Average Case $(s, T)$-Weak Tractability of $L_2$-Approximation with Weighted Covariance Kernels

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Average Case $(s, T)$-Weak Tractability of $L_2$-Approximation with Weighted Covariance Kernels

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Jia Chen^*,

Huichao Yan

Jia Chen^*,

Huichao Yan

This version is not peer-reviewed

Submitted:

29 October 2024

Posted:

30 October 2024

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Abstract

We study multivariate $L_2$-approximation problem $\textrm{APP}_d$ defined over a Banach space in the average case setting. The space is equipped with a zero-mean Gaussian measure with weighted covariance kernel which depends on parameter sequences \texorpdfstring{$\a=\{\az_j\}_{j\in\N}$ and $\ba=\{\bz_j\}_{j\in\N}$}{} with $1 0$. In this paper two interesting weighted covariance kernels are considered, which model the importance of the covariance kernels. Under the absolute error criterion or the normalized error criterion, we discuss \texorpdfstring{$(s,t)$}{}-weak tractability of the \texorpdfstring{$L_2$}{}-approximation problem $\textrm{APP}=\{\textrm{APP}_d\}_{d\in\N}$ with the above two weighted covariance kernels for some positive numbers $s$ and $t$ in the average case setting, where \texorpdfstring{$(s,t)$}{}-weak tractability means that how the information complexity depends on $d$ and $\va^{-1}$ for large dimension $d$ and small threshold $\va$. In particular, for all $s>0$ and $t\in(0,1)$ we find the matching sufficient and necessary condition on the parameter sequences \texorpdfstring{$\a=\{\az_j\}_{j\in\N}$ and $\ba=\{\bz_j\}_{j\in\N}$}{} to obtain average case \texorpdfstring{$(s,t)$}{}-weak tractability under the absolute error criterion or the normalized error criterion.

Keywords:

Subject: Computer Science and Mathematics - Mathematics

1. Introduction

This paper is devoted to studying d-variate problems

S_{d}

with huge d. This is a hot topic in computational finance (see [1]) and computational chemistry (see [2]). We consider multivariate problem

S = {S_{d} : F_{d} \to G_{d}}_{d \in N}

in the average case setting, where

F_{d}

is a Banach space with a zero-mean Gaussian measure

μ_{d}

, and

G_{d}

is a Hilbert space. We approximate

S_{d}

by arbitrarily n continuous linear functionals. Let

ε \in (0, 1)

and

d \in N

. In the average case setting, for the absolute error criterion (ABS) or the normalized error criterion (NOR), information complexity

n^{X} (ε, S_{d})

is defined as the minimal number of continuous linear functionals to approximate the multivariate problem

S_{d}

with the threshold less than

ε {CRI}_{d}^{1 / 2}

, where

X \in {ABS, NOR}

and

{CRI}_{d} : = \{\begin{matrix} 1, for X = ABS, \\ \int_{F_{d}} {∥ S_{d} (f) ∥}_{G_{d}}^{2} μ_{d} (d f), for X = NOR . \end{matrix}

In 1994, the notion of tractability was first introduced to describe the behavior of the information complexity

n^{X} (ε, S_{d})

when d tends to infinity and

ε

tends to zero (see [3]). If the information complexity

n^{X} (ε, S_{d})

is the function of d and

ε^{- 1}

for large d and small

ε

, then the problem S is called algebraic tractability. In the average case setting, there are many papers discussing the algebraic tractability such as strongly polynomial tractability, polynomial tractability, quasi-polynomial tractability, unform weak tractability and

(s, t)

-weak tractability; see [4,5,6].

Recently, some authors are interested in algebraic tractability of multivariate approximation problems from Banach spaces equipped with zero-mean Gaussian measures with weighted covariance kernels in the average case setting. The weights can model how important the covariance kernels are. Some special weights were investigated such as analysis Korobov weights, Korobov weights, Euler weights, Wiener weights and Gaussian weights. In the average case setting, under ABS or NOR [7,8,9] discussed the

L_{2}

-approximation problem defined over a Banach space whose covariance kernel has analysis Korobov weight. [7] obtained the compete sufficient and necessary conditions for weak tractability, strongly polynomial tractability, polynomial tractability, quasi-polynomial tractability and uniform weak tractability; [8,9] got the compete sufficient and necessary conditions for

(s, t)

-weak tractability. For the

L_{2}

-approximation problem from a Banach space whose covariance kernel has Korobov weight, the matching sufficient and necessary conditions for weak tractability, strongly polynomial tractability and polynomial tractability under NOR in [10], quasi-polynomial tractability under NOR in [10,11], uniform weak tractability under ABS or NOR in [12], strongly polynomial tractability and polynomial tractability under ABS in [13], and

(s, t)

-weak tractability under ABS or NOR in [9] were studied in the average case setting. In the average case setting, for the

L_{2}

-approximation problem with Euler covariance kernel and Wiener covariance kernel under NOR, the sufficient and necessary conditions for weak tractability, strongly polynomial tractability, polynomial tractability and quasi-polynomial tractability in [14], uniform weak tractability in [15], and

(s, t)

-weak tractability in [9,16] were gotten. For the

L_{2}

-approximation problem with Gaussian covariance kernel under ABS or NOR, the matching necessary and sufficient conditions for strongly polynomial tractability and polynomial tractability in [17], quasi-polynomial tractability in [18], and weak tractability, uniform weak tractability and

(s, t)

-weak tractability in [19] were obtained in the average case setting.

It is interesting that different methods are used to solve

(s, t)

-weak tractability for different fixed s and t in the same article. Hence, in this paper we study

(s, t)

-weak tractability of multivariate

L_{2}

-approximation problems defined over Banach spaces with two different weighted covariance kernels in the average case setting. Those two weights come from the ideas of analysis Korobov weights (see [7,8,9]), Korobov weights (see [9,10,11,12,13]), and Gaussian ANOVA weights (see [20,21,22,23]). For ABS or NOR we obtain a compete sufficient and necessary condition on

(s, t)

-weak tractability of the above two

L_{2}

-approximation problems with all

s > 0

and

t \in (0, 1)

We summarize the contents of this paper as follows. In Section 2 we present a general multivariate approximation problem equipped with a zero-mean Gaussian measure in the average case setting, and give some definitions about algebraic tractability. Section 3 discusses

L_{2}

-approximation problems with weighted covariance kernels in the average case setting. Section 3.1 and Section 3.2 introduce a variant of the korobov covariance kernel and a variant of the Gaussian ANOVA covariance kernel, respectively. In Section 4 we investigate sufficient and necessary conditions for

(s, t)

-weak tractability of the

L_{2}

-approximation problems with the above two weighted covariance kernels for all

s > 0

and

t \in (0, 1)

in the average case setting, and then give the proof. Section 5 provides a summary of this paper.

2. Average Case Algebraic Tractability of Multivariate Approximation Problems

First, some notions on the paper: we define

N = {1, 2, \dots}

N_{0} = {0, 1, \dots}

{ln}^{+} x = max {1, ln x}

⌈ a ⌉

for the smallest integer not less than a.

We recall some concepts of multivariate approximation problems from functions defined over Banach spaces with zero-mean Gaussian measures in the average case setting; see [4].

We consider multivariate approximation problem

S_{d} : F_{d} \to G_{d}

for each

d \in N

, where

F_{d}

is a Banach space equipped with a zero-mean Gaussian measure

μ_{d}

, and

G_{d}

is a Hilbert space with an inner product

{〈 \cdot, \cdot 〉}_{G_{d}}

. For every

f \in F_{d}

we approximate

S_{d} (f)

by algorithm of the form

A_{n, d} (f) = Φ_{n, d} (L_{1} (f), \dots, L_{n} (f)),

(1)

where

L_{1}, L_{2}, \dots, L_{n}

are continuous linear functionals on

F_{d}

, and

Φ_{n, d} : R^{n} \to G_{d}

is an arbitrary mapping. We set

A_{0, d} = 0 .

In this paper we approximate

S_{d}

in the average case setting. The average case error

e (A_{n, d})

of the algorithm

A_{n, d}

is defined as

e (A_{n, d}) : = {(\int_{F_{d}} {∥ S_{d} (f) - A_{n, d} (f) ∥}_{G_{d}} μ_{d} (d f))}^{\frac{1}{2}} .

