1. Introduction
Ill-posed problems [
1,
2] include linear [
3] and nonlinear [
4,
5,
6,
7] ill-posed problems. With the development of applied science, studies on nonlinear ill-posed problems are attracting increased attention, and a variety of methods have emerged. Some of the most widely known methods are Tikhonov regularization [
8,
9,
10,
11,
12,
13] and Landweber iteration [
14,
15,
16,
17]. The purpose of this study was to explore a compression multiscale Galerkin method for solving nonlinear ill-posed integral equations via Landweber iterations.
Landweber iteration is a regularization method when the iteration is terminated by the generalized discrepancy principle. Hanke et al. [
14] used Landweber iteration to solve nonlinear ill-posed problems for the first time, and proved the convergence and convergence rate of the Landweber iteration. However, they did not consider the case of finite dimensions. Based on the gradient method, Neubauer [
18] presented a new iterative method that greatly reduces iteration number. The convergence and convergence rate of the new method were proven. Neubauer did not consider the case of finite dimensions either. However, in numerical simulations and practical applications, we should consider regularization methods of finite dimension to solve nonlinear ill-posed problems. For this, Jin and Scherzer provided their results [
16,
19] in this field. In previous studies [
14,
16,
18,
19], the important parameter in the generalized discrepancy principle was greater than two. This is not satisfactory. Hanke [
15] derived a smaller parameter for the generalized discrepancy principle that may be close to one under certain conditions. To obtain a better approximation solution, we hope the parameter in the generalized discrepancy principle is as close to one as possible.
From the development of Landweber iteration, the estimate of the parameter in the generalized discrepancy principle depends on an unknown constant, but the selection range of the parameter was not optimal in previous studies [
14,
16,
18,
19]. Therefore, the approximate solution obtained by this parameter is also not optimal. To obtain a better approximate solution, this paper introduces the radius
of the field and the relaxation factor
under the strong Scherzer condition to improve the selection range of the parameter. Simultaneously, by combination with the generalized discrepancy principle, the convergence rates of the Landweber iteration are proven in finite dimensional space in this paper.
This paper is organized as follows. In
Section 2, we outline some lemmas and propositions of Landweber iteration under the strong Scherzer condition. In
Section 3, we describe the matrix form of the Landweber iteration discretized using the multiscale Galerkin method, and the convergence of the Landweber iteration is proven in finite dimensions. In
Section 4, we develop a compression multiscale Galerkin method for Landweber iteration to solve nonlinear ill-posed integral equations, which leads to an optimal approximate solution under certain conditions. A multiscale compression algorithm is proposed to solve nonlinear ill-posed integral equations. Finally, in
Section 5, we provide numerical example to verify the theoretical results.
2. Landweber Iteration
In this section, we will describe some lemmas and propositions of Landweber iteration for solving nonlinear integral equations in detail. These results are based on [
14,
15].
Suppose that
is a bounded domain with
. Let spaces
and
denote Hilbert spaces. For the sake of simplicity, inner products and norms of Hilbert space are denoted as
and
, respectively, unless otherwise specified. The nonlinear compact operator
is defined by
where
is nonlinear mapping. We focus on the following nonlinear integral equation,
where
is given and
is the unknown to be determined. Note that
represents the range of the operator
F and
represents the domain of the operator
F. Without loss of generality, we assume that the nonlinear operator
F satisfies the following two local properties.
The
derivative of nonlinear operator
F is denoted by:
with
, which satisfies
where
denotes the neighborhood of
with radius
.
The
derivative of nonlinear operator
F satisfies the strong Scherzer condition. In other words, a bounded linear operator
exists:
, such that
holds for any elements
, where the linear operator
satisfies
for
.
The accurate data
y in Equation (
1) may not be known; instead, we have noisy data
, satisfying:
where
is a given small number. As
F is a compact operator defined on an infinity dimensional space, (
1) is an ill-posed problem. In other words, the solution does not depend continuously on the right-hand side. The Landweber iteration [
3,
4] is one of the prominent methods in which a sequence of iterative solutions
is defined by
from
, where
is a relaxation factor. If the noise is free, i.e., the right-hand side is
y, then we replace
with
. Note that
is not the solution of problem (
1); it is only a given initial function used to solve (
1).
