Let be a sequence of independent random processes, where . Suppose that every random element has zero mean and a covariance function . Let be the covariance operator with the covariance kernel for every . Then for any and , we have
We consider the random field
with the following zero-mean covariance function
and covariance operator
In this paper, we investigate the average case approximation of , as a random element of the space equipped with inner product and norm , by a finite rank random field.
The th minimal 2-average case error, for , is defined by
Here is the class of all linear algorithms with rank defined by
The so-called initial average case error is given by
We use either the absolute error criterion (ABS) or the normalized error criterion (NOR). For any , we study information complexity of approximation of the random fields defined by
Let . First we consider average case tractability of . Various notions of tractability have been discussed for multivariate problems. We recall some of the basic tractability notions (see [7, 8, 9, 10]).
For , say is
strongly polynomially tractable (SPT) iff there exist non-negative numbers and such that for all ,
The infimum of satisfying the above inequality is called the exponent of strong polynomial tractability and is denoted by .
polynomially tractable (PT) iff there exist non-negative numbers and such that for all ,
quasi-polynomially tractable (QPT) iff there exist two constants such that for all ,
uniformly weakly tractable (UWT) iff for all ,
weakly tractable (WT) iff
This paper is devoted to studying average case tractability of the additive random field under ABS and NOR. For additive random fields similar problems were investigated in [2, 5, 6, 11] in various settings for the homogeneous case and in [3, 4] for the non-homogeneous case. Here, the homogeneous case means that approximated additive random fields are constructed (in a special way) from copies of one marginal process, while the non-homogeneous case means that the random fields are composed of a whole sequence of marginal random processes with generally different covariance functions. Specifically, the authors in  obtained the growth of for arbitrary fixed and for the non-homogeneous case and gave application to the additive random fields with marginal random processes corresponding to the Korobov kernels.
It should be noted, however, that all these works deal only with NOR. In this paper, we consider average case tractability of the problem of the additive random fields with marginal random processes corresponding to the Korobov kernels under ABS and NOR. We shall show that the problem is always polynomially tractable for ABS or NOR. Obviously, PT implies all QPT, UWT, WT. We also give sufficient and necessary conditions for which is SPT for ABS or NOR.
The paper is organized as follows. In Section 2 we give preliminaries about the additive random fields with marginal random processes corresponding to the Korobov kernels and introduce main results, i.e., Theorems 2.1-2.3. Section 3 is devoted to proving Theorems 2.1-2.3.
2. Preliminaries and main results
), the eigenvalues, are generally unknown or not easily depend on . However, under the following condition, we can explicitly describe the eigenvalues .
For every there exist such that for all . We denote by
the eigenvalues corresponding to eigenvector 1. Letand be the non-increasing sequence of the remaining eigenvalues and the corresponding sequence of eigenvectors of , respectively. It is known that the family of eigenvectors
is an orthogonal system in for every , see . Hence the identical 1 is an eigenvector of with the eigenvalue , and the pairs , for all and , are the remaining eigenpairs of .
Let and be the non-increasing sequence of the eigenvalues and the corresponding sequence of eigenvectors of defined by (1.3). Then the average case information complexity can also be described in terms of eigenvalues of by
Particularly, we study additive random fields with marginal random processes corresponding to the Korobov kernels. Let , be a zero-mean random field with the following covariance function
Here , and . Let be the covariance operator with kernel of , and for any and ,
The eigenpairs of the covariance operator are known, see . The identical 1 is an eigenvector of with the eigenvalue . The other eigenpairs of have the following form:
for any .
Suppose that is a sequence of independent zero-mean random fields with covariance functions , respectively. Let , , be the sequence of zero-mean random fields with the covariance functions
where , , and the parameters , for all , and .
Let be the covariance operator of . We have
for any and . Then the identical 1 is an eigenvector of with the eigenvalue
and the remaining eigenvalues and eigenvectors are
respectively. Let be the sequence of non-increasing rearrangement of the eigenvalues of . Then we have
and for any and ,
where , is the Riemann zeta-function.
In the sequel we always assume that the sequences satisfy
In this paper, we consider the tractability of the problem
under ABS and NOR, where the sequences satisfy (2.6). Our main results can be formulated as follows.
Let the sequences satisfy (2.6). Then the problem
(i) is always PT for ABS or NOR;
(ii) is SPT for ABS iff
The exponent of SPT is
In order to investigate the strong polynomial tractability of the problem for NOR, we consider two cases.
Let satisfy (2.6) and
where is a constant. Then for NOR the problem is SPT iff
The exponent of SPT is
Let satisfy (2.6). Further assume that
and there exists a constant such that for every ,
Then the problem is SPT for NOR iff
The exponent of SPT is
In particular, if for , then the problem is SPT for NOR iff
In , Khartov and Zani investigated for arbitrary fixed and of the above problem with the parameters
where , , and . They obtained the following results.
(1) For and either , or , . Then
(2) For , and ,
(3) For , and , then
We remark that in some sense the above results give the tractability results of the problem for NOR under the condition (2.11). Specifically, the above result (1) corresponds to SPT of for NOR, and results (2) and (3) relate to PT of for NOR, but more explicitly. Comparing with , we obtain the tractability of the problem with general parameters satisfying (2.6) for ABS and NOR. Also we get the exponent of SPT for ABS and NOR.
For the above problem , let be the additive random fields with marginal random processes corresponding to the Korobov covariance functions
with parameters satisfying
Let and be the sequences of non-increasing rearrangement of the eigenvalues of the covariance operators and , respectively. Then for some ,
From the above equalities we obtain that
It follows from (2.1) and the inequality that for ,
On the other hand, we note that for ,
where . This gives that
which means that
Hence we have for ,
It follows that the problems
have the same tractability for ABS and the same exponent of SPT for ABS if the problem is SPT for ABS.
3. Proofs of Theorems 2.1-2.3
Proof of Theorem 2.1.
It follows that for ,
where in the last inequality we used for all . This forces
Due to [7, Theorem 6.1] we obtain that is PT for ABS. Therefore the problem is always PT for ABS or NOR.
(ii) From Remark 2.5, it is sufficient to prove that the problem
Assume that (2.7) holds, i.e.,
We want to prove that is SPT for ABS. Indeed, by (3.1) we have for any ,
Next, we shall prove that for any ,
Indeed, for such , there exists a for which
there exists a such that
It follows that
for any . Hence we obtain
for any . We note that is equivalent to
due to the monotonicity of the function
On the other hand, assume that is SPT for ABS. Then there exist positive and such that
where we used
in the above inequalities. Obviously, by (3) we have
It follows from (3) that
where , which yields that
Letting we get
Hence if is SPT for ABS, then we have