For any

n \in N_{0}

the nth minimal average case error is defined to be

e (n, S_{d}) : = inf_{A_{n, d}} e (A_{n, d}),

where the infimum is taken over all linear algorithms

A_{n, d}

of the form (1). Then for

n = 0

the error

e (0, S_{d}) = {(\int_{F_{d}} {∥ S_{d} (f) ∥}_{G_{d}} μ_{d} (d f))}^{\frac{1}{2}}

is called the average case initial error. If there exists an algorithm

A_{n, d}^{□}

of the form (1) such that

e (A_{n, d}^{□}) = e (n, S_{d}),

we call

A_{n, d}^{□}

the nth optimal algorithm of

S_{d}

Let

ε \in (0, 1)

and

d \in N

. Under the absolute error criterion (ABS) or the normalized error criterion (NOR) we define the information complexity

n^{X} (ε, S_{d})

for

X \in {ABS, NOR}

\begin{matrix} n^{X} (ε, S_{d}) & : = min \{n \in N_{0} : e (n, S_{d}) \leq ε {CRI}_{d}^{1 / 2}\}, \end{matrix}

where

{CRI}_{d} : = \{\begin{matrix} 1, for X = ABS, \\ e^{2} (0, S_{d}), for X = NOR . \end{matrix}

Recall some definitions of algebraic tractability (see [4,5,6]). Let

S = {S_{d}}_{d \in N}

. For

X \in {ABS, NOR}

, we say that

S is strongly polynomially tractable iff there are non-negative numbers C and p such that

$n^{X} (ε, S_{d}) \leq C ε^{- p} for all ε \in (0, 1), d = 1, 2, \dots .$
S is polynomially tractable iff there are non-negative numbers C, p, and q such that

$n^{X} (ε, S_{d}) \leq C d^{q} ε^{- p} for all ε \in (0, 1), d = 1, 2, \dots .$
S is quasi-polynomially tractable iff there are numbers $C > 0$ and $t > 0$ such that

$n^{X} (ε, S_{d}) \leq C exp (t (1 + ln d) (1 + ln ε^{- 1})) for all ε \in (0, 1), d = 1, 2, \dots .$
S is uniform weakly tractable iff for all $s, t > 0$ ,

$lim_{ε^{- 1} + d \to \infty} \frac{ln n^{X} (ε, S_{d})}{ε^{- s} + d^{t}} = 0 .$
S is weakly tractable iff

$lim_{ε^{- 1} + d \to \infty} \frac{ln n^{X} (ε, S_{d})}{ε^{- 1} + d} = 0 .$
S is $(s, t)$ -weakly tractable for fixed $s > 0$ and $t > 0$ iff

$lim_{ε^{- 1} + d \to \infty} \frac{ln n^{X} (ε, S_{d})}{ε^{- s} + d^{t}} = 0 .$

Obviously, we have the relationships between the above algebraic tractability notions:

\begin{matrix} strongly polynomial tractability & \Rightarrow polynomial tractability \\ \Rightarrow quasi - polynomial tractability \\ \Rightarrow uniform weak tractability \\ \Rightarrow (s, t) - weak tractability for all s, t > 0, \end{matrix}

and

\begin{matrix} weak tractability \Leftrightarrow (1, 1) - weak tractability . \end{matrix}

We will discuss the nth minimal average case error

e (n, S_{d})

and the information complexity

n^{X} (ε, S_{d})

more explicitly; see [4].

Let

C_{μ_{d}} : {(F_{d})}^{*} \to F_{d}

be the covariance operator of

μ_{d}

(see [4]), and

ν_{d} = μ_{d} S_{d}^{- 1}

be the induced measure of

μ_{d}

. Then the induced measure

ν_{d}

is a zero-mean Gaussian measure on the Borel sets of

G_{d}

with covariance operator

C_{ν_{d}} : G_{d} \to G_{d}

given by

C_{ν_{d}} = S_{d} C_{μ_{d}} {(S_{d})}^{*},

where

{(S_{d})}^{*}

is the operator dual to

S_{d}

. The eigenpairs

{(λ_{d, i}, η_{d, i})}_{i \in N}

C_{ν_{d}}

satisfy

C_{ν_{d}} η_{d, i} = λ_{d, i} η_{d, i} with λ_{d, 1} \geq λ_{d, 2} \geq \dots \geq 0,

where

{〈 η_{d, i}, η_{d, j} 〉}_{G_{d}} = δ_{i, j},

and

δ_{i, j} = \{\begin{matrix} 1, for i = j, \\ 0, for i \neq j . \end{matrix}

Then for

n \in N_{0}

, we have that the nth optimal algorithm

A_{n, d}^{□}

S_{d}

satisfies

A_{n, d}^{□} = \sum_{i = 1}^{n} {〈 S_{d} (f), η_{d, i} 〉}_{G_{d}} η_{d, i},

and the nth minimal average case error

e (n, S_{d})

has the form

e (n, S_{d}) = e (A_{n, d}^{□}) = {(\sum_{i = n + 1}^{\infty} λ_{d, i})}^{\frac{1}{2}};

see [4]. Hence for the absolute error criterion (ABS) or the normalized error criterion (NOR) the information complexity has the form

\begin{matrix} n^{X} (ε, S_{d}) & = min \{n \in N_{0} : \sum_{i = n + 1}^{\infty} λ_{d, i} \leq ε^{2} {CRI}_{d}\}, \end{matrix}

(2)

where

{CRI}_{d} = \{\begin{matrix} 1, for X = ABS, \\ \sum_{i = 1}^{\infty} λ_{d, i}, for X = NOR . \end{matrix}

(3)

It is obvious that the algebraic tractability of

S = {S_{d}}

depends on the behavior of the eigenvalues

{λ_{d, i}}_{i \in N}

. Next, we present some relationships between the information complexity

n^{X} (ε, S_{d})

and the eigenvalues

{λ_{d, i}}_{i \in N}

for ABS and NOR in the average setting.

For any

τ \in (0, 1)

, since

λ_{d, j} \leq {(\frac{\sum_{i = 1}^{j} λ_{d, i}^{τ}}{j})}^{\frac{1}{τ}} \leq {(\frac{\sum_{i = 1}^{\infty} λ_{d, i}^{τ}}{j})}^{\frac{1}{τ}},

we have by (2) that

\begin{matrix} ε^{2} {CRI}_{d} & < \sum_{j = n^{X} (ε, S_{d})}^{\infty} λ_{d, j} \leq {(\sum_{i = 1}^{\infty} λ_{d, i}^{τ})}^{\frac{1}{τ}} \sum_{j = n^{X} (ε, S_{d})}^{\infty} \frac{1}{j^{\frac{1}{τ}}} \\ \leq {(\sum_{i = 1}^{\infty} λ_{d, i}^{τ})}^{\frac{1}{τ}} \int_{n^{X} (ε, S_{d}) - 1}^{\infty} \frac{1}{x^{\frac{1}{τ}}} \\ = {(\sum_{i = 1}^{\infty} λ_{d, i}^{τ})}^{\frac{1}{τ}} \frac{τ}{(1 - τ) {(n^{X} (ε, S_{d}) - 1)}^{\frac{1 - τ}{τ}}} . \end{matrix}

It means

n^{X} (ε, S_{d}) \leq ⌈{(\frac{τ}{1 - τ} \frac{{(\sum_{i = 1}^{\infty} λ_{d, i}^{τ})}^{\frac{1}{τ}}}{{CRI}_{d}})}^{\frac{τ}{1 - τ}} ε^{\frac{- 2 τ}{1 - τ}}⌉ .

