1. Introduction
The tractability of multivariate problems
has become a very active research field (see [
1,
2,
3]), with many scholars devoted to studying the behavior of the information complexity
, which changes as the variable
tends to zero and
d goes to infinity. As before, we define information complexity as the minimal number of continuous linear functionals needed to seek an
-approximation of the operator
, and the considered problems are related to a zero-mean Gaussian measure under the normalized error criterion and in an average-case setting. Note that
is a Hilbert space and that
is a Banach space equipped with Gaussian measure
with a zero mean value. The algorithm
is considered an
—approximation of
if
The tractability concepts on multivariate problems were first proposed in 1994 by professor H.Woźni-kowski (see [
4]). In general, a problem is considered intractable if
is an exponential function of the variable
or
d. Otherwise, it is considered tractable. Until now, various tractability concepts have been studied for many multidimensional approximation problems in different error settings. Among these numerous studies, multivariate approximation and integration are the most extensively studied and important issues. In brief, we now recall some of the basic tractability concepts (see [
1,
5,
6,
7]).
For multivariate problems , we state the following:
Based on the above tractability definition, we can easily obtain the following logical relation:
We note that many papers have studied the various concepts of tractability on the approximation of integrated Euler and Wiener processes (see [
8,
9,
10,
11,
12,
13]). Usually, the covariance kernel of the Gaussian measure used in these papers is the tensor product of the one-dimensional kernels corresponding to these random processes with nondecreasing and non-negative smoothness parameters
. In this regard, we investigate a more special case of the covariance kernel (for more details, see
Section 2), which is essentially different to that in previous papers. We finally obtain matching necessary and sufficient conditions such that the above concepts of tractability hold, and the proofs employ several techniques and methods for the general covariance kernel. It should be noted that
-WT and
-WT have not been discussed before. In this regard, we prove that the considered multivariate problem is not
-WT. For
-WT, we provide a sufficient condition, and whether this condition is a necessary condition for matching remains an open question.
Following is an outline of this paper. We first provide some basic concepts and background information about the multivariate problem of the integrated Euler and Wiener processes in
Section 2. The proofs of our main results are given and proven in
Section 3 and
Section 4.
2. Euler and Wiener Integrated Processes
In the following, we use , , and to represent the sets of positive integers, non-negative integers, and real numbers, respectively. Furthermore, we denote their —ary Cartesian powers as , , and for each . Additionally, we define for .
A linear multivariate problem is defined as a sequence of continuous linear operators , where is a Hilbert space, and is a separable Banach space equipped with Gaussian measure with a zero mean value.
Now, let
be a Gaussian measure with a zero mean value induced on space
by operator
and measure
on
. On the other side, let
be the covariance operator of measure
. Then,
becomes a self-adjoint, non-negative definite operator and has finite trace (see [
1]). The eigenpairs of
are defined as
, which satisfy
For each
, we use information-based algorithms
to approximate
, where
,
is an arbitrary measurable mapping, and
are continuous linear functionals on
. As a special case, we define
.
The average-case approximation error for the algorithm
of the form (3) is defined by
Then, the
n-th minimal average-case error for
is given by
where the infimum is taken over by all algorithms of the form (3). In fact, this can be achieved by the
n-th optimal algorithm
For
, it is easy to see that
and the
initial error is given by (see [
1])
Using the above preparation knowledge, we give the definition of information complexity. For
and the absolute or normalized error criterion, the information complexity of
is defined as
where
Hence, the information complexity with the normalized error criterion can be expressed by
while, for the absolute error criterion,
Now, we introduce the approximation problems
,
It should be noted that the continuous real function space
is equipped with Gaussian measure
with a zero mean value, and its covariance kernel corresponds to two random processes, i.e., Euler and Wiener integrated process. In the following, we provide basic knowledge on and detail the important properties of Euler and Wiener integrated processes. For
, let
be a standard Wiener process, i.e., a Gaussian measure with a zero mean value and its covariance kernel
For
, the two sequences of random processes
and
in the interval
are recursively defined on parameter
r by
and
Usually, we refer to
as the univariate Euler integrated process and
as the univariate integrated Wiener process.
The Gaussian measure corresponding to random processes
and
is focused on a series of functions that are
r times continuously differentiable but have different boundary conditions. For the covariance kernel of
, it is represented by
usually referred to as the Euler kernel. Furthermore, this kernel can be expressed by Euler polynomials (see [
8]). For the covariance kernel of
, it is denoted as
called the Wiener kernel.
