Новий метод для відновлення мішаних похідних вищих порядків функцій двох змінних
The problem of numerical recovering high-ordermixed derivatives of bivariate functions with finite smoothness is studied. On the basis of the truncation method, an algorithm for numerical differentiation is constructed, which guarantees a high order of approximation accuracy.
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Інститут прикладних проблем механіки і математики ім. Я. С. Підстригача НАН України
2023
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Репозитарії
Physico-mathematical modeling and informational technologies| _version_ | 1867479635216826368 |
|---|---|
| author | Semenova, Yevgeniya Solodky, Serhii G. |
| author_facet | Semenova, Yevgeniya Solodky, Serhii G. |
| author_institution_txt_mv | [
{
"author": "Yevgeniya Semenova",
"institution": "Candidate of Physical and Mathematical Sciences, Institute of Mathematics, National Academy of Sciences of Ukraine, 3, Tereschenkivska Str., 01024, Kiev"
},
{
"author": "Serhii G. Solodky",
"institution": "Doctor of Physical and Mathematical Sciences, Institute of Mathematics, National Academy of Sciences of Ukraine, 3, Tereschenkivska Str., 01024, Kiev"
}
] |
| author_sort | Semenova, Yevgeniya |
| baseUrl_str | http://www.fmmit.lviv.ua/index.php/fmmit/oai |
| collection | OJS |
| datestamp_date | 2025-02-21T17:32:19Z |
| description | The problem of numerical recovering high-ordermixed derivatives of bivariate functions with finite smoothness is studied. On the basis of the truncation method, an algorithm for numerical differentiation is constructed, which guarantees a high order of approximation accuracy. |
| first_indexed | 2026-06-09T01:09:24Z |
| format | Article |
| fulltext |
17
doi.org/10.15407/fmmit2023.36.017
A new method for recovering high-order mixed derivatives of
bivariate functions
Yevgeniya Semenova1, Serhii Solodky2
1 Candidate of Physical and Mathematical Sciences, Institute of Mathematics, National Academy of
Sciences of Ukraine, 3, Tereschenkivska Str., 01024, Kiev, e-mail: semenovaevgen@gmail.com
2 Doctor of Physical and Mathematical Sciences, Institute of Mathematics, National Academy of
Sciences of Ukraine, 3, Tereschenkivska Str., 01024, Kiev, e-mail: solodky@imath.kiev.ua
The problem of numerical recovering high-order mixed derivatives of bivariate functions with
finite smoothness is studied. On the basis of the truncation method, an algorithm for numerical
differentiation is constructed, which guarantees a high order of approximation accuracy.
Ключові слова: Numerical differentiation, Legendre polynomials, truncation method
Introduction. Many research activities on the problem of stable numerical
differentiation have been taking place due to the importance of this tool in such areas
of science and technology as finance, mathematical physics, image processing,
analytical chemistry, viscous elastic mechanics, reliability analysis, pattern
recognition, and many others. Among this investigation, we highlight [1], which is
the first publication on numerical differentiation in terms of the theory of ill-
posed problems. Further research [1] has been continued in numerous publications on
numerical differentiation for covering different classes of functions and the types of
proposed methods. Despite the abundance of works on this topic, the problem of
recovery of high-order derivatives was considered only in a few publications. In
particular, the results of [2] have opened a perspective for further investigation of
numerical methods for the recovery of high-order derivatives. Namely, as the main
criteria of the method's efficiency have been taken its ability to achieve the optimal
order of accuracy by using a minimal amount of discrete information. Note that
particular these aspects of numerical differentiation remain still insufficiently studied.
The present paper continues the research of [2], [3] and proposes a numerical method
for recovering the high-order mixed derivatives of smooth bivariate functions. The
method is not only stable to small perturbations of the input data, but achieves a high
order of accuracy with economical use of discrete information, and also has a simple
numerical implementation.
