Multiscale Compression Algorithm for Solving Nonlinear Ill-Posed Integral Equations via Landweber Iteration

In this paper, Landweber iteration with a relaxation factor is proposed to solve nonlinear ill-posed integral equations. A compression multiscale Galerkin method that retains the properties of the Landweber iteration is used to discretize the Landweber iteration. This method leads to the optimal convergence rates under certain conditions. As a consequence, we propose a multiscale compression algorithm to solve nonlinear ill-posed integral equations. Finally, the theoretical analysis is verified by numerical results.

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 ρ of the field and the relaxation factor ω 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.

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 Ω ⊂ R d is a bounded domain with d ≥ 1. Let spaces X and Y 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 F : X → Y is defined by where Φ(s, t, x) is nonlinear mapping. We focus on the following nonlinear integral equation, where y ∈ R(F) is given and x ∈ D(F) is the unknown to be determined. Note that R(F) represents the range of the operator F and D(F) represents the domain of the operator F. Without loss of generality, we assume that the nonlinear operator F satisfies the following two local properties. (H1) The Fréchet derivative of nonlinear operator F is denoted by: where B ρ (x 0 ) denotes the neighborhood of x 0 with radius ρ. (H2) The Fréchet derivative of nonlinear operator F satisfies the strong Scherzer condition. In other words, a bounded linear operator R(x,x) exists: X → Y, such that holds for any elements x,x ∈ B ρ (x 0 ) ⊂ D(F), where the linear operator R(x,x) satisfies for c 0 > 0.
The accurate data y in Equation (1) may not be known; instead, we have noisy data y δ ∈ Y, satisfying: where δ ≥ 0 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 {x δ l } is defined by If the noise is free, i.e., the right-hand side is y, then we replace x δ l with x l . Note that x 0 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.
holds, then any solutionx * ∈ B ρ (x 0 ) of Problem (1) satisfies and vice versa. Here, N (·) represents the null space of an operator.
Let Problem (1) be solvable on B ρ (x 0 ) and let be a solution set of Problem (1) on B ρ (x 0 ). From [18], a unique local minimum norm solution x † exists for x 0 , i.e., Proposition 1. Let x * ∈ B ρ/2 (x 0 ) be a solution to Problem (1). If condition (H2) holds, then a unique local minimum norm solution x † existsfor x 0 .
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].
Proof. Letx * ∈ B ρ/2 (x 0 ) be a solution to Problem (1), and From Proposition 2, e l is monotonically decreasing and converges to a constant ε ≥ 0. Now, we prove that e l is a Cauchy sequence. Without loss of generality, we assume that there exists a positive integer N such that integers k > l ≥ N. By the Minkowski inequality, we have where the integer i ∈ [l, k] we chose satisfies for any integer j ∈ [l, k]. Therefore, proving that e l is a Cauchy sequence, we only need to prove that e k − e i → 0 and e i − e l → 0 when l → ∞. We prove the first: e k − e i → 0. From the definition of inner products and norms, we know that By Condition (H2) and Equation (6), we can conclude that: Combining the monotonicity of the sequence e l and Equation (14), we have This means that lim l→∞ e k − e i = 0. Similarly, we can prove that lim l→∞ e i − e l = 0. So, sequences {e k } and {x k } are Cauchy sequences. From lim l→∞ y − F(x l ) = 0, the limit x * * of the sequence {x k } is also a solution to Problem (1). Using Equation (3), we can obtain N (F (x † )) ⊂ N (F (x l )). Therefore, it follows from Equation (6) that From Proposition 1 and the above, we can conclude that x * * = x † .

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 N denote a set of natural numbera, and define N 0 := {0} ∪ N. Suppose there is a nested and consistently dense space sequences {X n : n ∈ N 0 } ⊂ X, i.e., We further assume that a subspace W n exists satisfying for n ∈ N: X n = X n−1 ⊕ ⊥ W n with W 0 := X 0 . Therefore, we conclude the multiscale and orthogonal subspace for n ∈ N. For the specific structure of this space, refer to Chapter 4 in [20]. We need to pay attention to spaces W 0 and W 1 , which are two polynomial function spaces of degree ≤ r. For n > 1, subspace W n can be generated by subspace W 1 . We define w(n) :=dimW n ∼ µ n and s(n) :=dimX n ∼ µ n+1 for some positive integer µ > 1. Let the indicator set U n : Thus, we have Assume that P n is the linear orthogonal projection from X onto X n , and a positive constant c exists such that where H r (Ω) denotes the linear subspace of X, which is equipped with the norm · H r = · X + D r · X . Here, D r is some linear operator acting from H r → X and c denotes a generic constant. For convergence of projection operator P n , the following condition is needed.
(H3) Assume that k(·, ·, ·) ∈ C r (Ω × Ω) 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 (H3) holds, then there exists a constant c independent of n such that Proof. See Lemma 2.1 of [22].
We apply the multiscale Galerkin method to solve iterative Scheme (6) for the free noise case, i.e., finding x n(l),l+1 ∈ X n(l) such that x n(l),l+1 = x n(l−1),l + ωP n(l) F (x n(l−1),l ) * (y − F(x n(l−1),l )), l = 0, 1, 2, · · · (16) holds, where n(l) denotes the number n depending on iteration l, x n(−1),0 = x 0 ∈ D(F) is a given initial function, and the space X n(−1) is the selected initial space. From the definition of space X n(l) , the approximate solution x n(l),l+1 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 x * ∈ B ρ/2 (x 0 ) be a solution to Problem (1). Assume that Conditions (H1)-(H3) and Equation (7) hold. If ω ≤ 1−2c 0 ρ 1−c 0 ρ and there exists positive integer n(l), for every positive integer l: with c denoting a generic constant and x n(l−1),l as in (16) for δ = 0, then Proof. From Conditions (H1) and (H2), and iteration scheme (16), we can conclude that for 0 ≤ l ≤ k. 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 0 < l ≤ k.
From the above analysis, we know that if lim l,n(l)→∞ 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 (H1)-(H3), (7) and (19) hold. If Problem (1) is solvable on B ρ/2 (x 0 ), then the approximate solution x n(l−1),l converges to x † as l → ∞.
Proof. From Lemma 2 and Proposition 3, we can obtain the result.