For
, the corresponding tensor product kernels are represented by
with a sequence of nondecreasing, non-negative integers
, and
We note that the tensor product kernels
and
are essentially different to those in [
8,
9,
11,
12]. For the multivariate problem APP, paper [
14] obtained the eigenvalues of the covariance operators of the induced measure corresponding to the above two random processes, i.e.,
where
for all
, and
where, for
,
as
implies that there exists
and
such that
for any
.
Note that, for all
,
,
and
It is proven in [
8] that
where, for
,
as
implies that
and
as
. Furthermore, from (10) and (11), one has
Detailed information about the multivariate approximation of integrated Euler and Wiener processes is provided in [
8].
Remark 1. Let be an approximation problem, and the eigenvalues of the covariance operators arewhere and in the following Hence, from (9) and (4), we know that , and then and APP have the same tractability properties for the normalized error criterion. Specially, for , 3. Tractability of Euler Integrated Process
In this section, we study the various concepts of tractability on the Euler integrated process and give matching necessary and sufficient conditions, except for -WT.
Theorem 1. Consider the multivariate approximation problem APP for the Euler integrated process. Then, for the normalized error criterion,
- (i)
SPT holds if PT holds ifor equivalently ifIf so, then the exponent of SPT is - (ii)
- (iii)
- (iv)
(s,t)-WT with and always holds.
- (v)
(s,1)-WT with holds if WT holds if - (vi)
(s,t)-WT holds with and if - (vii)
APP is not -WT with .
- (viii)
If APP is -WT with , then (18) holds with . However, ifthen APP is -WT with .
Proof of part (i). Based on the logical relationship between SPT and PT, we can easily observe that, to prove
, it is enough to show
We first prove for some . Indeed, let , and then, for some , there exists a positive integer such that when . Therefore, and, thus, whenever . Note that, in this case, , which yields .
Now, we turn to prove that
By Chapter 6 in [
1], we know that PT holds if there exist
and
such that
Furthermore, APP is SPT if (21) holds with
, and the exponent of SPT is given by
Now and in the remainder of this section, we let
. Take
; then, by (7) and (9), we have
It follows from the proof of Theorem 1 in [
8] that, for all
,
Therefore, for
,
where
, and both are uniformly bounded,
Now, we assume that
for some
. By combining (24) and (25), we have
where we use the fact that
for
. Therefore, (21) holds with
, and we conclude that
implies that SPT holds. Furthermore, from the above proof, we know that, if
, then (21) holds with
. Thus, (22) implies that
The relation on SPT⇒PT is trivial.
We now suppose that PT holds. From (21) and (24), we have
for some
and
. By
, one has
where, based on (25) and (6), we know that
Hence, we obtain
By taking logarithms on the above relation, one has
where, in the first inequality above, we adopt the fact that
for
. Therefore, this means that
Then, from the above inequality, we can easily obtain
. By taking the logarithms of this inequality, we obtain
and this clearly implies that
as claimed. Therefore, all statements in (20) are equivalent. We also notice that the first inequality in (27) implies that
. Moreover, it is obvious to see that (21) holds only if
. It follows from (22) that
Thus, by (26) and (28), we obtain (15). The proof of
is complete. □
Proof of part (ii). From Theorem 2 in [
10], we know that APP is QPT if there exists
such that
Regarding sufficiency, we first prove that (16) implies (29) with
. From (9), we have
where
. We now divide the last product into two products
For
, by (14) and (23), we see that, for
,
; then, we have
Clearly, (16) implies that
.
We now consider the product
. From the proof of Theorem 1 in [
8], we know that there exists
such that
Therefore, we easily obtain
and, by (16), we find that
Therefore,
which implies that (29) holds. Hence, APP is QPT.
Regarding necessity, assume that QPT holds. Now, we let
and then, by (9), one has
Using the proof of (20) in [
10], we obtain
Then, by the above equality, one has
where
. From Corollary 4 in [
10], we know that, if quasi-polynomial tractability holds, then
Since APP is QPT, by combining (30) and (31), we obtain
Since
, it is easy to see that all terms in the sums over
j are positive. By omitting all terms for
in the last condition, we have
Furthermore, due to
, we obtain
Obviously, the above condition is equivalent to (16). □
Proof of part (iii). Assume that APP is UWT. Note that
Thus, from Lemma 5 in [
10] and the above inequality, we have
Taking the logarithm of the above inequality yields
where, in the second inequality, we use the fact that
for
. Due to APP being UWT and fixed
, by (1), we obtain
The above equations mean that, for all
, we have
for sufficiently large
d. It follows that
as claimed.
