the same or almost the same as the power of arbitrary linear functionals. We summarize and supply a few new estimates on the power function. We also indicate eight open problems related to the power function since this function has not yet been studied for many cases. We believe that the open problems will be of interest to a general audience of mathematicians.

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arXiv2010

The Curse of Dimensionality for Monotone and Convex Functions of Many Variables

We study the integration and approximation problems for monotone and convex bounded functions that depend on $d$ variables, where $d$ can be arbitrarily large. We consider the worst case error for algorithms that use finitely many function values. We prove that these problems suffer from the curse of dimensionality. That is, one needs exponentially many (in $d$) function values to achieve an error $ε$.

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