About the Tikhonov Regularization Method for the Solution of
Incorrect Problems
V. M. RYABOV, I. G. BUROVA
Department of Computational Mathematics,
St. Petersburg State University,
7-9 Universitetskaya Embankment, St. Petersburg,
RUSSIA
Abstract: - From time to time, papers are published containing gross errors when solving integral equations of
the first kind. This paper is devoted to the analysis of these errors. The paper considers Tikhonov’s weak and
operator regularization. To construct a solution to the integral equation, the local splines of the Lagrangian type
of the second order of approximation, as well as the local splines of the Hermitian type of the fourth order of
approximation of the first level, are used. The results of numerical experiments are presented.
Key-Words: integral equation of the first kind, regularization, local splines.
Received: October 8, 2022. Revised: May 17, 2023. Accepted: June 9, 2023. Published: June 28, 2023.
1 Introduction
From time to time, articles are published containing
gross errors when solving integral equations of the
first kind. This paper is devoted to the analysis of
these errors. As is known, the Fredholm equation of
the first kind
and the Volterra equation of the first kind
belong to ill-posed tasks. These problems are
unstable on the right-hand side and, therefore, are
ill-posed. In other words, there are arbitrarily small
perturbations of the right-hand side, which
correspond to large perturbations of the solution.
Therefore, the solutions to these problems are
unstable. In addition, in a number of cases, even if a
solution exists for some right-hand sides, there are
small changes in the right-hand side for which the
solution does not exist.
Therefore, when solving the Fredholm or the
Volterra integral equations of the first kind, one
often has to resort to regularizing the solution. Two
approaches to solving this problem should be noted.
The first and easiest way is as follows. The integral
equation is reduced to solving a system of linear
algebraic equations. This system of equations
usually has a large conditional number or can be
degenerated. Therefore, it is necessary to carry out
the regularization of the system of equations
according to Tikhonov. Here we consider the second
variant of regularization, namely, the operator
regularization.
2 Operator Regularization
Given the equation
where . In the general case, the spaces
and are the Hilbert spaces.
2.1 The Case of Systems of Linear Algebraic
Equations
The systems of linear algebraic equations (SLAEs)
can be degenerate or ill-posed and we know
approximations such that
.
Existence solutions to the original and perturbed
SLAEs are not assumed. The concepts of pseudo-
solutions and normal solutions are introduced, [1],
[2], [3], [4].
To find a normal solution, we introduce the
functional
. (2)
The minimum point of this functional is found as a
solution to the equation Euler, [3]
(3)
A solution to this equation exists and is unique due
to the symmetry and positive definiteness of the
matrix
for .
The set of solutions to the Euler equation for all
is bounded, [3], and due to its compactness, one
can choose a convergent sequence from it. In [3], a
method was indicated for choosing the
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regularization parameter α which tends to be zero.
This method also depends on the parameter
which tends to be zero, at which the selected
sequence converges to the normal solution of the
original exact task. In the case of a symmetric
positive definite matrix, the Euler equation (3) (i.e.
regularization) should be carried out in the form [1],
[2], [5], [6] . (4)
2.2 The Case of Fredholm Integral Equations of
the First Kind
Equation (1) has the form
(5)
where is the kernel of the integral equation.
2.2.1 Case ]
Suppose ]. The kernel of the
operator and the right side are known with
errors and . In this case, the regularization is
carried out in the form
. (6)
and the convergence of solutions to Eq. (6) is
guaranteed for a consistent tending to zero
parameters to the exact solution of equation
(5) (under the assumption of its existence) in the
metric , [4].
2.2.2 Case
Suppose Therefore,
. Errors are possible only in
setting the right side of the equation. Let us assume
that the desired solution belongs to some compact
subset of the space of continuous functions. We use
this as a priori of information about the solution in
setting the stabilizing functional included in
(2):
The domain of the definition of this functional
consists of functions uniformly bounded and
equicontinuous on . By the Arzelà–Ascoli
theorem set
for any
is compact in space . Euler's equation for the
functional
has the form [3],
(7)
where . The
operator
is symmetric and a positive
definite, and the solutions of Eq. (7) belong to a
compact set and converge to the normal solution of
the integral equation as the parameters and tend
to zero, [3].
