Adaptive Refinement of the Variational Grid Method
Yu. K. DEM'YANOVICH, N. A. LEBEDINSKAYA, A. N. TEREKHOV
Mathematics & Mechanics Faculty,
Saint Petersburg State University,
7/9 Universitetskaya Embankment,
Saint Petersburg,
RUSSIA
Abstract: - In this paper, we have developed a new approach to the local improvement of an approximate
solution which has been obtained with the finite element method. The proposed improvement is adaptive, and
it minimizes the energy functional. The discussed algorithm can be implemented analytically as well as
numerically. The proposed approach can be applied to the adaptive refinement of the previously calculated
variational grid approximation. The proposed method allows you to expand the approximate space by adding
new coordinate functions. The implementation of a local approximation requires a small amount of arithmetic
operations. The number of arithmetic operations does not depend on the number of nodes of the previously
calculated approximation. Examples of the implementation of the method are considered for a problem with
strong degeneracy and for a problem without degeneracy.
Key-Words: - adaptive methods, refinement calculations, variational grid methods
Received: March 4, 2022. Revised: November 2, 2022. Accepted: December 3, 2022. Published: December 15, 2022.
1 Introduction
Research in the field of artificial intelligence covers
many areas of the development of Science and
practice. These studies give opportunities to
optimize efforts aimed at the effective solution of
difficult problems. Here we give examples of some
studies in the mentioned area. Paper [1] predicts the
future of excitation energy transfer with artificial
intelligence-based Quantum dynamics. In paper [2]
the authors propose an algorithm with a faster
convergence rate for federated edge learning. In
paper [3] the authors propose a new algorithm of
machine learning techniques. Based on the recent
advancements in music structure analysis, the
authors of paper [4] automate the evaluation process
by introducing a collection of metrics. The
development of a disagreement-based online
learning algorithm is given in [5]. In [6] the authors
develop a deep learning-based approach to model.
Let us mention some works in which it is possible to
effectively use learning systems and the means of
artificial intelligence. Paper [7] is devoted to the
solution of the nonstationary, integro-differential
equation with a degenerate elliptic differential
operator. It seems that the adaptive methods
developed with the help of artificial intelligence,
greatly simplifies the solution of problems of
mathematical physics. Article [8] investigates the
approximate solution to a nonlinear Volterra
integro-differential equation. In article [9], the
authors have proposed a highly efficient and
accurate collocation method. It can also be assumed
that the mentioned means would be useful in solving
the problems considered in works [10] and [11].This
work is devoted to the variational grid method for
one-dimensional boundary value problems for
differential equations of the second order. Such
problems are often encountered in the study and
modeling of various phenomena in physics and
technology. The numerical solution of such
problems with appropriate accuracy can be very
labor intensive. The numerical solution of these
tasks may require significant computer resource
systems (with respect to memory, time and
computational accuracy). Special difficulties arise in
the case when solving degenerate equations. S.G.
Michlin conducted many studies of these problems
(see [12] - [13]). Computational stability issues are
investigated in [12]. A comprehensive study of the
approximation in variational grid methods in the
case of uniform grids is given in the monograph
[13]. A generation to irregular grids is available
(see [14]). In well-known works, only the global
improvement of the approximation was considered.
In this case, large computing system resources
(memory and run time) are required. The proposed
local improvement requires fewer resources. The
local improvement can be done analytically or
numerically.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2022.17.58
Yu. K. Dem'yanovich, N. A. Lebedinskaya, A. N. Terekhov
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The aim of this work is to develop adaptive
refinement methods based on the previously
calculated variation-grid approximation. At the
same time, an adaptive expansion of the projection
space is discussed.
Let us dwell on the advantages of the proposed
approach. The refinement under consideration is
obtained as a result of a small number of arithmetic
operations. The last one does not depend on the
total number of nodes, used in obtaining the
previously calculated numerical solution. The
proposed approach supports adaptability, i.e. it
automatically chooses the best version of the
algorithm. The solution is refined locally. The
refinement process involves only a few nodes of the
original grid and their corresponding values of the
approximate solution. The proposed method does
not require global recalculation of the previously
obtained approximate solution.