1. Description of the problem
Let {𝜑𝑘(𝑡)}𝑘=0
∞ be the system of Legendre polynomials orthonormal on [−1,1] as
UDC 519.653
mailto:solodky@imath.kiev.ua
Yevgeniya Semenova, Serhii Solodky
A new method for recovering high-order mixed derivatives of bivariate functions
18
𝜑𝑘(𝑡) = 𝑘 + 1/2(2𝑘𝑘!)−1
𝑑𝑘
𝑑𝑡𝑘
[(𝑡2 − 1)𝑘], 𝑘 = 0,1,2,….By 𝐿2 = 𝐿2(𝑄) we mean
space of square-summable on 𝑄 = [−1,1]2 functions 𝑓(𝑡, 𝜏) with inner product
⟨𝑓, 𝑔⟩ = ∫−1
1
∫−1
1
𝑓(𝑡, 𝜏)𝑔(𝑡, 𝜏)𝑑𝜏𝑑𝑡 and corresponding norm
∥ 𝑓 ∥𝐿2
2 = ∑
𝑘,𝑗=0
∞
|⟨𝑓, 𝜑𝑘,𝑗⟩|
2 < ∞,
where ⟨𝑓, 𝜑𝑘,𝑗⟩ are Fourier-Legendre coefficients of 𝑓. Let ℓ𝑝, 1 ≤ 𝑝 ≤ ∞, be the
space of numerical sequences 𝑥 = {𝑥𝑘,𝑗}𝑘,𝑗∈ℕ0 , ℕ0 = {0}⋃ℕ, such that
∥ 𝑥 ∥ℓ𝑝:= ( ∑
𝑘,𝑗∈ℕ0
|𝑥𝑘,𝑗|
𝑝)
1
𝑝 < ∞, 1 ≤ 𝑝 < ∞,
and ∥ 𝑥 ∥ℓ∞= 𝑠𝑢𝑝
𝑘,𝑗∈ℕ0
|𝑥𝑘,𝑗| < ∞, 𝑝 = ∞ . We introduce the class of functions
𝐿𝑠,2
𝜇
(𝑄) = {𝑓 ∈ 𝐿2(𝑄): ∥ 𝑓 ∥𝑠,𝜇
𝑠 : = ∑
𝑘,𝑗=0
∞
(𝑘 ⋅ 𝑗
¯
)𝑠𝜇|⟨𝑓, 𝜑𝑘,𝑗⟩|
𝑠 ≤ 1}, where 𝜇 > 0 ,
1 ≤ 𝑠 < ∞, 𝑘
¯
= 𝑚𝑎𝑥{1, 𝑘}, 𝑘 = 0,1,2,…. Note that 𝐿𝑠,2
𝜇
is a generalization of a
class of bivariate functions with dominating mixed partial derivatives. Moreover, let
𝐶 = 𝐶(𝑄) be the space of continuous on 𝑄 bivariate functions.
We represent a function 𝑓(𝑡, 𝜏) from 𝐿𝑠,2
𝜇
as
𝑓(𝑡, 𝜏) = ∑
𝑘,𝑗=0
∞
⟨𝑓, 𝜑𝑘,𝑗⟩𝜑𝑘(𝑡)𝜑𝑗(𝜏),
and by its mixed derivative we mean the following series
𝑓(𝑟,𝑟)(𝑡, 𝜏) = ∑
𝑘,𝑗=𝑟
∞
⟨𝑓, 𝜑𝑘,𝑗⟩𝜑𝑘
(𝑟)
(𝑡)𝜑𝑗
(𝑟)
(𝜏), 𝑟 = 1,2,…. (1)
assume that instead of the exact values of the Fourier-Legendre coefficients ⟨𝑓, 𝜑𝑘,𝑗⟩
only their perturbations are known with the error level 𝛿 in the metric of ℓ𝑝,
1 ≤ 𝑝 ≤ ∞.
More accurately, we assume that there is a sequence of numbers
𝑓𝛿 = {⟨𝑓𝛿 , 𝜑𝑘,𝑗⟩}𝑘,𝑗∈ℕ0
such that for 𝜉 = {𝜉𝑘,𝑗}𝑘,𝑗∈ℕ0, where 𝜉
𝑘,𝑗
= ⟨𝑓 − 𝑓𝛿, 𝜑
𝑘,𝑗
⟩, and for some 1 ≤ 𝑝 ≤ ∞
the relation
∥ 𝜉 ∥ℓ𝑝≤ 𝛿, 0 < 𝛿 < 1, (2)
is true.
The research of this work is devoted to recovering the derivative (1) of
functions from 𝐿𝑠,2
𝜇
. Our goal is to achieve the best order of accuracy by using the
minimal number of perturbed Fourier-Legendre coefficients⟨𝑓𝛿 , 𝜑𝑘,𝑗⟩.
ISSN 1816-1545 Фізико-математичне моделювання та інформаційні технології
2023, вип. 36, 17-21
19
2. Truncation method
It should be noted that at the moment a number of approaches were developed for
numerical differentiation. All these methods are accepted to divide into three groups
(see [4]): difference methods, interpolation methods and regularization methods. As
is known, the first two types of methods have their advantage in the simplicity of
implementation, but they guarantee satisfactory accuracy only in the case of exactly
given input data about the differentiable function. At the same time, regularization
methods give stable approximations to the desired derivatives in the case of perturbed
input data but most of them (for example, the Tikhonov method and its various
variations) are quite complicated for numerical realization in view of their integral
form and require hard-to-implement rules for determination of regularization
parameters (see [4]). Recently in [2] a concise numerical method, called the
truncation method, has been proposed as a stable and simple approach to numerical
differentiation of multivariable functions. The essence of this method is to replace the
Fourier series (1) with a finite Fourier sum using perturbed data ⟨ , , ⟩. In the
truncation method, to ensure the stability of the approximation and achieve the
required order accuracy, it is necessary to choose properly the discretization
parameter, which here serves as a regularization parameter. So, the process of
regularization in the method under consideration consists in matching the
discretization parameter with the perturbation level of the input data. The simplicity
of implementation is the main advantage of this method.