(4)

We note that the fact

ln ⌈ x ⌉ \leq |ln (2 x)|

holds for

x > 0

. Indeed, we have

ln ⌈ x ⌉ < ln (x + 1) \leq ln (2 x)

for

x \geq 1

, and

ln ⌈ x ⌉ = 0

for

0 < x < 1

, which deduces

ln ⌈ x ⌉ \leq |ln (2 x)|

for

x > 0

. Combing the inequality (4) and the above fact, we have

\begin{matrix} ln n^{X} (ε, S_{d}) & \leq ln ⌈{(\frac{τ}{1 - τ} \frac{{(\sum_{i = 1}^{\infty} λ_{d, i}^{τ})}^{\frac{1}{τ}}}{{CRI}_{d}})}^{\frac{τ}{1 - τ}} ε^{\frac{- 2 τ}{1 - τ}}⌉ \\ \leq |ln (2 {(\frac{τ}{1 - τ} \frac{{(\sum_{i = 1}^{\infty} λ_{d, i}^{τ})}^{\frac{1}{τ}}}{{CRI}_{d}})}^{\frac{τ}{1 - τ}} ε^{\frac{- 2 τ}{1 - τ}})| \\ = |\frac{τ}{1 - τ} (ln \frac{τ}{1 - τ} + ln ({(\sum_{i = 1}^{\infty} λ_{d, i}^{τ})}^{\frac{1}{τ}} / {CRI}_{d}) + 2 ln (ε^{- 1})) + ln 2| . \end{matrix}

(5)

3. $L_{2}$ -Approximation with Weighted Covariance Kernels

Let

H_{d, α, β} ({[0, 1]}^{d})

be a Banach space equipped with a zero-mean Gaussian measure

μ_{d}

with weighted covariance kernel

\begin{matrix} K_{W_{d, α, β}} (x, y) = \int_{H_{d, α, β}} f (x) f (y) μ_{d} (d f) = \sum_{h \in N_{0}^{d}} W_{d, α, β} (h) exp (2 π i h \cdot (x - y)) \end{matrix}

(6)

for

x, y \in {[0, 1]}^{d}

, where

i = \sqrt{(- 1)}

h = (h_{1}, \dots, h_{d}) \in N_{0}^{d}

u \cdot v = \sum_{i = 1}^{d} u_{i} v_{i}

u = (u_{1}, \dots, u_{d}) \in R^{d}

v = (v_{1}, \dots, v_{d}) \in R^{d}

, and

W_{d, α, β}

is the weight of the covariance kernel

K_{W_{d, α, β}}

. Here,

α = {α_{j}}_{j \in N}

and

β = {β_{j}}_{j \in N}

are parameter sequences satisfying

1 < α_{1} \leq α_{2} \leq \dots and 1 \geq β_{1} \geq β_{2} \geq \dots > 0 .

(7)

Then the covariance operator of

μ_{d}

is given by

C_{μ_{d}} : {(H_{d, α, β} ({[0, 1]}^{d}))}^{*} \to H_{d, α, β} ({[0, 1]}^{d})

In this paper we discuss the

L_{2}

-approximation problem

APP = {{APP}_{d}}_{d \in N}

\begin{matrix} {APP}_{d} : H_{d, α, β} ({[0, 1]}^{d}) \to L_{2} ({[0, 1]}^{d}) with {APP}_{d} (f) = f, \end{matrix}

(8)

for

f \in H_{d, α, β} ({[0, 1]}^{d})

. Then the covariance operator

C_{ν_{d}} : L_{2} ({[0, 1]}^{d}) \to L_{2} ({[0, 1]}^{d})

of the induced measure

ν_{d} = μ_{d} {APP}_{d}^{- 1}

has the form

(C_{ν_{d}} f) (x) = \int_{{[0, 1]}^{d}} K_{W_{d, α, β}} (x, y) f (y) d y for x, y \in {[0, 1]}^{d} .

(9)

By (6) and (9) we have that the eigenvalue sequence

{λ_{d, i}}_{i \in N}

with

λ_{d, 1} \geq λ_{d, 2} \geq \dots \geq 0

C_{ν_{d}}

is the sequence

{W_{d, α, β} (h)}_{h \in N_{0}^{d}}

We will consider product weights

W_{d, α, β} (h) = \prod_{j = 1}^{d} W_{α_{j}, β_{j}} (h_{j}) for h = (h_{1}, \dots, h_{d}) \in N_{0}^{d} .

Then for any

τ > 0

we have

\sum_{i = 1}^{\infty} λ_{d, i}^{τ} = \sum_{h \in N_{0}^{d}} W_{d, α, β}^{τ} (h) = \sum_{h \in N_{0}^{d}} \prod_{j = 1}^{d} W_{α_{j}, β_{j}}^{τ} (h_{j}) = \prod_{j = 1}^{d} \sum_{h = 0}^{\infty} W_{α_{j}, β_{j}}^{τ} (h) .

(10)

We note that the weighted covariance kernels are restricted by their weights. So it is worth to investigate the weights. There are many papers discussing the Korobov weights (see [9,10,11,12,13]), the analysis Korobov weights (see [7,8,9]), and the Gaussian ANOVA weights (see [20,21,22,23]). According to ideas of the above weights we introduce two weights, which have faster decay rates than the Korobov weights and the Gaussian ANOVA weights, respectively.

3.1. A Variant of the Korobov Covariance Kernel

In this subsection we introduce a weighted covariance kernel

K_{W_{d, α, β}}

with the weight

W_{d, α, β}

given by a variant of the korobov weight

ρ_{d, α, β}

, where

α = {α_{j}}_{j \in N}

and

β = {β_{j}}_{j \in N}

satisfy (7). The weight

ρ_{d, α, β}

is given as product form,

ρ_{d, α, β} (k) : = \prod_{j = 1}^{d} ρ_{α_{j}, β_{j}} (k_{j}), k = (k_{1}, \dots, k_{d}) \in N_{0}^{d},

where

ρ_{α_{j}, β_{j}} (k_{j})

are univariate weights,

ρ_{α, β} (k) : = \{\begin{matrix} 1, for k = 0, \\ β A^{k^{⌈ α ⌉}}, for k \geq 1, \end{matrix}

for fixed

A \in (0, 1)

α \in (1, + \infty)

and

β \in (0, 1]

. The idea of the weight

ρ_{d, α, β}

comes from the korobov weight (see [9,10,11,12,13]), and the analysis korobov weight (see [7,8,9]).

The references [9,10,11,12,13] consider the

L_{2}

-approximation problem

APP = {{APP}_{d}}_{d \in N}

satisfying (8) from the Banach space

H_{d, α, β} ({[0, 1]}^{d})

equipped with a zero-mean Gaussian measure

μ_{d}

whose weighted covariance kernel

K_{W_{d, α, β}}

of the form (6) has the kovobov weight

W_{d, α, β} = r_{d, α, β}

of the form

r_{d, α, β} (k) : = \prod_{j = 1}^{d} r_{α_{j}, β_{j}} (k_{j}), k = (k_{1}, \dots, k_{d}) \in N_{0}^{d},

with

r_{α, β} (k) : = \{\begin{matrix} 1, for k = 0, \\ \frac{β}{k^{α}}, for k \geq 1, \end{matrix}

for

α \in (1, + \infty)

and

β \in (0, 1]

, where the parameter sequences

α = {α_{j}}_{j \in N}

and

β = {β_{j}}_{j \in N}

satisfy (7). Using ABS or NOR the references [9,10,11,12,13] have solved the algebraic tractability of the above problem

APP

, and got the following results:

For ABS or NOR, strongly polynomial tractability holds iff polynomial tractability holds iff

$\underset{k \to \infty}{lim inf} \frac{ln \frac{1}{β_{k}}}{ln k} > 1 .$
For NOR, quasi-polynomial tractability holds iff

$sup_{d \in N} \frac{1}{{ln}^{+} d} \sum_{k = 1}^{d} β_{k} {ln}^{+} \frac{1}{β_{k}} < \infty .$
For ABS or NOR, weak tractability holds iff

$lim_{k \to \infty} β_{k} = 0 .$
For ABS or NOR, uniform weak tractability holds iff

$lim_{k \to \infty} β_{k} k^{p} = 0 for all p \in (0, 1) .$
For ABS or NOR, $(s, t)$ -weak tractability with $s > 0$ and $t > 1$ always holds.
For ABS or NOR, $(s, t)$ -weak tractability with $s > 0$ and $t \in (0, 1)$ holds iff

$lim_{k \to \infty} k^{1 - t} β_{k} {ln}^{+} \frac{1}{β_{k}} = 0 .$

Another covariance kernel is the analysis korobov covariance kernel, which is famous for its fast exponentially decaying weight. The analysis korobov weight is given as

ω_{d, α, β} (k) : = \prod_{j = 1}^{d} ω_{α_{j}, β_{j}} (k_{j}), k = (k_{1}, \dots, k_{d}) \in N_{0}^{d},

with

ω_{α, β} (k) : = \{\begin{matrix} 1, for k = 0, \\ ω^{β k^{α}}, for k \geq 1, \end{matrix}

for fixed

ω \in (0, 1)

α > 0

and

β > 0

, where the parameter sequences

α = {α_{j}}_{j \in N}

and

β = {β_{j}}_{j \in N}

satisfy

inf_{k \in N} α_{k} > 0 and 0 < β_{1} \leq β_{2} \leq \dots .