On the contrary, assume that
. This implies that, for every
, there exists positive integer
such that, for all
, we have
i.e.,
By (23), we know that, for all
and all
, one has
Now, we use the similar method in [
11]. For a fixed
, we set
and
. Therefore, it is easy to see that
and
. Observe that, for
, one has
On the one hand, by (35),
for
and the above equation, we obtain
On the other hand, by the proof of Theorem 8 in [
10], we see that
Then, for all
, (37) yields
Therefore, by combining (36) and (38), we finally find that (1) holds; i.e., APP is UWT. □
Proof of part (iv). We now turn to
-WT with
and
. Instead, by replacing
with
t in (35) and then combining
and
, by (35), one obtains
Furthermore, from (37), we find that, for all
s,
Thus, for and , from (39) and (40), we know that (2) holds; i.e., APP is -WT with and . □
Proof of part (v). We consider
-WT with
. Now, we first assume that (17) holds. Then, by letting
in (35) first and then using
, we find that, for any
, one has
Thus, from (40) with , , and (41), we find that (2) holds; i.e., APP is -WT with .
Conversely, suppose that APP is
-WT with
. Notice that, if
, then by (6), we know that
for sufficiently large
d. Therefore, from (34), we obtain
The above inequality contradicts
-WT with
, and, thus, (17) holds. Obviously, the same proof works for WT, which is just the case of
-WT with
. The proof of
is complete. □
Proof of part (vi). We consider
-WT with
and
. Firstly, by using a method similar to the proof of Lemma 2.4 in [
9], we can easily find that
for
. In the following, we set
Now, we consider the necessity. The assumption
-WT with
and
holds, which implies that
-WT holds with
; therefore, we have (17). By applying (42) with
and
, we obtain
where, in the second inequality, we use the conclusion
for
. From (43), we have
Since APP is
-WT, by (44) and (45), we find that
which implies that
Then, by continuing with the same technique as in the proof of Theorem 2.1 in [
9], we have
Note that
; hence, (18) holds.
We turn to sufficiency.
; thus, (4) and (5) imply that
. Furthermore, from Remark 1, we know that
. Hence, for
and
, in order to show that APP is (s,t)-WT for the normalized error criterion, it suffices to prove that
is
-WT for the absolute error criterion. Note that, for
, one has
Thus, for any
,
,
where we use
for
in the last inequality.
From the proof of Theorem 2.1 in [
9], we know that, for
,
Now, let
, where
and
are given in (43). Note that, by combining (46) and (47), we have
Due to
therefore,
Note that
thus, by the above equality and (18), we have
On the one hand, by using (45), we have
On the other hand, from (49), one finds that
Hence, for and , from (48), (50), and (51) we know that (2) holds; i.e., is -WT with and for the absolute error criterion. □
Proof of part (vii). It follows from (7) that, for
,
. Hence, through statement (1) in Theorem 3.1, in [
5], we know that APP is not
-WT. □
Proof of part (viii). We note that the necessity is completely similar to the proof of necessity in
, and we omit the details. In the following, we consider the sufficiency. Similar to the discussion in
, it suffices to prove that
is
-WT for the absolute error criterion. Now, we let
. Note that, in this case,
By using (52) and the same method in (44) and then from (45), we obtain
By (19), we know that
and, hence,
However, by
and (19), we have
Therefore, from (55), one obtains
Hence, for and , from (53), (54), and (56), we know that (2) holds; i.e., is -WT with for the absolute error criterion. This completes our proof. □
4. Tractability of Wiener Integrated Process
Now, we consider the various notions of tractability on the Wiener integrated process and derive matching necessary and sufficient conditions, except for -WT.
Theorem 2. Consider the multivariate approximation problem APP for the Wiener integrated process. Then, for the normalized error criterion,
- (i)
- (ii)
- (iii)
- (iv)
(s,t)-WT with and always holds.
- (v)
(s,1)-WT with holds if WT holds if (17) holds.
- (vi)
(s,t)-WT holds with and if - (vii)
APP is not -WT with .