In the case of an arbitrary bounded domain of the coordinate plane[ ,∞)
[ ,∞), the truncation method for differentiating functions of two variables has the
form
( , )
( , ) = ∑
( , )∈
⟨ ,
,
⟩
( )
( )
( )
( ). By ( ) we mean the
number of points that make up .
In order to increase the efficiency of the approach under study, we take a
hyperbolic cross as the domain of the following form = : = {( , ): ⋅ ≤
− 1, , = ,… , − 1}, ( ) = ( ). Then our version of the proposed
truncation method can be written as
( , )
( , ) = ∑
, , ≤ −1
⟨ ,
,
⟩
( )
( )
( )
( ). (3)
3. Error estimate in – metric
Theorem 1. Let ∈ ,2
, 1 ≤ < ∞, > 2 − 1/ + 1/2, and let the condition
(2) be satisfied. Then for ( −1 1/ −1/
1
)
1
1/ 1/
it holds
∥ ( , ) −
( , )
∥ 2≤ ( 1/ −1/
1
)
2 1/ 1/2
1/ 1/
/2−1/
1
.
Yevgeniya Semenova, Serhii Solodky
A new method for recovering high-order mixed derivatives of bivariate functions
20
Corollary 1. In the considered problem, the truncation method
( , )
(3) achieves the
accuracy
(( 1/ −1/
1
)
2 1/ 1/2
1/ 1/
/2−1/
1
)
on the class ,2
, > 2 − 1/ + 1/2, and requires
( ) ( −1
1
)
1
−1/ 1/
perturbed Fourier-Legendre coefficients.
Remark 1 Let's consider the standard variant of the truncation method with =
: = [ , ] [ , ] . It is easy to verify that such an approach guarantees the
accuracy (
2 1/ 1/2
2 2/ 1/ 1/2) on the class ,2
, > 2 − 1/ + 1/2 , and requires
( ) 2
−
2
2
2
1
1
2 perturbed Fourier-Legendre coefficients. Comparison
of the estimates found above with the corresponding estimates for the method
( , )
(3) (see Corollary 1) demonstrates that (3) is more efficient both in terms of accuracy
and the amount of discrete information used.
3. Error estimate in – metric
Theorem 2. Let ∈ ,2
, 1 ≤ < ∞, > 2 − 1/ + /2, and let the condition (2)
be satisfied. Then for ( −1 1/ −1/
1
)
1
1/ 1/
it holds ∥ ( , ) −
( , )
∥ ≤
( 1/ −1/
1
)
2 1/ /2
1/ 1/
2−1/
1
.
Corollary 2. In the considered problem, the truncation method
( , )
(3) achieves the
accuracy (( 1/ −1/
1
)
2 1/ /2
1/ 1/
2−1/
1
) on the class ,2
, > 2 − 1/ +
/2 , and requires ( ) ( −1
1
)
1
1/ 1/ perturbed Fourier-
Legendre coefficients.
Remark 2 Consider the standard variant of the truncation method with = : =
[ , ] [ , ] . It is easy to verify that this approach guarantees the accuracy
(
2 1/ /2
2 2/ 1/ /2) on the class ,2
, > 2 − 1/ + /2 , and requires ( )
2
−
2
2
2
1
2 perturbed Fourier-Legendre coefficients. Comparison of the
estimates found above with the corresponding estimates for the method
( , )
(3) (see
Corollary 2) demonstrates that (3) is more efficient both in terms of accuracy and the
amount of discrete information used.
Remark 3. The method
( , )
(3) was studied earlier (see [5]) for the problem of
numerical differentiation of functions from ,2
in the case of = 1 and = = 2.
ISSN 1816-1545 Фізико-математичне моделювання та інформаційні технології
2023, вип. 36, 17-21
21
In addition, for the recovery of mixed derivatives (2,2), this method was considered
in [3]. Thus, Theorems 1 and 2 generalize previously known results for the case of
arbitrary , , .
Remark 4. It is easy to show that method (3) is optimal in terms of the informational
complexity of the numerical differentiation problem.
Conclusions. In the work, a new approach to the numerical recovering mixed
derivatives of any order of bivariate functions is investigated. For the proposed
method, accuracy estimates are found in integral and uniform metrics, and the
amount of discrete information used is calculated.