Similar to Proposition 2.1 of [
14], we provide the following properties.
Lemma 1. Let be a solution to Problem (1). Assume that Condition holds. Ifholds, then any solution of Problem (1) satisfiesand vice versa. Here, represents the null space of an operator. Proof. From Condition
, we have
for any
, where
and
. From Equations (
3) and (
4), we obtain
From Equation (
7), the conclusion is established. □
Let Problem (
1) be solvable on
and let
be a solution set of Problem (
1) on
. From [
18], a unique local minimum norm solution
exists for
, i.e.,
Proposition 1. Let be a solution to Problem (1). If condition holds, then a unique local minimum norm solution existsfor . Proof. See Proposition 2.1 of [
18]. □
Proposition 2. Let be a solution to Problem (1). Assume that Conditions , , and (7) hold. Ifholds, then we haveand Proof. From Conditions
,
, and (
6), using the induction method, we can conclude that
with
. Combining Equations (
8) and (
9) and Condition (
12), we have
and
It follows from Equation (
7) that Equation (
13) holds. Combining the above inequality and Equation (
13), we have
Then, Equation (
14) holds. □
To illustrate the convergence of the Landweber iteration with the noise-free case, the following lemma is given. The proof of this lemma refers to Theorem 2.3 in [
14].
Lemma 2. Assume that Conditions and , and Equations (7) and (12) hold. If Problem (1) is solvable on , then the approximate solution converges to . Proof. Let
be a solution to Problem (
1), and
From Proposition 2,
is monotonically decreasing and converges to a constant
. Now, we prove that
is a Cauchy sequence. Without loss of generality, we assume that there exists a positive integer
N such that integers
. By the Minkowski inequality, we have
where the integer
we chose satisfies
for any integer
. Therefore, proving that
is a Cauchy sequence, we only need to prove that
and
when
. We prove the first:
. From the definition of inner products and norms, we know that
By Condition
and Equation (
6), we can conclude that:
Combining the monotonicity of the sequence
and Equation (
14), we have
This means that
. Similarly, we can prove that
. So, sequences
and
are Cauchy sequences. From
, the limit
of the sequence
is also a solution to Problem (
1). Using Equation (
3), we can obtain
. Therefore, it follows from Equation (
6) that
Thus,
. Because
, we have
From Proposition 1 and the above, we can conclude that . □
3. A Multiscale Galerkin Method of Landweber Iteration
The multiscale Galerkin method is a classical and effective projection method (cf. [
20]), and is often used in integral equations (cf. [
21,
22,
23,
24]). We next discuss using the multiscale Galerkin method to discrete the iteration scheme (
6). The purpose of this section is to analyze the convergence of the multiscale Galerkin method for Landweber iterations. Here, we only provide a brief description of the multiscale Galerkin method. For a more in-depth understanding of its specific structure and numerical implementation, please refer to [
22,
25,
26].
Let
denote a set of natural numbera, and define
. Suppose there is a nested and consistently dense space sequences
, i.e.,
We further assume that a subspace
exists satisfying for
:
with
. Therefore, we conclude the multiscale and orthogonal subspace
for
. For the specific structure of this space, refer to Chapter 4 in [
20]. We need to pay attention to spaces
and
, which are two polynomial function spaces of degree
. For
, subspace
can be generated by subspace
. We define
dim
and
dim
for some positive integer
. Let the indicator set
with
. Assume that the family base function of space
is
for
, i.e.,
Assume that
is the linear orthogonal projection from
onto
, and a positive constant
c exists such that
where
denotes the linear subspace of
, which is equipped with the norm
. Here,
is some linear operator acting from
and
c denotes a generic constant. For convergence of projection operator
, the following condition is needed.
Assume that
holds. Then, a positive constant
c exists such that:
Some related lemmas are outlined in the following that are similar to the case using Tikhonov regularization in [
22,
24]. We omit their proofs.