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 y δ . 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 x δ n,l+1 ∈ X n such that x δ n,l+1 = x δ n,l + ωB l n (y δ − F(x δ n,l )), l = 0, 1, 2, · · · holds, where x δ n,0 = x 0 ∈ D(F) is a given initial function and B l n := P n F (x δ n,l ) * P n . Note that the above number n does not depend on l. From the definition of space X n , the approximate solution x δ n,l+1 can be denoted as Therefore, the iteration Scheme (16) is equivalent to the iteration system E n c l+1 n = E n c l n + ωB l n y l n , l = 0, 1, 2, · · · , where, E n := [ w ij , w i j : (i, j), (i , j ) ∈ U n ], c l+1 n := [c l+1 i j : (i , j ) ∈ U n ], B l n := [ F (x δ n,l )w ij , w i j : (i, j), (i , j ) ∈ U n ], y l n := [ y δ − F(x δ n,l ), w i j : (i , j ) ∈ U n ].
As Lemma 4 shows that most entries of B l n 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 byB l n = ∑ i∈Z n+1 P i F (x δ n,l )Q n−i with Q n−i = P n−i − P n−i−1 and P −1 = 0. Using the basis of space X n , the equivalent matrix form of operatorB l n isB We replace B l n withB l n in Equation (16). Then, a fast discrete scheme for the iterative scheme (6) with noise is established, i.e., x δ n,l+1 =x δ n,l + ωB l n (y δ − F(x δ n,l )), l = 0, 1, 2, · · · , wherex δ n,0 = x 0 ∈ D(F) 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 (H3) holds, then there exists a positive constant c r such that for any n ∈ N: Proof. See Lemma 2.3 of [22]. We have B l n −B l n ≤ c(n + 1)µ −rn/d . 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 x 0 and x 0 -minimum-norm solution x † [5,12]; and the last is the discrete error control criterion [16,19,22].
We next provide the proof of convergence rates for the compression multiscale Galerkin method of Landweber iteration under conditions (H1)-(H6).
Proof. Proof of the general case was provided in [4] and will not be repeated here.
Proof. We use the same notation as in the proof of Theorem 2. It follows from Equation (35) that where From Theorem 2, we have Combining the above and Proposition 4, we have with ζ 1 > 0 depending on ν, but being independent of l * . It follows from (H1)-(H6), and Equations (9) and (45) that Thus, by the interpolation inequality [3], we can conclude with C 1 > 0 depends on ν. From Equation (45) and Proposition 4, we have with C 2 > 0 depending on ν. If l * = 0, then the assertion holds. Otherwise, we apply (43) with l = l * − 1 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 X n (Algorithm 1), updating the iteration (Algorithm 2), and stopping criterion (Algorithm 3).

Algorithm 2 Updating iteration
Step 1. Compute vector c 0 n , y 0 n and matrix E n ,B 0 n . Step 2. Suppose that vector c l n has been obtained.
• ComputeB l n and y l n . • Solve c l+1 n ∈ X n from E n c l+1 n = E n c l n + ωB l n × y l n for l ∈ N 0 .

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] F(x) = y, where F : Here, L 2 [0, 1] denotes the linear space of all real-valued square integrable functions, and where D represents first-order differential operators. The Fréchet derivative of the operator F is To complete numerical calculations, we provide the concrete construction of the sequence space X n (cf. [25]). For the space X 0 , we choose the linear basis function as w 00 (t) = t, t ∈ [0, 1] and w 01 (t) = 1 − t, t ∈ [0, 1].
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 x δ n,l − x † H 1 [0,1] and the iteration step √ l. Through the results in Figure 1, we further verified Theorem 2. Figure 2 shows the close degree ofx δ n,l * and x † and their generalized derivatives.  Author Contributions: R.Z. designed the paper and completed the research; F.L. and X.L. proposed the compression strategy and completed some proofs. All authors have read and agreed to the published version of the manuscript.