2 Background
Consider the one-dimensional boundary value
problem
(2)
where is a measurable bounded function
The generalized solution of the problem (1) -- (2) is
the solution of the problem to a minimum of
functional
on the energy space of the operator (1) -- (2) (see
[1]). For an approximate solution of problem (1) --
(2) use the minimization of the functional (3) on
one or
another subspace (the Ritz methods, finite
elements, etc.). We take the space of piecewise
linear functions using a finite set of nodes on the
segment [0,1] and vanishing at the ends of this
segment. So, let the grid be given on the interval
The variant of the variational grid method under
consideration consists of functional minimization
(4)
as functions of the variables Here
To reduce the error of the approximate solution
for a given number of grid nodes, it is important to
find their successful location. In those parts of the
domain where the solution changes slowly, the
number of nodes may not be large. However, in
those parts of the domain where the solution
changes quickly, you should use significantly more
nodes. The measure of quality for the choice of
nodes is the value of the functional (4). In this way,
functional (4) is desirable to minimize both
variables and variables
while observing the condition
.
The space of the piecewise linear continuous
functions of the form
is denoted by . Here are arbitrary
numbers with and
. The space is a subspace of the energy
space.
The basis of the space consists of the
functions
for
for
for
where Minimizing the functional
(3) on the space
(5)
we obtain a certain version of the variational grid
method.
By enote the solution of problem (5).
Denote by he grid, obtained from the grid X
with adding the node We have
. It is clear to see that
,
where
for
for
for
Here denotes the linear span of objects
contained in curly braces. Let , and
the number ) satisfies the condition
I
Obviously the ratio
. Hence
I
Let be solution of the problem
I
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It is clear that finding the node in a certain
sense optimizes the approximation of the solution
of problem (1) -- (2).
The functional can be represented as
(6)
Let's raise the question of adaptively by adding a
new node in the advanced fixed
interval.
3 Case of One Node
At this point, we will assume . Let us
introduce the notation
So, in the case under consideration
represents a function of two variables. For this
function, we introduce notation , so that
(7)
From formula (7) we have
The minimum point of the quadratic form (8)
for fixed is the point defined by the
formula
.
The minimum of the mentioned quadratic form
is equal to
(9)
Relation (9) can be transformed into the form
In the case the solution to
problem (1) -- (2) is the function
. Substituting into (9), we
find . Calculating the
minimum with respect to the variable , we find the
minimum point
4 Adaptive Grid Refinement
Consider the question of the optimal location of the
added node in the variational grid method for
problem (1) - (2). Functional (6) should be
minimized. Adding the node to the interval
and the value of the approximation at
that node, we obtain the corresponding
extension. Our goal is to choose this node in
the best way, i.e. so that the resulting approximation
error of the desired solution is the smallest.
It is easy to see that it suffices to minimize the
function
Setting
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we have
Let us introduce the notation
(16)
(19)
In addition, let's put
Theorem 1. The next formulas are right
()(), (26)
where
, ()=,
(27)
()
+.
(28)
Proof. By relations (11) and (14) - (16) we have
From relations (12) and (17) - (19) we find
Formulas (29) -- (30) can be represented
Using formulas (13) and (31) -- (32), we obtain
representation (26) -- (28). The theorem has been
proven.
Remark 2. The minimum point of the quadratic
function (26) is the point
Thus
=
Consider the function depending
on , putting
From formulas (33) -- (34) we obtain the
relation
The problem of finding the optimal position of the
point goes to finding the minimum of the function
on the interval .
Let's consider several special cases.
5 Non-degenerate Boundary Value
Problem
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Consider the boundary value problem
The solution to this problem is the function
.
In the case under consideration we have
According to formulas (14) - (19) we have
,
,
,
,
Using formulas (36) - (39) in relations (31) - (32),
we find
In accordance with formulas (20) - (25) we have
Theorem 2. Relation
. (40)
is correct.
Proof. In view of relations (27) - (28) we obtain
B
.
Note that for , and
we have B. Expression B can be written
as
B [
].
Similarly, we find
From formula (33) we obtain relation (40). This
concludes the proof.
Corollary 1. For ,
the inequalities , B hold. In
addition, the formulas
are right.
The proof obviously follows from formula (40).
Theorem 3. The formula
holds.
