Numerical integration without smoothness assumption

03/30/2020
by   Vladimir Temlyakov, et al.
0

We consider numerical integration in classes, for which we do not impose any smoothness assumptions. We illustrate how nonlinear approximation, in particular greedy approximation, allows us to guarantee some rate of decay of errors of numerical integration even in such a general setting with no smoothness assumptions.

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1 Introduction

The paper is devoted to numerical integration. The goal is to obtain rates of decay of errors of numerical integration for functions from a given function class. Theoretical aspects of the problem of numerical integration are intensely studied in approximation theory and in discrepancy theory. A typical problem in that regard is to study numerical integration in a given smoothness class (see, for instance, [9], Ch. 6). It is a difficult area of research, related to discrepancy theory and other areas of research, with a number of outstanding open problems (see, for instance, [4], [7], [2], and [8]). In the case of classes of multivariate functions with mixed smoothness delicate number theoretical methods are used to build good cubature formulas. The main goal of this paper is to study numerical integration in much more general classes than smoothness classes. Clearly, we cannot expect that delicate methods developed for studying numerical integration in smoothness classes will apply to the case of general classes. It was observed in [4] that very general method of nonlinear approximation, in particular greedy approximation, may be successfully used in numerical integration. However, it is known (see [5], Section 2.7) that in general greedy approximation has a property of saturation. Usually, the saturation rate in , where is the number of iterations of a greedy algorithm. As a result in our applications of nonlinear approximation we cannot beat a barrier of .

We now proceed to a detailed description of our results. Numerical integration seeks good ways of approximating an integral

by an expression of the form

(1.1)

It is clear that we must assume that is integrable and defined at the points . Expression (1.1) is called a cubature formula (if , ) or a quadrature formula (if ) with knots and weights .

Some classes of cubature formulas are of special interest. For instance, the Quasi-Monte Carlo cubature formulas, which have equal weights , are discussed in this paper. We use a special notation for these cubature formulas

For a function class we introduce a concept of error of the cubature formula by

(1.2)

The quantity is a classical characteristic of the quality of a given cubature formula .

Let and let be a -periodic function, where is a dual to exponent. Consider the following class of functions

Note that the case of classes of multivariate functions with bounded mixed derivative corresponds to the function

where for a scalar

The following result (in a more general setting) is proved in [6] (see also [4] for previous results).

Theorem 1.1.

Let and let be a class of functions defined above. Assume that . Then for any there exists (provided by an appropriate greedy algorithm) a cubature formula such that

Proof of Theorem 1.1 is based on the theory of greedy algorithms in Banach spaces. That theory is well developed under assumption that the Banach space is uniformly smooth, which means , where is a modulus of smoothness of the space. It is well known that the space is not uniformly smooth. This is why the case is excluded in Theorem 1.1. In this paper we analyze an algorithm – the Averaging Search algorithm – which allows us to prove an analog of Theorem 1.1 in the case under additional assumptions on the kernel . We begin with the definition of the Averaging Search algorithm. This algorithm and its greedy version were analyzed in the recent paper [1].

Averaging Search algorithm. Let be a real -periodic function satisfying condition . We build a sequence ,…, of points from inductively. At the first step choose any . Suppose, and after steps of the algorithm we have built points ,…, . Then, at the th step we choose such that

(1.3)

Note that such always exists. Indeed, by our assumption we have

and, therefore, there exists satisfying (1.3).

We now proceed to the main result of this paper. Denote .

Theorem 1.2.

Suppose that and function is a real even function, satisfying the condition , , . Then for any there exists (provided by the Averaging Search algorithm applied to ) a set of points in such that for the cubature formula we have

Associate with a cubature formula and the function the following function (see [7])

(1.4)

where

The following result is obtained in [7].

Theorem 1.3.

Let and . Then there exists a set of points such that

We now formulate a corollary of Theorem 1.1, which complements Theorem 1.3. Let function , associated with a function and a cubature formula , be defined by (1.4).

Corollary 1.1.

Suppose that and function is a real even function, satisfying the condition , , . Then for any there exists (provided by the Averaging Search algorithm applied to ) a set of points in such that

We note that the Averaging Search algorithm is not a greedy type algorithm. The following greedy version of this algorithm has been studied in a very recent paper [1].

Greedy Averaging Search algorithm. Let be a real continuous -periodic function satisfying condition . We build a sequence ,…, of points from inductively. At the first step choose any . Suppose, and after steps of the algorithm we have built points ,…, . Then, at the th step we choose such that

(1.5)

Clearly, the Greedy Averaging Search algorithm is a realization of the Averaging Search algorithm and, therefore, Theorem 1.1 and Corollary 1.1 hold for points obtained by the Greedy Averaging Search algorithm. It is an interesting open problem, which is discussed in detail in [1], to understand if the Greedy Averaging Search algorithm can give better error bounds than .

2 Proof of Theorem 1.2

We begin with a simple identity, which was used in [3] in a context of numerical integration (see also [1]).

Lemma 2.1.

Let be a -periodic function with absolutely convergent Fourier series satisfying condition . For a given set of points , denote

Then

Proof.

For the reader’s convenience we present this simple proof here. We have

Performing summation with respect to and we obtain the required identity. ∎

We continue the proof of Theorem 1.1. By duality relation (see [4] and [9], p.254, (6.3.2)) we obtain

(2.1)

We have

(2.2)

By our assumption on the we obtain

Using Lemma 2.1 we obtain from here and (2.2)

(2.3)

We now set with obtained from the Averaging Search algorithm applied to . Then, we have

It remains to note that by the choice of we have

Acknowledgment. The work was supported by the Russian Federation Government Grant No. 14.W03.31.0031.

References