An Optimized Monte Carlo Approach for
Multidimensional Integrals Related to Intelligent
Systems
Venelin Todorov
*†
, Ivan Dimov
†
, Stefka Fidanova
†
, Rayna Georgieva
†
, Tzvetan Ostromsky
†
, Stoyan Poryazov
*
*
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
8 Acad. G. Bonchev Str., 1113 Sofia, Bulgaria
†
Institute of Information and Communication Technologies
Bulgarian Academy of Sciences
25A Acad. G. Bonchev Str., 1113 Sofia, Bulgaria
Email: vtodorov@math.bas.bg, venelin@parallel.bas.bg, ivdimov@bas.bg, stefka@parallel.bas.bg,
rayna@parallel.bas.bg, ceco@parallel.bas.bg, stoyan@math.bas.bg
Abstract—We study an optimized Monte Carlo algorithm for
solving multidimensional integrals related to intelligent systems.
Recently Shaowei Lin consider the difficult task of evaluating
multidimensional integrals with very high dimensions which
are important to machine learning for intelligent systems. Lin
multidimensional integrals with 3 to 30 dimensions, related to
applications in machine learning, will be evaluated with the
presented optimized Monte Carlo algorithm and some advantages
of the method will be analyzed.
I. I NTRODUCTION
T
EN YEARS ago Shaowei Lin in his works [4], [5]
consider the important problem of evaluating multidi-
mensional integrals used in intelligent systems. The first
multidimensional Lin integrals are of the form
Ω
p
u1
1
(x) ...p
us
s
(x)dx, (1)
and the second Lin integrals are of the form
Ω
e
-Nf (x)
φ(x)dx, (2)
where f (x) and φ(x) are multidimensional polynomials with
an integer N . Up to now multidimensional Lin integrals
(1) and (2) are computed unsatisfactory with deterministic
[10] and algebraic methods [9], and it is known that the
Monte Carlo (MC) methods [3], [7], [8] outperforms the
deterministic methods which suffer from the so called ,,curse
of dimensionality” [3] especially for higher dimensions.
Venelin Todorov is supported by the Bulgarian National Science Fund under
Project KP-06-M32/2 - 17.12.2019 ”Advanced Stochastic and Deterministic
Approaches for Large-Scale Problems of Computational Mathematics” and
Project KP-06-N52/5 ”Efficient methods for modeling, optimization and
decision making”. The work is also supported by the Bulgarian National
Science Fund under Project KP-06-N52/2 ”Perspective Methods for Quality
Prediction in the Next Generation Smart Informational Service Networks” and
by the Bilateral Project KP-06-Russia/17 ”New Highly Efficient Stochastic
Simulation Methods and Applications”.
The paper is organised as follows. The description of
the optimal stochastic approach is given in Section II. The
numerical study with Lin multidimensional integrals is given
in Section III. Finally some concluding remarks are given in
Section IV.
II. THE OPTIMAL STOCHASTIC APPROACH
We adapt the idea of the original MC method developed by
Atanassov and Dimov twenty years ago [1].
Let d and k be integers, d, k g 1. We consider the class
F
0
c W
k
('f '; U
d
) (3)
(sometimes abbreviated to W
k
) of real functions f defined
over the unit cube U
d
= [0, 1)
d
, possessing all the partial
derivatives
∂
r
f (x)
∂x
α1
1
...∂x
α
d
d
, α
1
+ ··· + α
d
= r f k, (4)
which are continuous when r<k and bounded in sup norm
when r = k. The semi-norm '·' on W
k
is defined as
'f ' = sup
∂
k
f (x)
∂x
α1
1
...∂x
αd
d
, α
1
+ ··· + α
d
= k, x c (x
1
,...,x
d
) * U
d
"
.
(5)
Now for n, s, k g 1 we construct a MC integration formula
depending on m g 1 and
(
s+k-1
s
)
points in [0, 1]
s
. Points
x
(r)
are exactly
(
s+k-1
s
)
and if for P (x) for the degree of the
polynom deg P f k is fulfilled P (x
(r)
)=0, then P c 0. If
N = n
s
for n g 1 we divide [0, 1]
s
into n
s
endless undercubes
K
j
, i.e.
[0, 1]
s
= c
n
s
i=1
K
j
and
K
j
=
s
i=1
[a
j
i
,b
j
i
),
b
j
i
2 a
j
i
=
1
n
,
Communication Papers of the of the 17
th
Conference on Computer
Science and Intelligence Systems pp. 101–104
DOI: 10.15439/2022F84
ISSN 2300-5963 ACSIS, Vol. 32
©2022, PTI 101