Proof. By relation (35) we have
The second term in (42) is defined by relation (41),
while the first term can be represented as
To find the critical point of function from
(42) we have
+
It is easy to check that this equation is satisfied by
the value
The study of the second derivative shows that
point is the minimum point of the function
on the segment . This completes
the proof.
Corollary 2. In the case under consideration, for
the optimal refinement of the original grid, the grid
intervals should be divided in half.
Theorem 4. Ratio
is true.
Proof. Assuming we have
+
Hence
In a similar way, we receive the presentation
After elementary transformations, we find
For function ), we similarly obtain
Using representation (42), we have
)
This implies the limit relation
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The limit relation is equivalent to formula (43).
This concludes the proof.
For , we have
.
Let us determine the function at the point
=. Extending the function by continuity to
the point , we have
) =-
Remark 2. Setting
, we get
so that according to
(10) and (19) we have
6 Degenerate Boundary Value
Problem
Consider a model problem with strong
degeneracy
where .
The generalized solution of the problem is
obtained by minimizing the functional (3) on the
energy space of the operator (44).
The solution of problem (44) is
=
().
For the construction of the variational grid method,
we use projection space which was proposed
in [13]. This space differs from space by
changing the coordinate function to a
function
,
),
)
).
As before, let's add the node to one of the
intervals ), where
In what follows, for convenience, we supply the
superscript for objects related to the strong
degeneration problem.
Let us proceed to the calculation of the functions
(14) -- (19) under the conditions
For the functions (14) -- (16) we successively
obtain
+q
,
Similarly, we find the functions (17) -- (19).
Now let's find expressions for ,
using formulas (20) - (22).. We have
,
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.
To find expressions for ,, , we use
formulas (23) -- (25),.
,
.
We introduce the notation
Theorem 5. In degenerate boundary value
problem (44) formula
is right. Here
Proof of the stated statement is similar to
Theorem 1's proof. The only difference is the
above calculation of the functions , ,
.
The critical point of the quadratic form (45) is
Thus
(46)
Consider the function
depending
on . We denote
From formulas (46) - (47) we obtain the relation
It is easy to see that in order to find the minimum
point of the function, one should find the roots of
the equation
. In this case, the last one
is a quadratic equation. We do not present the
corresponding calculations.
7 Discussion
In this paper, we have developed a new approach to
the local improvement of approximate solution
which has been obtained with the variational grid
method. The proposed method is adaptive, and it
minimizes the energy functional. The discussed
algorithm can be implemented analytically as well
as numerically. The algorithm contains a small
number of arithmetic operations. The number of the
mentioned operations does not depend on the
number of grid nodes of the original variational
grid method. The proposed approach can be applied
to the adaptive refinement of previously calculated
variational grid approximations.The proposed
method allows you to expand the approximate
space by adding new coordinate functions. The
implementation of a local approximation requires
a small amount of arithmetic operations. The
number of arithmetic operations does not depend on
the number of nodes of the previously calculated
approximation. Examples of the implementation of
the method are considered for problems with
strong degeneracy and for problems without
degeneracy. In this paper, we consider a new
approach to the refinement approximate solution
obtained with the variational grid method. This
approach is useful when you want to obtain a local
refinement of the solution. In other words, the
proposed approach is useful in cases where local
refinement is required for the previously obtained
approximate solution. The benefits of the new
approach are as follows: 1) the solution is refined
locally; 2) the refinement process involves only a
few nodes of the original grid and their
corresponding values of the approximate solution;
3) the proposed method does not require global
recalculation of the previously obtained
approximate solution; 4) the refinement under
consideration is obtained as a result of a small
number of arithmetic operations; 5) the last one
does not depend on the total number of nodes used
in obtaining the previously calculated numerical
solution; 6) the proposed approach supports
adaptability, i.e. it automatically chooses the best
version of the algorithm.
8 Conclusion
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DOI: 10.37394/23203.2022.17.58
Yu. K. Dem'yanovich, N. A. Lebedinskaya, A. N. Terekhov
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Volume 17, 2022
One-dimensional boundary value problems
are often found in the study and in the modeling of
various phenomena in physics and technology.
The numerical solution of such problems with
appropriate accuracy can be very labor intensive.