In the average case setting, the references [7,8,9] investigate the algebraic tractability of the

L_{2}

-approximation problem

APP = {{APP}_{d}}_{d \in N}

satisfying (8) from the Banach space

H_{d, α, β} ({[0, 1]}^{d})

equipped with a zero-mean Gaussian measure

μ_{d}

whose weighted covariance kernel

K_{W_{d, α, β}}

of the form (6) has the analysis kovobov weight

W_{d, α, β} = ω_{d, α, β}

. They obtained that (see [7,8,9]):

For ABS or NOR, strongly polynomial tractability holds iff polynomial tractability holds iff

$\underset{k \to \infty}{lim inf} \frac{β_{k}}{ln k} > \frac{1}{ln ω^{- 1}} .$
For ABS or NOR, weak tractability holds iff

$lim_{k \to \infty} β_{k} = \infty .$
For NOR, quasi-polynomial tractability holds iff

$sup_{d \in N} \frac{1}{{ln}^{+} d} \sum_{k = 1}^{d} β_{k} ω^{β_{k}} < \infty .$
For ABS or NOR, uniform weak tractability holds iff

$lim_{k \to \infty} ω^{β_{k}} k^{p} = 0 for all p \in (0, 1) .$
For ABS or NOR, $(s, t)$ -weak tractability with $s > 0$ and $t > 1$ always holds.
For ABS or NOR, $(s, t)$ -weak tractability with $s > 0$ and $t \in (0, 1)$ holds iff

$lim_{k \to \infty} k^{1 - t} β_{k} ω^{β_{k}} = 0 .$

Remark 1.

We note that the variant of the korobov weight

ρ_{d, α, β}

descends faster than the korobov weight

r_{d, α, β}

, but slower than the analysis korobov weight

ω_{d, α, β}

3.2. A Variant of the Gaussian ANOVA Covariance Kernel

In this subsection, we present a weighted covariance kernel

K_{W_{d, α, β}}

with the weight

W_{d, α, β}

given as a variant of the Gaussian ANOVA weight

σ_{d, α, β}

, where

α = {α_{j}}_{j \in N}

and

β = {β_{j}}_{j \in N}

satisfy (7). The weight

σ_{d, α, β}

is of product form, and determined by

σ_{d, α, β} (k) : = \prod_{j = 1}^{d} σ_{α_{j}, β_{j}} (k_{j}),

where

σ_{α_{j}, β_{j}} (k_{j})

are univariate weights,

σ_{α, β} (k) : = \{\begin{matrix} 1, for k = 0, \\ β A^{k!}, for 1 \leq k < ⌈ α ⌉, \\ β A^{k! / (k - ⌈ α ⌉)!}, for k \geq ⌈ α ⌉, \end{matrix}

for fixed

A \in (0, 1)

α \in (1, + \infty)

and

β \in (0, 1]

The weight

σ_{d, α, β}

is similar but different with the Gaussian ANOVA weight

ψ_{d, α, β}

given by

ψ_{d, α, β} (k) : = \prod_{j = 1}^{d} ψ_{α_{j}, β_{j}} (k_{j})

with

ψ_{α, β} (k) : = \{\begin{matrix} 1, & for k = 0, \\ \frac{β}{k!}, & for 1 \leq k < ⌈ α ⌉, \\ \frac{β (k - ⌈ α ⌉)!}{k!}, & for k \geq ⌈ α ⌉, \end{matrix}

for

α \in (1, + \infty)

and

β \in (0, 1]

, where

α = {α_{j}}_{j \in N}

and

β = {β_{j}}_{j \in N}

satisfy (7) (see [22,23]). In the worst case setting, the reference [21] and the reference [22] investigate the algebraic tractability of

APP = {{APP}_{d}}_{d \in N}

satisfying (8) defined over the reproducing kernel Hilbert space

H_{d, α, β} ({[0, 1]}^{d})

, where the reproducing kernel function has the Gaussian ANOVA weight

ψ_{d, α, β}

. But in the average case setting, there are no results about the algebraic tractability of the problem

APP = {{APP}_{d}}_{d \in N}

satisfying (8) from the Banach space

H_{d, α, β} ({[0, 1]}^{d})

equipped with a zero-mean Gaussian measure

μ_{d}

with the Gaussian ANOVA covariance kernel or the variant of the Gaussian ANOVA covariance kernel.

Remark 2.

Noting that the variant of the Gaussian ANOVA weight

σ_{d, α, β}

has faster decay rate than the Gaussian ANOVA weight

ψ_{d, α, β}

Remark 3.

Set

W_{α_{j}, β_{j}} (k) \in {ρ_{α_{j}, β_{j}} (k), σ_{α_{j}, β_{j}} (k)}

for all

j \in N

and

k \in N_{0}

. Then we have

W_{α_{j}, β_{j}} (k) \leq β_{j} A^{\frac{k^{⌈ α_{1} ⌉}}{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉}}} for all j, k \in N .

Especially, we have

W_{α_{j}, β_{j}} (0) = 1

for all

j \in N

Proof. (1) Set

W_{α_{j}, β_{j}} (k) = ρ_{α_{j}, β_{j}} (k)

for all

j, k \in N

. We have

ρ_{α_{j}, β_{j}} (k) \leq ρ_{α_{1}, β_{j}} (k) = β_{j} A^{k^{⌈ α_{1} ⌉}} \leq β_{j} A^{\frac{k^{⌈ α_{1} ⌉}}{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉}}}

for all

j, k \in N

(2) Set

W_{α_{j}, β_{j}} (k) = σ_{α_{j}, β_{j}} (k)

for all

j, k \in N

. For

1 \leq k < ⌈ α_{j} ⌉

and

j \in N

we have

σ_{α_{j}, β_{j}} (k) = β_{j} A^{k!} \leq β_{j} A \leq β_{j} A^{{(\frac{k}{⌈ α_{j} ⌉})}^{⌈ α_{j} ⌉}} = β_{j} A^{\frac{k^{⌈ α_{j} ⌉}}{{⌈ α_{j} ⌉}^{⌈ α_{j} ⌉}}} .

On the other hand, for

k \geq ⌈ α_{j} ⌉

and

j \in N

we get

\begin{matrix} σ_{α_{j}, β_{j}} (k) & = β_{j} A^{k! / (k - ⌈ α_{j} ⌉)!} = β_{j} A^{k (k - 1) \dots (k - ⌈ α_{j} ⌉ + 1)} \\ \leq β_{j} A^{{(k - ⌈ α_{j} ⌉ + 1)}^{⌈ α_{j} ⌉}} = β_{j} A^{k^{⌈ α_{j} ⌉} {(1 - \frac{⌈ α_{j} ⌉ - 1}{k})}^{⌈ α_{j} ⌉}} \\ \leq β_{j} A^{k^{⌈ α_{j} ⌉} {(1 - \frac{⌈ α_{j} ⌉ - 1}{⌈ α_{j} ⌉})}^{⌈ α_{j} ⌉}} = β_{j} A^{\frac{k^{⌈ α_{j} ⌉}}{{⌈ α_{j} ⌉}^{⌈ α_{j} ⌉}}} . \end{matrix}

It follows that for all

j, k \in N

σ_{α_{j}, β_{j}} (k) \leq β_{j} A^{\frac{k^{⌈ α_{j} ⌉}}{{⌈ α_{j} ⌉}^{⌈ α_{j} ⌉}}} .

Since

σ_{α_{j}, β_{j}} (k) \leq σ_{α_{1}, β_{j}} (k)

for all

j, k \in N

, we further obtain

σ_{α_{j}, β_{j}} (k) \leq β_{j} A^{\frac{k^{⌈ α_{1} ⌉}}{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉}}} .