Lemma 3. If condition holds, then there exists a constant c independent of n such that Proof. See (3.10) of Lemma 3.1 in [
24]. □
Lemma 4. If condition holds, then there exists a constant c independent of n such that We apply the multiscale Galerkin method to solve iterative Scheme (
6) for the free noise case, i.e., finding
such that
holds, where
denotes the number
n depending on iteration
l,
is a given initial function, and the space
is the selected initial space. From the definition of space
, the approximate solution
can be denoted as
Therefore, the iteration Scheme (
16) can be expressed as
where
Format (
18) is unique to the multiscale Galerkin projection. This is one of our motivations for using multiscale Galerkin projection.
To show that the multiscale Galerkin method maintains the convergence of Landweber iteration, we imply the following property:
Proposition 3. Let be a solution to Problem (1). Assume that Conditions – and Equation (7) hold. If and there exists positive integer , for every positive integer l:with c denoting a generic constant and as in (16) for , then Proof. From Conditions
and
, and iteration scheme (
16), we can conclude that
for
. Now, we use the induction method to prove Equation (
20). Combined with Lemma 3 and Equation (
19), we can conclude that
Suppose that Equation (
20) is established for
.
holds. Similar to Equation (
21), we can derive
Therefore, (
20) is established. □
From the above analysis, we know that if
with
n depending on iterative steps
l. Then,
Note that the larger the iteration step
l, the larger the number of discrete layers
n. Due to the multiscale and orthogonal of spatial sequences (cf. Chapter 4 of [
20]), the multiscale Galerkin scheme is more suitable for this iterative process than the general Galerkin scheme.
Theorem 1. Assume that conditions –, (7) and (19) hold. If Problem (1) is solvable on , then the approximate solution converges to as . Proof. From Lemma 2 and Proposition 3, we can obtain the result. □
4. Rates of Convergence and Algorithm
In this section, the compression multiscale Galerkin method for Landweber iteration is used to solve Problem (
1) with noisy data
. Convergence rates of this method are proved under certain conditions.
We apply the multiscale Galerkin method to solve the iterative Scheme (
6) for the noisy case, i.e., finding
such that
holds, where
is a given initial function and
. Note that the above number
n does not depend on
l. From the definition of space
, the approximate solution
can be denoted as
Therefore, the iteration Scheme (
16) is equivalent to the iteration system
where,
As Lemma 4 shows that most entries of
are very small, these small entries can be neglected without affecting the overall accuracy of the approximation. To reduce the computational cost, the compression strategy is defined by
with
and
. Using the basis of space
, the equivalent matrix form of operator
is
with (cf. [
22])
We replace
with
in Equation (
16). Then, a fast discrete scheme for the iterative scheme (
6) with noise is established, i.e.,
where
is a given initial function and
To analyze the convergence rates of the compression multiscale Galerkin Scheme (
23), we need the following estimates.
Lemma 5. If Condition holds, then there exists a positive constant such that for any :where . Proof. See Lemma 2.3 of [
22]. We have
. Combining this result and Lemma 3, the assertion is proved. □
To ensure the convergence rate of the approximate solution, we need some conditions: one is the stopping criterion [
14,
15,
18], which is a generalized discrepancy principle; another is the smoothness condition of the initial function
and
-minimum-norm solution
[
5,
12]; and the last is the discrete error control criterion [
16,
19,
22].
A positive integer
exists such that
where the discrete number
n depends on the noise level
for
and
satisfies
Let
be a
-minimum-norm solution of Problem (
1) that satisfies
There exists a positive integer
that satisfies the following condition,
with
.
Condition is a posteriori parameter selection criterion, which leads to the appropriate approximate solution. Condition is a necessary condition for the order optimal convergence rates. Condition ensures that the projection error does not affect the iterative process.
We next provide the proof of convergence rates for the compression multiscale Galerkin method of Landweber iteration under conditions (H1)–(H6).