Known numerical methods for solving such
problems are reduced to solving algebraic systems
of a higher order. This may require significant
computer system resources (with respect to
memory, time and computational accuracy).
Special difficulties arise when solving degenerate
equations. They can be overcome by the
modification of the projection space. To implement
a local refinement requires a small amount of
arithmetic, which does not depend on the dimension
of the projection space. The illustrative examples
given in the sixth and seventh sections of this
paper show the effectiveness of the proposed
approach. In these examples, the integrals were
calculated explicitly, because they were tabular
integrals. In the case of a complex structure of the
coefficients of equation (1), it is proposed to use the
approximate methods for calculated integrals.
Similar studies are expected to be carried out on
other boundary value problems in the future.
Acknowledgment:
The authors are highly grateful and indebted to St.
Petersburg University for financially supporting
the preparation of this paper (Pure ID 93852135,
92424538).
References:
[1] Ullah, A., Dral, P.O. Predicting the future of
excitation energy transfer in light-harvesting
complex with artificial intelligence-based quantum
dynamics, Nature Communications 13(1), 2022,
pp.19-30.
[2] Dinh, C.T., Tran, N.H., Nguyen, T.D., (...), Zhou,
B.B., Zomaya, A.Y. DONE: Distributed
Approximate Newton-type Method for
Federated Edge Learning, IEEE Transactions on
Parallel and Distributed Systems 33(11), 2022,
pp. 2648-2660.
[3] Jiao, Y., Gu, Y. Communication-Efficient
Decentralized Subspace Estimation, IEEE
Journal on Selected Topics in Signal Processing
16(3), 2022, pp. 516-531.
[4] De Berardinis, J., Cangelosi, A., Coutinho, E.
Measuring the Structural Complexity of Music:
From Structural Segmentations to the Automatic
Evaluation of Models for Music Generation,
IEEE/ACM Transactions on Audio Speech and
Language Processing, 30, 2022, pp. 1963-1976..
[5] Huang, B., Salgia, S., Zhao, Q. Disagreement-
Based Active Learning in Online Settings,
IEEE Transactions on Signal Processing, 70,
2022, pp. 1947-1958.
[6] Rahman, M.H., Xie, C., Sha, Z. Predicting
sequential design decisions using the function-
behavior-structure design process model and
recurrent neural networks, Journal of
Mechanical Design, Transactions of the ASME,
143(8), 2022, pp.081-706.
[7] Cern, D. Wavelet Method for Sensitivity Analysis
of European Options under Merton Jump-
Diffusion Model, AIP Conference Proceedings
2425, 2022, pp.110004.
[8] Erfanian, M., Zeidabadi, H. Solving of Nonlinear
Volterra Integro-Differential Equations in the
Complex Plane with Periodic Quasi-wavelets,
International Journal of Applied and
Computational Mathematics, 7(6) , 2021, pp.2-21.
[9] Priyadarshi, G., Rathish Kumar, B.V. Parameter
identification in multidimensional hyperbolic
partial differential equations using wavelet
collocation method, Mathematical Methods in
the Applied Sciences, 44(11) , 2021, pp. 9079-
9095.
[10] I. G. Burova, Application Local Polynomial and
Non-polynomial Splines of the Third Order of
Approximation for the Construction of the
Numerical Solution of the Volterra Integral
Equation of the Second Kind, WSEAS
Transactions on Mathematics, vol. 20, 2021, pp.
9-23.
[11] E. N. Menshov, Approximation of Families f
Characteristics of Energy Objects in the Class of
Linear Differential Equations, WSEAS
Transactions on Computers, vol. 20, 2021, pp.
239-246.
[12] Michlin S.G. Some Theorems on the Stability of
Numerical Processes, Atti d. Lincei. Classe fis.,
mat. e nat., fasc.2, 1982, pp. 1-32.
[13] Michlin S.G. Approximation auf dem Kubischen
Gitter. Berlin, 1976.
[14] Dem'yanovich Yu. K. Approximation on
Manifold,WSEAS Transactions on
MATHEMATICS, 20, 2021, pp. 62-73.
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WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2022.17.58
Yu. K. Dem'yanovich, N. A. Lebedinskaya, A. N. Terekhov
E-ISSN: 2224-2856
534
Volume 17, 2022