Therefore, by (1) and (2) we have

W_{α_{j}, β_{j}} (k) \leq β_{j} A^{\frac{k^{⌈ α_{1} ⌉}}{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉}}}

for

W_{α_{j}, β_{j}} (k) \in {ρ_{α_{j}, β_{j}} (k), σ_{α_{j}, β_{j}} (k)}

and all

j, k \in N

(3) For all

j \in N

it is obvious from

ρ_{α_{j}, β_{j}} (0) = σ_{α_{j}, β_{j}} (0) = 1

that

W_{α_{j}, β_{j}} (0) = 1

. □

Remark 4.

Set

W_{α_{j}, β_{j}} (k) \in {ρ_{α_{j}, β_{j}} (k), σ_{α_{j}, β_{j}} (k)}

for all

j \in N

and

k \in N_{0}

. Then for all

j \in N

due to

ρ_{α_{j}, β_{j}} (1) = σ_{α_{j}, β_{j}} (1) = β_{j} A

, we have

W_{α_{j}, β_{j}} (1) = β_{j} A

4. Average Case $(s, t)$ -Weak Tractability of $L_{2}$ -Approximation with the Two Weighted Covariance Kernels and the Main Result

In this section, we consider

(s, t)

-weak tractability of the

L_{2}

-approximation

APP = {{APP}_{d}}_{d \in N}

satisfying (8) defined over the Banach space

H_{d, α, β} ({[0, 1]}^{d})

with a zero-mean Gaussian measure

μ_{d}

in the average case setting. Here, the covariance kernel

K_{W_{d, α, β}}

with weight

W_{d, α, β}

of the measure

μ_{d}

is given by (6), and the parameter sequences

α = {α_{j}}_{j \in N}

and

β = {β_{j}}_{j \in N}

satisfy (7). In this paper, we consider two product weights: the variant of the korobov weight

ρ_{d, α, β}

and the variant of the Gaussian ANOVA weight

σ_{d, α, β}

Let

W_{d, α, β} = \prod_{j = 1}^{d} W_{α_{j}, β_{j}} (h_{j})

with

W_{α_{j}, β_{j}} (k) \in {ρ_{α_{j}, β_{j}} (k), σ_{α_{j}, β_{j}} (k)}

for all

j \in N

and

k \in N_{0}

. Then from Lemma 3 we have

W_{α_{j}, β_{j}} (0) = 1 \geq W_{α_{j}, β_{j}} (k) for all j, k \in N,

(11)

which yields

λ_{d, 1} = W_{d, α, β} (0) = \prod_{j = 1}^{d} W_{α_{j}, β_{j}} (0) = 1 .

(12)

We conclude from (3), (10) with

τ = 1

and (11) that

{CRI}_{d} = \{\begin{matrix} 1, for X = ABS, \\ \prod_{j = 1}^{d} (1 + \sum_{k = 1}^{\infty} W_{α_{j}, β_{j}} (k)), for X = NOR . \end{matrix}

(13)

By (2) and (13) we obtain

\begin{matrix} n^{NOR} (ε, {APP}_{d}) \leq n^{ABS} (ε, {APP}_{d}) . \end{matrix}

(14)

Theorem 1.

Let the parameter sequences

α = {α_{j}}_{j \in N}

and

β = {β_{j}}_{j \in N}

satisfy (7). Consider the

L_{2}

-approximation APP from the space

H_{d, α, β} ({[0, 1]}^{d})

with the covariance weight

W_{d, α, β} \in {ρ_{d, α, β}, σ_{d, α, β}}

in the average case setting. For any

t \in (0, 1)

and

s > 0

(s, t)

-weak tractability holds under ABS or NOR iff

lim_{j \to \infty} β_{j} j^{1 - t} = 0 .

Proof.

Necessity. Let

s > 0

and

t \in (0, 1)

. Assume that

(s, t)

-weak tractability holds for ABS or NOR.

By the inequality (14), we only need to assume that

(s, t)

-weak tractability holds for NOR. Due to the definition of the information complexity (2) for NOR, we have

{CRI}_{d} - \sum_{i = 1}^{n^{NOR} (ε, {APP}_{d})} λ_{d, i} = \sum_{i = n^{NOR} (ε, {APP}_{d}) + 1}^{\infty} λ_{d, i} \leq ε^{2} {CRI}_{d},

which means

(1 - ε^{2}) {CRI}_{d} \leq \sum_{i = 1}^{n^{NOR} (ε, {APP}_{d})} λ_{d, i} .

We further deduce from (13) and (12) that

\begin{matrix} (1 - ε^{2}) \prod_{j = 1}^{d} (1 + \sum_{k = 1}^{\infty} W_{α_{j}, β_{j}} (k)) & = (1 - ε^{2}) {CRI}_{d} \leq \sum_{i = 1}^{n^{NOR} (ε, {APP}_{d})} λ_{d, i} \\ \leq n^{NOR} (ε, {APP}_{d}) λ_{d, 1} = n^{NOR} (ε, {APP}_{d}) . \end{matrix}

(15)

From (15) we get

\begin{matrix} ln n^{NOR} (ε, {APP}_{d}) \geq ln (1 - ε^{2}) + \sum_{j = 1}^{d} ln (1 + \sum_{k = 1}^{\infty} W_{α_{j}, β_{j}} (k)) . \end{matrix}

(16)

Set

ε = \frac{1}{2}

. It follows from the assumption, inequality (16) and Remark 4 that

\begin{matrix} 0 = lim_{d \to \infty} \frac{ln n^{NOR} (\frac{1}{2}, {APP}_{d})}{2^{s} + d^{t}} & \geq lim_{d \to \infty} \frac{ln \frac{3}{4} + \sum_{j = 1}^{d} ln (1 + \sum_{k = 1}^{\infty} W_{α_{j}, β_{j}} (k))}{2^{s} + d^{t}} \\ \geq lim_{d \to \infty} \frac{ln \frac{3}{4} + \sum_{j = 1}^{d} ln (1 + W_{α_{j}, β_{j}} (1))}{2^{s} + d^{t}} \\ = lim_{d \to \infty} \frac{ln \frac{3}{4} + \sum_{j = 1}^{d} ln (1 + β_{j} A)}{2^{s} + d^{t}} \\ = lim_{d \to \infty} \frac{\sum_{j = 1}^{d} ln (1 + β_{j} A)}{d^{t}} . \end{matrix}

Due to the fact

ln (1 + x) > \frac{x}{2}

for all

x \in (0, 1)

β_{j} A \in (0, 1)

for all

j \in N

, and Stolz theorem, we further have

\begin{matrix} 0 = lim_{d \to \infty} \frac{ln n^{NOR} (\frac{1}{2}, {APP}_{d})}{2^{s} + d^{t}} & \geq lim_{d \to \infty} \frac{\sum_{j = 1}^{d} \frac{β_{j} A}{2}}{d^{t}} \\ = \frac{A}{2} lim_{d \to \infty} \frac{β_{d}}{d^{t} - {(d - 1)}^{t}} \\ = \frac{A}{2 t} lim_{d \to \infty} β_{d} d^{1 - t} \geq 0 . \end{matrix}

It yields

lim_{d \to \infty} β_{d} d^{1 - t} = 0

for any

t \in (0, 1)

Sufficiency. Assume that

lim_{d \to \infty} β_{d} d^{1 - t} = 0

for any

t \in (0, 1)

. We will prove that

(s, t)

-weak tractability holds for ABS or NOR.

By the inequality (14), we only need to prove that

(s, t)

-weak tractability holds for ABS. We set

τ_{k} = max \{\frac{1}{{ln}^{+} (β_{k}^{- 1})}, \frac{1}{ln (2 k + 1)}\} for k \in N .

