Abstract
Full Text
UDC 518.12+517.392
MATHEMATICS
I. M. SOBOL’
APPLICATION OF HAAR SERIES TO ERROR ESTIMATION IN THE COMPUTATION OF INFINITE-DIMENSIONAL INTEGRALS
(Presented by Academician A. N. Tikhonov on 24 IX 1966)
We estimate the error of the approximation
[
\int_{K_\infty} f(X)\,dX \approx \frac{1}{N}\sum_{\nu=1}^{N} f(X_\nu),
\tag{*}
]
where (K_\infty) is the infinite-dimensional unit cube; (X=(x_1,x_2,\ldots,x_n,\ldots)) is a point of this cube. The measure on (K_\infty) is defined by means of the infinite product of Lebesgue measures ((dX=dx_1dx_2\ldots dx_n\ldots)).
In (1) a sequence of points ({X_\nu^}) was constructed, called a generalized Halton sequence, whose points can be used as nodes in (()), and an error estimate for such an approximation was obtained for certain classes of functions.
In § 2 of the present paper the error of the approximation (()) is estimated for broader classes of functions (S_p) and (\bar H_\alpha) (the classes (H_\alpha) are analogues of the Lipschitz classes (\operatorname{Lip}\alpha), (0<\alpha\le 1)). A new sequence of points ({X_\nu^{}}), called a generalized (ЛП_\tau)-sequence, is constructed, for which estimates are obtained that are somewhat better than for ({X_\nu^}). On the classes (S_p) the order of these estimates turns out to be the best possible. The concept of a grid ({X_1,\ldots,X_N}) with a multiplicative estimate of nonuniformity is introduced. For such grids an estimate of the approximation ((*)) is obtained in general form.
The paper uses the method of Haar series, which made it possible to obtain error estimates on the classes (S_p) and (H_\alpha) in the (n)-dimensional case ((^{2-4})). However, in order to be able to pass to the limit as (n\to\infty), it was necessary to improve these estimates somewhat. This is done in § 1.
In what follows, the symbol (K_{i_1\ldots i_s}) denotes the (s)-dimensional face of the cube (K_\infty) on which (x_{i_1},\ldots,x_{i_s}) range from 0 to 1, while all the remaining (x_i\equiv 1).
§ 1. The case of (n) variables.
1.1. Expansion of a function in “different-dimensional” terms.
Consider a function (f(P)), where (P=(x_1,\ldots,x_n)), defined in the (n)-dimensional unit cube (K\equiv K_{12\ldots n}). Write its expansion in the Fourier–Haar series, explicitly separating the first function of the Haar system (\chi_1(x)\equiv 1). Let
[
C_{k_1\ldots k_s}^{i_1\ldots i_s}
=
\int_K f(P)\chi_{k_1}(x_{i_1})\cdots \chi_{k_s}(x_{i_s})\,dP,
]
[
(dP=dx_1\cdots dx_n),
]
where (1\le i_1<i_2<\cdots<i_s\le n,\ 1\le s\le n). Then
[
f(P)=c_0+\sum_{k_1,\ldots,k_s}^{\wedge}\sum C_{k_1\ldots k_s}^{i_1\ldots i_s}
\chi_{k_1}(x_{i_1})\cdots \chi_{k_s}(x_{i_s}),
\tag{1}
]
where the symbol (\hat{\sum}) denotes summation over all different (s)-index quantities for (s=1,2,\ldots,n):
[
\hat{\sum} T_{i_1\ldots i_s}
=
\sum_{i_1=1}^{n} T_{i_1}
+
\sum_{1\leq i_1<i_2\leq n}\sum T_{i_1 i_2}
+\cdots+
T_{12\ldots n}.
]
In (1), as throughout what follows, the indices (k_\sigma) vary from (2) to (\infty). Each of the sums following the sign (\hat{\sum}) depends only on the (s) variables (x_{i_1},\ldots,x_{i_s}).
1.2. Estimation of the integration error. Choose an integration net (\Sigma), consisting of arbitrary points (P_1,\ldots,P_N) of the cube (K), and estimate the error
[
\delta(f;\Sigma)=\frac{1}{N}\sum_{\nu=1}^{N} f(P_\nu)-\int_K f(P)\,dP .
\tag{2}
]
Suppose that the series (1) converges uniformly, and substitute it into (2). After transformations entirely analogous to the transformations of Sec. 1 from ((^3)), we obtain the estimate
[
|\delta(f;\Sigma)| \leq \frac{1}{N}\hat{\sum} A_p^{i_1\ldots i_s}(f)\,\Phi_q^{i_1\ldots i_s}(\Sigma),
\tag{3}
]
in which (1/p+1/q=1);
[
A_p^{i_1\ldots i_s}(f)
=
\sum_m 2^{(m_1-1)/2+\cdots+(m_s-1)/2}
\left{\sum_j |C_k^i|^p\right}^{1/p},
]
and (\Phi_q^{i_1\ldots i_s}(\Sigma)) is the (s)-dimensional discrepancy of the projections of the points (P_1,\ldots,P_N) onto (K_{i_1\ldots i_s}) ((^3)). Here
(i=(i_1,\ldots,i_s)), (k=(k_1,\ldots,k_s)), (m=(m_1,\ldots,m_s)),
(j=(j_1,\ldots,j_s));
(1\leq m_\sigma<\infty), (1\leq j_\sigma\leq 2^{m_\sigma-1}).