3 Problem Solution
It is stated in [7], that the "regularization" of Eq. (5)
by the shift
(8)
and the simplest iterations as lead to the
solution of the original equation, and it is said that
this was allegedly proven in the works of [3], [7],
[8], which is completely wrong. In addition, paper,
[7], gives examples of constructing "regularized"
solutions in the form of a series, which are diverged,
but they successfully sum up the row. Note that in
this case no requirements are made regarding the
kernel and the right side of the equation! The same
error is repeated in the works of the authors of the
paper, [9].
In the next section, for the Fredholm integral
equation of the first kind, two methods of
regularizations will be carried out. Weak
regularization will be carried out first, i.e. the
integral equation is reduced to solving a regularized
system of linear algebraic equations (the Euler
equation). Thus we obtain the regularized solution.
While obtaining the system of equations, we used
the spline approximation of the function .
Further, in the second method, operator
regularization will be carried out using local splines
of the Hermitian type of the fourth order of
approximation of the first level.
4 Numerical Experiments
Consider the Fredholm integral equations of the first
kind
Two approaches to the regularization of the solution
were noted in section 2. The first and easiest way is
as follows. The integral equation is reduced to
solving a system of linear algebraic equations. This
method belongs to weak regularization and is quite
simple to implement. We use this method for
solving the next integral equation.
Example. Consider the integral equation
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where the right part is constructed using the
exact solution .
So we have the expression
The plot of the function is given in Fig. 1.
Fig. 1: The plot of the function
The plot of the function is given in Fig. 2.
Fig. 2: The plot of the function
First, we do regularization in the simplest way
4.1 The First Regularization Method
Let a grid of ordered nodes be constructed on
the interval Let be integers,
, and the basis spline be such
that . First, we
approximate the unknown function with the
expression:
where are the unknown coefficients.
. On the interval we approximate
the function by the following expression:
,
where the basis splines are as follows:
We call them splines of the second order of
approximation (following Professor Mikhlin). So
they are the local splines of the second order of
approximation of the Lagrangian type.
We will use the norm of the vector of the form:
Let The theorem of approximation of
the function with the splines is the following:
This splines were used, in papers, [5], [10].
Now we have the equation
Next, we put = and we can obtain the system of
equations in the form
The integral
can be
calculated exactly or by an appropriate quadrature
formula. Now we solve the resulting system of
linear algebraic equations. and after that we solve
the regularization system of linear algebraic
equations (the Euler equation) according to
Tikhonov. And we obtain the regularized solution.
Fig. 3 shows the exact solution and the solution
obtained by solving the system of equations
(without regularization). Similarly, Fig. 4 presents
the plot of the error of the solution without
regularization.
Fig. 3: The plot of the function and the solution
of the system of equations without regularization
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Fig. 4: The plot of the error of the solution without
regularization
The next step is the following. We make a
regularization according to Tikhonov. The results of
the calculation are shown in Fig. 5 and Fig. 6. The
plot of the function and the solution after the
regularization are given in Fig.5. The plot of the
error of the solution after regularization is given in
Fig.6. Note, that the numbers of the grid nodes are
marked along the abscissa axis.
Fig. 5: The plot of the function and the solution
after the regularization
Fig. 6: The plot of the error of the solution after the
regularization
4.2 The Second Regularization Method
Now consider the second regularization method.
The second method of regularization refers to
variational methods and is as follows. We assume
that the kernel of the integral equation is continuous,
and the homogeneous integral equation has only a
zero solution. Let be normed spaces. More
specifically, let be the Sobolev space , and
be a Hilbert space. The essence of regularization
according to Tikhonov is as follows. Instead of the
equation where
we consider the modified problem:
where the Tikhonov stabilizer has the form
are continuous and , The Euler
equation for this variational problem will be the
following integro-differential equation:
with the kernel , and the right
side that are as follows:
,
and with the boundary conditions:
We consider a special case: The
function on each grid interval is replaced by
the expression using the basis splines of the
fourth order of approximation and the first level.