(17)

Obviously,

lim_{k \to \infty} β_{k} = 0

and thus

lim_{k \to \infty} τ_{k} = 0

, i.e.,

τ_{k} \in (0, 1)

for sufficiently large k. Set

τ = 1 - τ_{d}

. It follows from inequality (5) for ABS that

\begin{matrix} lim_{ε^{- 1} + d \to \infty} \frac{ln n^{ABS} (ε, {APP}_{d})}{ε^{- s} + d^{t}} & \leq lim_{ε^{- 1} + d \to \infty} \frac{|\frac{1 - τ_{d}}{τ_{d}} (ln \frac{1 - τ_{d}}{τ_{d}} + 2 ln (ε^{- 1}) + \frac{ln (\sum_{i = 1}^{\infty} λ_{d, i}^{1 - τ_{d}})}{1 - τ_{d}}) + ln 2|}{ε^{- s} + d^{t}} . \end{matrix}

(18)

Note that

\begin{matrix} 0 \leq lim_{ε^{- 1} + d \to \infty} \frac{|\frac{1 - τ_{d}}{τ_{d}} (ln \frac{1 - τ_{d}}{τ_{d}} + 2 ln (ε^{- 1})) + ln 2|}{ε^{- s} + d^{t}} \\ \leq lim_{ε^{- 1} + d \to \infty} \frac{|(ln (2 d + 1) - 1) (ln (ln (2 d + 1) - 1) + 2 ln (ε^{- 1})) + ln 2|}{ε^{- s} + d^{t}} \\ \leq lim_{d \to \infty} \frac{(ln (2 d + 1) - 1) ln (ln (2 d + 1) - 1)}{d^{t}} + lim_{ε^{- 1} + d \to \infty} \frac{2 (ln (2 d + 1) - 1) ln (ε^{- 1})}{ε^{- s} + d^{t}} \\ = lim_{ε^{- 1} + d \to \infty} \frac{2 (ln (2 d + 1) - 1) ln (ε^{- 1})}{ε^{- s} + d^{t}} \\ \leq lim_{ε^{- 1} + d \to \infty} \frac{{(ln (2 d + 1) - 1)}^{2} + {(ln (ε^{- 1}))}^{2}}{ε^{- s} + d^{t}} \\ \leq lim_{d \to \infty} \frac{{(ln (2 d + 1) - 1)}^{2}}{d^{t}} + lim_{ε^{- 1} \to \infty} \frac{{(ln (ε^{- 1}))}^{2}}{ε^{- s}} = 0, \end{matrix}

and thus in the inequality (18) we only need to prove

\begin{matrix} lim_{ε^{- 1} + d \to \infty} \frac{|\frac{1}{τ_{d}} ln (\sum_{i = 1}^{\infty} λ_{d, i}^{1 - τ_{d}})|}{ε^{- s} + d^{t}} = lim_{ε^{- 1} + d \to \infty} \frac{\frac{1}{τ_{d}} ln (\sum_{i = 1}^{\infty} λ_{d, i}^{1 - τ_{d}})}{ε^{- s} + d^{t}} = 0, \end{matrix}

(19)

where we used

ln (\sum_{i = 1}^{\infty} λ_{d, i}^{1 - τ_{d}}) \geq ln λ_{d, 1}^{1 - τ_{d}} = 0

by (12).

From (10) with

τ = 1 - τ_{d}

and Lemma 3 we have

\begin{matrix} ln (\sum_{i = 1}^{\infty} λ_{d, i}^{1 - τ_{d}}) & = ln (\prod_{j = 1}^{d} (1 + \sum_{k = 1}^{\infty} {(W_{α_{j}, β_{j}} (k))}^{1 - τ_{d}})) \\ = \sum_{j = 1}^{d} ln (1 + \sum_{k = 1}^{\infty} {(W_{α_{j}, β_{j}} (k))}^{1 - τ_{d}}) \\ \leq \sum_{j = 1}^{d} ln (1 + \sum_{k = 1}^{\infty} β_{j}^{1 - τ_{d}} A^{\frac{(1 - τ_{d}) k^{⌈ α_{1} ⌉}}{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉}}}) . \end{matrix}

(20)

Since

0 < A^{\frac{(1 - τ_{d}) k^{⌈ α_{1} ⌉}}{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉}}} \leq A^{\frac{(1 - τ_{1}) k^{⌈ α_{1} ⌉}}{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉}}}

and

\sum_{k = 1}^{\infty} A^{\frac{(1 - τ_{1}) k^{⌈ α_{1} ⌉}}{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉}}}

is convergent. It means that there exists a constant

C > 0

such that

\sum_{k = 1}^{\infty} A^{\frac{(1 - τ_{d}) k^{⌈ α_{1} ⌉}}{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉}}} < C

(21)

for all

d \in N

. By (20) and (21) we have

\begin{matrix} ln (\sum_{i = 1}^{\infty} λ_{d, i}^{1 - τ_{d}}) \leq \sum_{j = 1}^{d} ln (1 + C β_{j}^{1 - τ_{d}}) \leq C \sum_{j = 1}^{d} β_{j}^{1 - τ_{d}}, \end{matrix}

(22)

where in the last inequality we used

ln (1 + x) \leq x

for

x \geq 0

. Since (17), we have

τ_{j}^{- 1} \leq {ln}^{+} β_{j}^{- 1} \leq 1 + ln β_{j}^{- 1}

and thus

β_{j} \leq e^{1 - τ_{j}^{- 1}}

for all

j \in N

. We further get for all

1 \leq j \leq d

that

\begin{matrix} β_{j}^{1 - τ_{d}} \leq β_{j}^{1 - τ_{j}} \leq e^{(1 - τ_{j}^{- 1}) (1 - τ_{j})} = e^{1 - τ_{j}^{- 1}} e^{- τ_{j} (1 - τ_{j}^{- 1})} . \end{matrix}

(23)

We note that

\begin{matrix} e^{1 - τ_{j}^{- 1}} & = e e^{- τ_{j}^{- 1}} = \frac{e}{min {e^{{ln}^{+} β_{j}^{- 1}}, e^{ln (2 j + 1)}}} \\ \leq \frac{e}{min {e^{ln β_{j}^{- 1}}, e^{ln (2 j + 1)}}} = e max {β_{j}, \frac{1}{2 j + 1}}, \end{matrix}

(24)

and

\begin{matrix} e^{- τ_{j} (1 - τ_{j}^{- 1})} = e^{- τ_{j} + 1} \leq e . \end{matrix}

(25)

Combing (23), (24) and (25), we have

\sum_{j = 1}^{d} β_{j}^{1 - τ_{d}} \leq e^{2} \sum_{j = 1}^{d} max {β_{j}, \frac{1}{2 j + 1}} .

(26)

From (22) and (26) and we have

\begin{matrix} 0 & \leq lim_{ε^{- 1} + d \to \infty} \frac{\frac{1}{τ_{d}} ln (\sum_{i = 1}^{\infty} λ_{d, i}^{1 - τ_{d}})}{ε^{- s} + d^{t}} \leq lim_{d \to \infty} \frac{ln (\sum_{i = 1}^{\infty} λ_{d, i}^{1 - τ_{d}})}{τ_{d} d^{t}} \\ \leq lim_{d \to \infty} \frac{C \sum_{j = 1}^{d} β_{j}^{1 - τ_{d}}}{τ_{d} d^{t}} \leq lim_{d \to \infty} \frac{C e^{2} \sum_{j = 1}^{d} max {β_{j}, \frac{1}{2 j + 1}}}{τ_{d} d^{t}} . \end{matrix}

(27)

Next, we will prove

lim_{d \to \infty} \frac{\sum_{j = 1}^{d} max {β_{j}, \frac{1}{2 j + 1}}}{τ_{d} d^{t}} = 0 .

It follows from (17) that

\begin{matrix} 0 \leq lim_{d \to \infty} \frac{\sum_{j = 1}^{d} \frac{1}{2 j + 1}}{τ_{d} d^{t}} \leq lim_{d \to \infty} \frac{\int_{j = 0}^{d} \frac{1}{2 x + 1} d x}{τ_{d} d^{t}} \leq lim_{d \to \infty} \frac{\frac{1}{2} {ln}^{2} (2 d + 1)}{d^{t}} = 0, \end{matrix}

i.e.,

lim_{d \to \infty} \frac{\sum_{j = 1}^{d} \frac{1}{2 j + 1}}{τ_{d} d^{t}} = 0 .