- (viii)
If APP is -WT with , then (59) holds with . However, ifthen APP is -WT with .
Proof of part (i). Based on the logical relationship between SPT and PT, it is obvious that, in order to prove
, it is suffice to show
Now, we let
for
, and, thus, for
,
where
Since
as
,
is finite for all
d if
for all
. Therefore,
implies that, in what follows, we need to consider
.
Suppose that APP is PT; then, by (21), we know that
. It is easy to see that
, which, in turn, implies that
. In fact, otherwise,
at least exponentially increases on
d. However, from (12), for
, we see that there exists
such that
Thus, by dropping the sums over
j in the numerator and above inequality, we have
By taking logarithms to the above inequality and from (13), we have
Therefore, the above inequality implies
, and there exists
and
such that
We now let
and then have
Furthermore, it is easy to prove that the above inequality is equivalent to (57).
Now, we assume that (57) holds, and this implies that
for some
. For
, by combining this fact and (13), we can conclude that
Thus, it follows (21) and (62) that APP is SPT and obviously PT. □
Proof of part (ii). Due to the similarity between this proof and the Euler case, we only sketch it and need to study (29) and (32) for the Wiener eigenvalues. For condition (29), we take
,
and then choose
such that
. Hence, for all such
, we obtain
, and then we can use the uniform convergence result given in (12). Therefore, in (29),
We first suppose that (58) holds. Then,
and (58) implies that
is uniformly bounded in
d. Furthermore, the factor
can be analyzed exactly as for the Euler case, and, from [
8], we have
Note that assumption (58) implies
; thus,
is also uniformly bounded in
d. Therefore, condition (29) holds, which implies that APP is QPT.
Suppose now that APP is QPT. We use (32) and its consequence (33), which is equivalent to (58). The proof is complete. □
Proof of part (iii). Firstly, we assume that APP is UWT. As before, by taking the logarithm of (34) but replacing
with
, we have
Hence, for fixed
, by (1), we have
This means that, for all
, we have
for sufficiently large
d. Therefore, the above inequality implies that
as claimed.
Conversely, we assume that
. That is, for every
, there exists a positive integer
such that, for all
, we have
i.e.,
Note that, for all
and all
, we have
Hence, it follows (64), (12), and (13) that, for
,
We now use the similar method in [
11] to fix
and set
and
. Thus, it is obvious to see that
and
. By (63), we know that, for
,
Therefore, by (65) and (66), we have
We finally continue with the same proof method as in of Theorem 1, and, by (31), we find that (1) holds; i.e., APP is UWT. □
Proof of part (iv). By (12) and
, we obtain
such that
for all
. Therefore, by (40), (67), and
, we easily observe that (2) holds, which means that
-WT holds with
and
. □
Proof of part (v). We first assume that (17) holds, and then we have . Based on this fact, (65), and (40), we directly find that APP is -WT with .
We now plan to consider necessity, which is verified by the proof of contradiction. Let . By following the same procedure as in the Euler case, we can demonstrate that is an exponential function of d, which contradicts -WT. It is obvious that the same works for WT, and we complete the proof of . □
Proof of part (vi). The necessity is completely similar to that of the proof in of Theorem 1. We only need to replace the value of by and obtain (59).
On the contrary, now we consider the sufficiency. Suppose that condition (59) holds; then, by proof
of Theorem 1, we know that, in order to prove that
-WT holds with
and
, we only need to verify for
that there is a constant
such that
. In fact, by (12) and
, we find that there exists a
such that
Thus, for
, by the above estimate, one has
The proof of
is complete. □
Proof of part (vii). From (8), we know that, for
,
. Therefore, statement (1) in Theorem 3.1 of [
5] yields that APP is not
-WT. □
Proof of part (viii). The necessity is completely similar to the proof of necessity in of Theorem 1; thus, we omit the details.
Now, we suppose that (60) holds. Therefore, we can continue the same way as in
of Theorem 1. Firstly, it suffices to prove that
is
-WT for the absolute error criterion, and we let
. Similar to the proof process in (53), by using (68) and the same method in (44) and then from (45), we have
Note that, by (60), we have
, and, thus, (54) holds. However, the result
and (60) imply that
Therefore, from (70) and (13), one has
Hence, for and , by (69), (54), and (71), we know that (2) holds, and this also means that is (s,0)-WT with for the absolute error criterion. This completes our proof. □