Proposition 4. For any , and , two positive numbers and exist satisfyingandwhere and both depend on ν. Proof. Proof of the general case was provided in [
4] and will not be repeated here. □
Theorem 2. Assume that conditions –, (5) and Equations (7) hold. If is small enough and the relaxation factors ω satisfythen there exists a positive number such thatandholds with , where and obtained from (23). We havewith . Proof. For convenience, let
,
, and
. It follows from Condition
and Equation (
23) that
where
. For
, this yields the closed expression for the error
and consequently,
with
. For
, we next turn to the estimates of
and
. Using Equation (
8) implies that
Combining Equations (
5), (
9), and (
25), we have:
with
. It follows from Equations (
4) and (
37) that
Similarly, it follows with Lemma 5 and Equation (
37) that
Now, we use induction to show that for all
hold with
. Here,
For
, Equation (
40) is always true. We assume that Equation (
40) holds for all
with
. Thus, we have to verify Equation (
40) for
. Combining
and
, we have [
14,
16]
From the above and Equations (
35) and (
36), we can conclude:
and
For
, applying Equations (
29), (
30) and (
38)–(
40), we can conclude
Using Equation (
37), we can conclude that for
By combining with Equation (
9), it follows that
Thus, using Proposition 4, we can obtain
and
Combining Condition
and Equation (
41), it follows that
and
with
. Therefore, it follows that
with
. Similarly, we have
This means that a constant
exists that depends on
,
, and
, but is independent of
,
k and
. Thus, we can obtain
and
By combining
(
31) and (
42), we can conclude
with
. Thus,
where:
If
is sufficiently small, namely
, then
. Therefore, Equations (
32) and (
33) are true. From Equations (
33) and (
37), Assertion (
34) holds. □
Theorem 3. If the conditions of Theorem 2 hold, thenwhere is a constant and depends on ν. Proof. We use the same notation as in the proof of Theorem 2. It follows from Equation (
35) that
where
Combining the above and Proposition 4, we have
with
depending on
, but being independent of
. It follows from (
H1)–(
H6), and Equations (
9) and (
45) that
Thus, by the interpolation inequality [
3], we can conclude
with
depends on
.
From Equation (
45) and Proposition 4, we have
with
depending on
. If
, then the assertion holds. Otherwise, we apply (
43) with
to obtain:
Therefore, Equation (
44) holds. □
For the convenience of numerical calculation, we wrote the above analysis process as an algorithm. This algorithm includes three parts: constructing space (Algorithm 1), updating the iteration (Algorithm 2), and stopping criterion (Algorithm 3).
Algorithm 1 Constructing space |
Step 1. Give initial function , perturbed level , and constants r, . Determine constants , , , and . |
Step 2. Solve n from inequality
|
with . |
Step 3. Construct multiscale bases for . |
Algorithm 2 Updating iteration |
Step 1. Compute vector , and matrix , . |
Step 2. Suppose that vector has been obtained. |
● Compute and . |
● Solve from
|
for .
|
Algorithm 3 Stopping criterion |
Step 1. Restore by bases and vector . |
Step 2. Continue iteration until the approximate solution satisfies: |
5. Numerical Experiment
This section provides a numerical example of nonlinear integral equations using the compression multiscale Galerkin method of Landweber iteration. The purpose was to verify the theoretical results.
Consider nonlinear integral equations [
9]
, where
is defined as
Here,
denotes the linear space of all real-valued square integrable functions, and
denotes
where
D represents first-order differential operators. The
derivative of the operator
F is
To complete numerical calculations, we provide the concrete construction of the sequence space
(cf. [
25]). For the space
, we choose the linear basis function as
For space
, we choose the linear basis function as
Next, we can construct all the base functions of the subspace
for
through the following recursive formula [
25],
Therefore, we can obtain a linear system (
18) using these bases. Here,
and
.
Let
be a
-minimum-norm solution of
. Thus, we have
with
and
. We choose initial function
for
. Therefore, we can obtain
We can conclude that the rates of convergence are
. From the definition of linear operator
, we know that
Thus, we take
,
and
. Let
and
with:
for
. From Equation (
31), we know that the relaxation factor
, then we choose
. When constant
,
, and noise level
, it follows from Equation (
28) that
.
All our numerical experiments were conducted in MATLAB (Sun Yat-sen University, Guangzhou) on a computer with a 3.0 GHz CPU and 8 GB memory. The numerical results in
Table 1 show that the compression multiscale Galerkin method for Landweber iteration can effectively solve nonlinear ill-posed integral equations. The results in
Table 1 are consistent with the assertion of Theorem 3.
Figure 1 shows the relationship between the error
and the iteration step
. Through the results in
Figure 1, we further verified Theorem 2.
Figure 2 shows the close degree of
and
and their generalized derivatives.