(28)

Due to

lim_{d \to \infty} d^{1 - t} β_{d} = 0

for any

t \in (0, 1)

, we have

lim_{d \to \infty} d^{1 - t / 2} β_{d} = 0

. This means that there exists a positive number

N \in N

such that

β_{j} \leq j^{t / 2 - 1}

for all

j > N

. It follows that

\begin{matrix} \sum_{j = 1}^{d} β_{j} & = \sum_{j = 1}^{N} β_{j} + \sum_{j = N + 1}^{d} β_{j} \leq N + \sum_{j = N + 1}^{d} j^{t / 2 - 1} \\ \leq N + \sum_{j = 2}^{d} j^{t / 2 - 1} \leq N + \int_{1}^{d} x^{t / 2 - 1} d x \\ = N + \frac{2 d^{t / 2} - 2}{t} \leq N + \frac{2 d^{t / 2}}{t} \end{matrix}

for sufficiently large d, which yields by (17) that

\begin{matrix} 0 & \leq lim_{d \to \infty} \frac{\sum_{j = 1}^{d} β_{j}}{τ_{d} d^{t}} \leq lim_{d \to \infty} \frac{ln (2 d + 1) \sum_{j = 1}^{d} β_{j}}{d^{t}} \\ \leq lim_{d \to \infty} \frac{ln (2 d + 1) (N + \frac{2 d^{t / 2}}{t})}{d^{t}} \\ = lim_{d \to \infty} \frac{N ln (2 d + 1)}{d^{t}} + lim_{d \to \infty} \frac{2 d^{t / 2} ln (2 d + 1)}{t d^{t}} \\ = lim_{d \to \infty} \frac{2 ln (2 d + 1)}{t d^{t / 2}} = 0, \end{matrix}

i.e.,

lim_{d \to \infty} \frac{\sum_{j = 1}^{d} β_{j}}{τ_{d} d^{t}} = 0 .

(29)

We conclude from (28) and (29) that

lim_{d \to \infty} \frac{\sum_{j = 1}^{d} max {β_{j}, \frac{1}{2 j + 1}}}{τ_{d} d^{t}} = 0 .

Then we have by (27) that

lim_{ε^{- 1} + d \to \infty} \frac{\frac{1}{τ_{d}} ln (\sum_{i = 1}^{\infty} λ_{d, i}^{1 - τ_{d}})}{ε^{- s} + d^{t}} = 0,

and thus (19) holds. Hence

(s, t)

-weak tractability holds for any

t \in (0, 1)

and

s > 0

for ABS or NOR. Therefore, we finish the proof. □

Example 1.

An example for

(s, t)

-weak tractability with

s > 0

and

t \in (0, 1)

Assume that

α_{j} = 2 j

and

β_{j} = \frac{1}{j + 2}

satisfy (7) for all

j \in N

. Obviously, we have

lim_{j \to \infty} β_{j} j^{1 - t} = lim_{j \to \infty} \frac{j^{1 - t}}{j + 2} = 0 .

Next, we will prove that the problem APP defined over the space

H_{d, α, β} ({[0, 1]}^{d})

with the covariance weight

W_{d, α, β} \in {ρ_{d, α, β}, σ_{d, α, β}}

(s, t)

-weakly tractable for

s > 0

and

t \in (0, 1)

under ABS or NOR. By the inequality (14), we only need to prove that

(s, t)

-weak tractability holds for

s > 0

and

t \in (0, 1)

under ABS.

Let

s > 0

and

t \in (0, 1)

. Choose

τ_{d}^{'} = \frac{1}{ln (d + 2)}, for d \in N .

Set

τ = 1 - τ_{d}^{'}

in the inequality (5) for ABS. Then we have

\begin{matrix} 0 & \leq lim_{ε^{- 1} + d \to \infty} \frac{ln n^{ABS} (ε, {APP}_{d})}{ε^{- s} + d^{t}} \leq lim_{ε^{- 1} + d \to \infty} \frac{|\frac{1 - τ_{d}^{'}}{τ_{d}^{'}} (ln \frac{1 - τ_{d}^{'}}{τ_{d}^{'}} + 2 ln (ε^{- 1}) + \frac{ln (\sum_{i = 1}^{\infty} λ_{d, i}^{1 - τ_{d}^{'}})}{1 - τ_{d}^{'}}) + ln 2|}{ε^{- s} + d^{t}} \\ \leq lim_{ε^{- 1} + d \to \infty} \frac{|\frac{1 - τ_{d}^{'}}{τ_{d}^{'}} (ln \frac{1 - τ_{d}^{'}}{τ_{d}^{'}} + 2 ln (ε^{- 1})) + ln 2|}{ε^{- s} + d^{t}} + lim_{ε^{- 1} + d \to \infty} \frac{|\frac{1}{τ_{d}^{'}} ln (\sum_{i = 1}^{\infty} λ_{d, i}^{1 - τ_{d}^{'}})|}{ε^{- s} + d^{t}} . \end{matrix}

Since

\begin{matrix} 0 \leq lim_{ε^{- 1} + d \to \infty} \frac{|\frac{1 - τ_{d}^{'}}{τ_{d}^{'}} (ln \frac{1 - τ_{d}^{'}}{τ_{d}^{'}} + 2 ln (ε^{- 1})) + ln 2|}{ε^{- s} + d^{t}} \\ = lim_{ε^{- 1} + d \to \infty} \frac{|(ln (d + 2) - 1) (ln (ln (d + 2) - 1) + 2 ln (ε^{- 1})) + ln 2|}{ε^{- s} + d^{t}} \\ \leq lim_{d \to \infty} \frac{(ln (d + 2) - 1) ln (ln (d + 2) - 1) + ln 2}{d^{t}} + lim_{ε^{- 1} + d \to \infty} \frac{2 (ln (d + 2) - 1) ln (ε^{- 1})}{ε^{- s} + d^{t}} \\ = lim_{ε^{- 1} + d \to \infty} \frac{2 (ln (d + 2) - 1) ln (ε^{- 1})}{ε^{- s} + d^{t}} \\ \leq lim_{ε^{- 1} + d \to \infty} \frac{{(ln (d + 2) - 1)}^{2} + {(ln (ε^{- 1}))}^{2}}{ε^{- s} + d^{t}} \\ \leq lim_{d \to \infty} \frac{{(ln (d + 2) - 1)}^{2}}{d^{t}} + lim_{ε^{- 1} \to \infty} \frac{{(ln (ε^{- 1}))}^{2}}{ε^{- s}} = 0, \end{matrix}

i.e.,

lim_{ε^{- 1} + d \to \infty} \frac{|\frac{1 - τ_{d}^{'}}{τ_{d}^{'}} (ln \frac{1 - τ_{d}^{'}}{τ_{d}^{'}} + 2 ln (ε^{- 1})) + ln 2|}{ε^{- s} + d^{t}} = 0,

next, we only need to prove

lim_{ε^{- 1} + d \to \infty} \frac{|\frac{1}{τ_{d}^{'}} ln (\sum_{i = 1}^{\infty} λ_{d, i}^{1 - τ_{d}^{'}})|}{ε^{- s} + d^{t}} = 0 .

It follows from (10) with

τ = 1 - τ_{d}^{'}

and Lemma 3 that

\begin{matrix} ln (\sum_{i = 1}^{\infty} λ_{d, i}^{1 - τ_{d}^{'}}) & = ln (\prod_{j = 1}^{d} (1 + \sum_{k = 1}^{\infty} {(W_{α_{j}, β_{j}} (k))}^{1 - τ_{d}^{'}})) \\ = \sum_{j = 1}^{d} ln (1 + \sum_{k = 1}^{\infty} {(W_{α_{j}, β_{j}} (k))}^{1 - τ_{d}^{'}}) \\ \leq \sum_{j = 1}^{d} ln (1 + \sum_{k = 1}^{\infty} β_{j}^{1 - τ_{d}^{'}} A^{\frac{(1 - τ_{d}^{'}) k^{⌈ α_{1} ⌉}}{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉}}}) \\ = \sum_{j = 1}^{d} ln (1 + \frac{1}{{(j + 2)}^{1 - τ_{d}^{'}}} \sum_{k = 1}^{\infty} A^{\frac{(1 - τ_{d}^{'}) k^{2}}{4}}) . \end{matrix}

(30)

We note that

A^{\frac{(1 - τ_{d}^{'}) k^{2}}{4}} = A^{\frac{(1 - \frac{1}{ln (d + 2)}) k^{2}}{4}} \leq A^{\frac{(1 - \frac{1}{ln 3}) k^{2}}{4}}

and

\sum_{k = 1}^{\infty} A^{\frac{(1 - \frac{1}{ln 3}) k^{2}}{4}}

is convergent. Then there exists a constant

C > 0

such that

\sum_{k = 1}^{\infty} A^{\frac{(1 - τ_{d}^{'}) k^{2}}{4}} \leq \sum_{k = 1}^{\infty} A^{\frac{(1 - \frac{1}{ln 3}) k^{2}}{4}} \leq C .