In ((^3)) the norm (|f|_p=\hat{\sum} A_p^{i_1\ldots i_s}(f)) was used. Therefore the estimate (3) was coarsened and written in the form
[
|\delta(f;\Sigma)| \leq N^{-1}|f|_p\,\varphi_q(\Sigma),
\quad
\text{where }
\varphi_q(\Sigma)=\max \Phi_q^{i_1\ldots i_s}(\Sigma).
]
Below only the simplest quantities (\Phi_\infty^{i_1\ldots i_s}(\Sigma)) are used; we shall call them the discrepancies of the net (\Sigma). It is easy to prove that for any net
[
\Phi_q^{i_1\ldots i_s}(\Sigma) \leq N^{1/q}\Phi_\infty^{i_1\ldots i_s}(\Sigma).
\tag{4}
]
1.3. Classes of functions (S_p) and (H_\alpha).
Definition. A function (f(x_1,\ldots,x_n)) belongs to the class (S_p(L_{i_1\ldots i_s})), (1\leq p<\infty), if for any (1\leq i_1<i_2<\cdots<i_s\leq n) and (1\leq s\leq n)
[
A_p^{i_1\ldots i_s}(f) \leq L_{i_1\ldots i_s}.
\tag{5}
]
The constants ({L_{i_1\ldots i_s}}) are called the defining constants of the class.
As in ((^2,^3)), it is proved that if (f\in S_p), then the series (1) converges uniformly in (K), and estimate (3) with (L_{i_1\ldots i_s}) in place of (A_p^{i_1\ldots i_s}(f)) is an exact estimate on (S_p) (for any prescribed net (\Sigma)).
Definition. A function (f(x_1,\ldots,x_n)) belongs to the class (H_\alpha(L_{i_1\ldots i_s})), (0<\alpha\leq 1), if for any (1\leq i_1<i_2<\cdots<i_s\leq n) and (1\leq)
(\leqslant s \leqslant n) in the cube (K)
[
\left|\Delta_{\xi_{i_1}}\cdots \Delta_{\xi_{i_s}} f(P)\right|
\leqslant
L_{i_1\ldots i_s}\left(\frac{\alpha+1}{2}\right)^s
\left|\xi_{i_1}\cdots \xi_{i_s}\right|^\alpha .
\tag{6}
]
Just as in ((^2)), it is proved that if (f \in H_\alpha(L_{i_1\ldots i_s})) and (\alpha p>1), then
[
A_p^{i_1\ldots i_s}(f)
\leqslant
L_{i_1\ldots i_s}\left(2^{1+\alpha}-2^{1+1/p}\right)^{-s}.
\tag{7}
]
From (3), (4), and (7) one can obtain an estimate of (|\delta(f;\Sigma)|) on the classes (H_\alpha), quite analogous to estimate (8) from ((^3)).
§ 2. The case of infinitely many variables
2.1. Classes of functions
The classes (S_p(L_{i_1\ldots i_s})) and (H_\alpha(L_{i_1\ldots i_s})) are defined in the same way as in the (n)-dimensional case: the corresponding inequalities (5) or (6) must be satisfied in (K_\infty) for arbitrary (1\leqslant i_1<i_2<\cdots<i_s), (1\leqslant s<\infty). In addition, it is assumed that at each point
(f(X)=\lim_{n\to\infty} f(x_1,\ldots,x_n,1,1,1,\ldots)), and passage to the limit is possible:
[
\int_{K_\infty} f(X)\,dX
=
\lim_{n\to\infty}\int_K f(x_1,\ldots,x_n,1,1,\ldots)\,dP .
]
2.2. Estimate of the error of integration
Choose a grid consisting of points (X_1,\ldots,X_N) in (K_\infty). The projections of these points onto (K_{i_1\ldots i_s}) form an (s)-dimensional grid, whose discrepancy shall be denoted by (\Phi_\infty^{i_1,\ldots,i_s}).
Suppose that there exist constants (B) and ({h_i}), (1\leqslant i<\infty), such that for any (i_1<\cdots<i_s)
[
\Phi_\infty^{i_1\ldots i_s}\leqslant Bh_{i_1}\cdots h_{i_s}.
\tag{8}
]
In this case we shall say that the grid admits a multiplicative estimate of discrepancies.