This approximation uses the values of the function
and its first derivative at the grid nodes:
,
where the basis splines have the form:
The basis splines of the fourth order of
approximation and the first level have the support:
supp supp
We call these the local splines of the
Hermitian type of the fourth order of approximation
of the first level.
We obtain these basis splines on the interval
solving the system of identities:
The approximation of the first derivative of a
function with the splines of the fourth order of
approximation is given with the relation:
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,
We can obtain the form of the basis splines on
in the next way. We solve a similar
system of equations on the interval :
Here we have the formula for the approximation the
function ,
,
Choosing the basis functions with common vertices,
we obtain formulas for basis splines when
We have the next expressions:
The plots of these basis splines when
are shown in Fig.7 and Fig. 8.
Fig. 7: The plot of the function
Fig. 8: The plot of the function
The plots of the first derivatives of these basis
splines when are shown
in Fig. 9 and Fig. 10.
Fig. 9: The plot of the function
Fig. 10: The plot of the function
Let The theorem of
approximation of the function
is the following:
, .
The proof follows from the formula for the
remainder term of the Hermite interpolation.
We have the equidistant set of nodes with the step
on the interval Now we can
approximate the Runge function with the splines of
the fourth order of approximation and the first level
on the interval The plot of the error
approximation of the Runge function
is given in Fig.11. The plot of the error
approximation of the first derivative Runge function
is given in Fig.12.
Fig. 11: The plot of the error approximation of the
Runge function with the splines of the fourth order
of approximation
Fig. 12: The plot of the error approximation of the
first derivative of the Runge function with the
splines of the fourth order of approximation
Now we can use the approximate values of the
first derivative. In the simplest way we can use the
approximate expressions with the error of the
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second order. In this case, the derivative is
replaced by the numerical differentiation formula:
And, the derivative is replaced by the
numerical differentiation formula:
Now we have the expression:
.
We apply the operator’s regularization. According
to Tikhonov’s regularization, the new kernel has the
form:
The new right side of the equation has the form
Now we solve the system of equations
where
(
and
The graphs of the exact solution and its
approximation after regularization are given in Fig.
13. The plot of the error of the solution after
regularization is given in Fig. 14.
Fig. 13: The plot of the function and the
solution after regularization
Fig. 14: The plot of the error of the solution after
regularization
Now we also use the next stabilizer:
(
The graphs of the exact solution and its
approximation after regularization are given in Fig.
15. The plot of the error of the solution after
regularization is given in Fig. 16.
Fig. 15: The plot of the function and the
solution after regularization
Fig. 16: The plot of the error of the solution after
regularization
4.3 The Inversion of the Laplace Transform
Consider the problem of inverting the Laplace
transform, i.e., finding the original from its
image , from the equation
Making a variable change
, we obtain
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Denote .
Thus, we get an integral equation of the first kind:
This equation can be solved by the methods
considered in this work.
5 Conclusion
The solution of integral equations of the first kind
requires great care. Without a deep knowledge of
the theory of operator regularization, it is
recommended to reduce the solution of the integral
equation to the solution of a system of linear
algebraic equations. Next, regularize the solution of
the system of equations using A.N.Tikhonov’s
theory. In the following works, we will consider the
problem of numerically solving the Laplace
transform by solving integral equations of the first
kind in more detail.
References:
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numerical solution of systems of linear
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388–393.
[2] V. M. Ryabov and others, Inverse and ill-posed
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[3] A. N. Tikhonov, V.Ya.Arsenin, Solutions of
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[4] A. B. Bakushinsky, A.V. Goncharsky,
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[8] D. L. Phillips, A technique for the numerical
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[9] E. K. Kulikov, A.A. Makarov., A method for
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[10] I. G. Burova, G.O. Alcybeev, Application of
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
I. G. Burova was responsible for the theory.
V. M. Ryabov was responsible for the theory.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The authors are gratefully indebted to St. Petersburg
University for their financial support in the
preparation of this paper (Pure ID 94029567, ID
104210003, 104625746), as well as to a resource
center for providing the package Maple.
Conflict of Interest
The authors have no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
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