We further get from (30) that

\begin{matrix} ln (\sum_{i = 1}^{\infty} λ_{d, i}^{1 - τ_{d}^{'}}) & \leq \sum_{j = 1}^{d} ln (1 + \frac{C}{{(j + 2)}^{1 - τ_{d}^{'}}}) \leq C \sum_{j = 1}^{d} \frac{1}{{(j + 2)}^{1 - τ_{d}^{'}}} \\ = C \sum_{j = 1}^{d} \frac{1}{{(j + 2)}^{1 - \frac{1}{ln (d + 2)}}} = C \sum_{j = 1}^{d} \frac{{(j + 2)}^{\frac{1}{l n (d + 2)}}}{j + 2} \\ \leq C \sum_{j = 1}^{d} \frac{{(d + 2)}^{\frac{1}{l n (d + 2)}}}{j + 2} = C e \sum_{j = 1}^{d} \frac{1}{j + 2} \\ \leq C e (\int_{0}^{d} \frac{1}{x + 2} d x) = C e (ln (d + 2) - ln 2), \end{matrix}

which conclude that

\begin{matrix} 0 & \leq lim_{ε^{- 1} + d \to \infty} \frac{|\frac{1}{τ_{d}^{'}} ln (\sum_{i = 1}^{\infty} λ_{d, i}^{1 - τ_{d}^{'}})|}{ε^{- s} + d^{t}} \\ = lim_{ε^{- 1} + d \to \infty} \frac{|C e ln (d + 2) (ln (d + 2) - ln 2)|}{ε^{- s} + d^{t}} \\ \leq lim_{d \to \infty} \frac{|C e ln (d + 2) (ln (d + 2) - ln 2)|}{d^{t}} \\ \leq lim_{d \to \infty} \frac{|C e ln (d + 2) (ln (d + 2))|}{d^{t}} = 0, \end{matrix}

i.e.,

lim_{ε^{- 1} + d \to \infty} \frac{|\frac{1}{τ_{d}^{'}} ln (\sum_{i = 1}^{\infty} λ_{d, i}^{1 - τ_{d}^{'}})|}{ε^{- s} + d^{t}} = 0 .

Hence we have

lim_{ε^{- 1} + d \to \infty} \frac{ln n^{ABS} (ε, {APP}_{d})}{ε^{- s} + d^{t}} = 0,

which yields that

(s, t)

-weak tractability holds for ABS. Therefore, APP is

(s, t)

-weakly tractable for any

s > 0

and

t \in (0, 1)

under ABS or NOR.

Example 2.

An example not for

(s, t)

-weak tractability with any

s > 0

and

t \in (0, 1)

Assume that

α_{j} = 2^{j}

and

β_{j} = \frac{1}{ln (3 j)}

for all

j \in N

. Obviously, we have

lim_{j \to \infty} β_{j} j^{1 - t} = lim_{j \to \infty} \frac{j^{1 - t}}{ln (3 j)} = \infty .

Next, we will prove that the problem APP defined over the space

H_{d, α, β} ({[0, 1]}^{d})

with the covariance weight

W_{d, α, β} \in {ρ_{d, α, β}, σ_{d, α, β}}

is not

(s, t)

-weakly tractable for any

s > 0

and

t \in (0, 1)

under ABS or NOR. Due to the inequality (14), we only need to prove that

(s, t)

-weak tractability does not hold for any

s > 0

and

t \in (0, 1)

under NOR.

Let

s > 0

and

t \in (0, 1)

. We conclude from inequality (16) with

ε = \frac{1}{2}

and Remark 4 that

\begin{matrix} lim_{d \to \infty} \frac{ln n^{NOR} (\frac{1}{2}, {APP}_{d})}{2^{s} + d^{t}} & \geq lim_{d \to \infty} \frac{ln \frac{3}{4} + \sum_{j = 1}^{d} ln (1 + \sum_{k = 1}^{\infty} W_{α_{j}, β_{j}} (k))}{2^{s} + d^{t}} \\ \geq lim_{d \to \infty} \frac{ln \frac{3}{4} + \sum_{j = 1}^{d} ln (1 + W_{α_{j}, β_{j}} (1))}{2^{s} + d^{t}} \\ = lim_{d \to \infty} \frac{ln \frac{3}{4} + \sum_{j = 1}^{d} ln (1 + β_{j} A)}{2^{s} + d^{t}} \\ = lim_{d \to \infty} \frac{\sum_{j = 1}^{d} ln (1 + β_{j} A)}{2^{s} + d^{t}} \\ = lim_{d \to \infty} \frac{\sum_{j = 1}^{d} ln (1 + \frac{A}{ln (3 j)})}{2^{s} + d^{t}} \\ = lim_{d \to \infty} \frac{ln (1 + \frac{A}{ln (3 d)})}{d^{t} - {(d - 1)}^{t}} \\ = lim_{d \to \infty} \frac{1}{t} d^{1 - t} ln (1 + \frac{A}{ln (3 d)}) \\ = lim_{d \to \infty} \frac{A d^{1 - t}}{t ln (3 d)} = + \infty, \end{matrix}

where in the fourth equality we used Stolz theorem. Hence APP is not

(s, t)

-weak tractable for any

s > 0

and

t \in (0, 1)

under ABS or NOR.

5. Conclusions

In this paper we study average case

(s, t)

-weak tractability with any

s > 0

and

t \in (0, 1)

for the

L_{2}

-approximation problem

APP = {{APP}_{d}}_{d \in N}

from the Banach space

H_{d, α, β}

equipped with a zero-mean Gaussian measure

μ_{d}

with the covariance kernel

K_{W_{d, α, β}}

with weight

W_{d, α, β}

, where

1 < α_{1} \leq α_{2} \leq \dots

and

1 \geq β_{1} \geq β_{2} \geq \dots > 0

are parameters. We obtain a compete result for

W_{d, α, β} \in {ρ_{d, α, β}, σ_{d, α, β}}

that APP is

(s, t)

-weakly tractable under ABS or NOR for any

s > 0

and

t \in (0, 1)

iff

lim_{j \to \infty} β_{j} j^{1 - t} = 0 .

We will further investigate other algebraic tractability notions about multivariate approximation problems from Banach spaces equipped with zero-mean Gaussian measures with different weighted covariance kernels and hope to get more good results.

Author Contributions

Conceptualization, J.C. and H.Y.; methodology, J.C. and H.Y.; validation, J.C.; formal analysis, J.C.; investigation, H.Y.; resources, H.Y.; data curation, J.C.; writing—original draft preparation, J.C.; writing—review and editing, J.C.; visualization, J.C.; supervision, J.C. and H.Y.; project administration, J.C. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the National Natural Science Foundation of China (Project no. 12001342), the Scientific and Technological Innovation Project of Colleges and Universities in Shanxi Province (Project no. 2022L438), the Basic Youth Research Found Project of Shanxi Datong University (Project no. 2022Q10), and the Doctoral Foundation Project of Shanxi Datong University (Project no. 2021-B-17, Project no. 2019-B-10).

Data Availability Statement

The original data were presented in this manuscript.

Acknowledgments

We are very grateful to all those for many useful suggestions on how to improve the drafts.

Conflicts of Interest

The authors declare no conflict of interest.

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Average Case $(s, T)$-Weak Tractability of $L_2$-Approximation with Weighted Covariance Kernels

Abstract

1. Introduction

2. Average Case Algebraic Tractability of Multivariate Approximation Problems

3. L 2 -Approximation with Weighted Covariance Kernels

3.1. A Variant of the Korobov Covariance Kernel

3.2. A Variant of the Gaussian ANOVA Covariance Kernel

4. Average Case ( s , t ) -Weak Tractability of L 2 -Approximation with the Two Weighted Covariance Kernels and the Main Result

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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3. $L_{2}$ -Approximation with Weighted Covariance Kernels

4. Average Case $(s, t)$ -Weak Tractability of $L_{2}$ -Approximation with the Two Weighted Covariance Kernels and the Main Result