Theorem. For a grid (X_1,\ldots,X_N) admitting the multiplicative estimate of discrepancies (8), on classes of functions (S_p) and (H_\alpha) whose defining constants decrease uniformly with respect to ({h_i}), the estimate holds
[
\left|
\frac{1}{N}\sum_{\nu=1}^{N} f(X_\nu)
-
\int_{K_\infty} f(X)\,dX
\right|
\leqslant
ABN^{-1/p}\exp\sum_{i=1}^{\infty} v_i,
\tag{9}
]
where (v_i=\varepsilon_i h_i) in the case of the class (S_p), while in the case of the class (H_\alpha) the parameter (p>1/\alpha) and
[
v_i=\varepsilon_i h_i(2^{1+\alpha}-2^{1+1/p})^{-1}.
]
Proof scheme. Use inequality (3) for the function (f(x_1,\ldots,x_n,1,1,\ldots)), then inequalities (5) or (7), (4), and (8). The right-hand side is estimated by means of Lemma 2 from ((^1)). Then pass to the limit as (n\to\infty).
2.3. Use of the generalized Halton sequence
As the grid (X_1,\ldots,X_N) we choose the points (X_1^,\ldots,X_N^). Since the projections of the points ({X_\nu^*}) onto (K_{i_1\ldots i_s}) form an (s)-dimensional Halton sequence, it is easy to show (cf. ((^1,^4))) that estimate (8) will be satisfied with (B=1), (h_i=\bar{\beta}_i\ln N+\bar{\gamma}_i). The asymptotics of the coefficients as (i\to\infty) is known: (\bar{\beta}_i\sim 4i), (\bar{\gamma}_i\sim 8i\ln i). Hence, for the applicability of the preceding theorem it is sufficient to require that the defining constants of the class decrease uniformly with respect to ({i\ln i}).
Choose an arbitrary (\varepsilon>0). One can choose (A) and ({\varepsilon_i}) so that (\sum \varepsilon_i\bar{\beta}_i=\varepsilon); then the order of estimate (9) on the classes (S_p) will be (N^{-1/p+\varepsilon}).
[
\text{* According to }(^1),\ \text{this means that }\
L_{i_1\ldots i_s}\leqslant A\varepsilon_{i_1}\cdots\varepsilon_{i_s}
\ \text{and, in addition, the series }\
\sum_{1}^{\infty}\varepsilon_i h_i
\ \text{converges.}
]
If (p) is fixed so that (0<\alpha-1/p<\varepsilon), and then (A) and ({\varepsilon_i}) are chosen so that
(\sum \varepsilon_i\beta_i=(2^{1+\alpha}-2^{1+1/p})(\varepsilon-\alpha+1/p)), then the order of the estimate (9) on the classes (H_\alpha) will be (N^{-\alpha+\varepsilon}).
We note that the orders of the estimates on the classes (S_p) and (H_\alpha), even in the (n)-dimensional case, cannot be better than (N^{-1/p}) and, respectively, (N^{-\alpha}).
2.4. Use of a generalized (\mathrm{LP}_\tau)-sequence
Definition. By a generalized (\mathrm{LP}\tau)-sequence we shall mean a sequence of points ({X\nu^{**}}) with coordinates
[
X_{\nu+1}^{**}=\bigl(p^{(1)}(\nu),\ p^{(2)}(\nu),\ldots,\ p^{(n)}(\nu),\ldots\bigr),
]
where ({p^{(n)}(\nu)}) is a sequence of type (DR) corresponding to the (n)-th monocyclic operator (5) (the operators are numbered so that their orders (\bar m_n) do not decrease).
As the grid (X_1,\ldots,X_N) we choose the points (X_1^{},\ldots,X_N^{}). Since the projections of the points ({X_\nu^{**}}) onto (K_{i_1,\ldots,i_s}) form an (s)-dimensional (\mathrm{LP}_\tau)-sequence, it follows from (5) that estimate (8) holds with (B=1/2), (h_i=2^{\bar m_i}), and also the asymptotic estimate for (\bar m_i) as (i\to\infty),
[
\bar m_i \leq \log_2 i+\log_2\bigl(\log_2 i\,\log_2\log_2 i\bigr)+O(1).
\tag{10}
]
Thus, (h_i\leq C i\ln i\ln\ln i), and for the applicability of our theorem it is sufficient to require that the defining constants of the class decrease uniformly with respect to ({i\ln i\ln\ln i}).
The order of the estimate (9) on the classes (S_p) turns out to be (N^{-1/p}). This is the best possible order.
Let us now consider the classes (H_\alpha). Let (0<\varkappa<\varepsilon). Fix (1/p=\alpha-\varkappa/\sqrt{\ln N}). After a suitable choice of (A) and ({\varepsilon_i}), the order of the estimate (9) will be equal to (N^{-\alpha+\varepsilon/\sqrt{\ln N}}).
Remark. In this subsection we had to require a more rapid decrease of the defining constants in comparison with Sec. 2.3. Apparently this is caused by the crudeness of estimate (10).
Received
20 IX 1966
References
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- I. M. Sobol’, DAN, 139, No. 4, 821 (1961).
- I. M. Sobol’, UMN, 21, No. 5, 271 (1966).