The authors acknowledge partial financial support due to the project
“Mathematical modelling of complex dynamical systems and processes caused by the
state security” (Reg. No. 0123U100853).
References
1. Dolgopolova, T.F., Ivanov, V.K. (1966) ‘On numerical differentiation’. Zh. Vychisl. Mat.and
Mat. Ph. 6(3). 223--232.
2. Semenova, E.V., Solodky, S.G., Stasyuk, S.A. (2022) ‘Application of Fourier Truncation Method
to Numerical Differentiation for Bivariate Functions’. Computational Methods in Applied
Mathematics. 22(2), 477-491.
3. Semenova, Y.V., Solodky, S.G. (2022) ‘Optimal Method for recovering mixed derivatives’.
Journal of Numerical Applied Mathematics. 2, 143-150.
4. Ramm, A.G., Smirnova, A.B.(2001) ‘On stable numerical differentiation’. Math. Comput. 70,
1131-1153.
5. Semenova Y.V., Solodky S.G. (2021) ‘Error bounds for Fourier-Legendre truncation method in
numerical differentiation’. Journal of Numerical and Applied Mathematics 137(3), 113–130.
Новий метод для відновлення мішаних похідних вищих
порядків функцій двох змінних
Євгенія Семенова, Сергій Солодкий
Досліджено задачу відновлення мішаних похідних вищих порядків функцій двох змінних зі
скінченною гладкістю. На основі методу зрізки побудовано алгоритм чисельного
диференціювання, який досягає високого порядку точності.
Received 14.03.23
|
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| institution | Physico-mathematical modeling and informational technologies |
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| language | Ukrainian |
| last_indexed | 2026-06-09T01:09:24Z |
| publishDate | 2023 |
| publisher | Інститут прикладних проблем механіки і математики ім. Я. С. Підстригача НАН України |
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| spelling | oai:ojs2.www.fmmit.lviv.ua:article-2682025-02-21T17:32:19Z A new method for recovering high-order mixed derivatives of bivariate functions Новий метод для відновлення мішаних похідних вищих порядків функцій двох змінних Semenova, Yevgeniya Solodky, Serhii G. Numerical differentiation, Legendre polynomials, truncation method The problem of numerical recovering high-ordermixed derivatives of bivariate functions with finite smoothness is studied. On the basis of the truncation method, an algorithm for numerical differentiation is constructed, which guarantees a high order of approximation accuracy. Досліджено задачу відновлення мішаних похідних вищих порядків функцій двох змінних зі скінченною гладкістю. На основі методу зрізки побудовано алгоритм чисельного диференціювання, який досягає високого порядку точності. Інститут прикладних проблем механіки і математики ім. Я. С. Підстригача НАН України 2023-06-13 Article Article application/pdf https://www.fmmit.lviv.ua/index.php/fmmit/article/view/268 PHYSICO-MATHEMATICAL MODELLING AND INFORMATIONAL TECHNOLOGIES; No. 36 (2023): ФІЗИКО-МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ТА ІНФОРМАЦІЙНІ ТЕХНОЛОГІЇ; 17-21 ФІЗИКО-МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ТА ІНФОРМАЦІЙНІ ТЕХНОЛОГІЇ; № 36 (2023): ФІЗИКО-МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ТА ІНФОРМАЦІЙНІ ТЕХНОЛОГІЇ; 17-21 2617-5258 1816-1545 10.15407/fmmit2023.36 uk https://www.fmmit.lviv.ua/index.php/fmmit/article/view/268/261 Авторське право (c) 2023 Yevgeniya Semenova, Serhii G. Solodky (Автор) |
| spellingShingle | Numerical differentiation Legendre polynomials truncation method Semenova, Yevgeniya Solodky, Serhii G. Новий метод для відновлення мішаних похідних вищих порядків функцій двох змінних |
| title | Новий метод для відновлення мішаних похідних вищих порядків функцій двох змінних |
| title_alt | A new method for recovering high-order mixed derivatives of bivariate functions |
| title_full | Новий метод для відновлення мішаних похідних вищих порядків функцій двох змінних |
| title_fullStr | Новий метод для відновлення мішаних похідних вищих порядків функцій двох змінних |
| title_full_unstemmed | Новий метод для відновлення мішаних похідних вищих порядків функцій двох змінних |
| title_short | Новий метод для відновлення мішаних похідних вищих порядків функцій двох змінних |
| title_sort | новий метод для відновлення мішаних похідних вищих порядків функцій двох змінних |
| topic | Numerical differentiation Legendre polynomials truncation method |
| topic_facet | Numerical differentiation Legendre polynomials truncation method |
| url | https://www.fmmit.lviv.ua/index.php/fmmit/article/view/268 |
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