Elsevier

Applied Numerical Mathematics

Volume 169, November 2021, Pages 122-145
Applied Numerical Mathematics

Accelerated homotopy perturbation iteration method for a non-smooth nonlinear ill-posed problem

https://doi.org/10.1016/j.apnum.2021.06.008Get rights and content

Abstract

In this paper, a fast iterative algorithm based on J-order homotopy perturbation method is proposed for the nonlinear ill-posed problem whose forward operator is not Gâteaux differentiable. The Bouligand subderivative of the forward operator is utilized to replace the Fréchet derivative in iteration system. The sequential subspace optimization technique is introduced to accelerate the convergence speed, which regards the correction term of homotopy perturbation as multiple search directions to update the new iterate. To this end, the current iteration is sequentially projected to the stripes whose width is determined by search directions, the nonlinearity of the forward operator and noise level. We present the convergence analysis based on the asymptotic stability estimates and a generalized tangential cone condition. Numerical experiments are performed to illustrate the effectiveness of the proposed method.

Introduction

In this paper, we consider a nonlinear ill-posed problemF(x)=yδ, where F:D(F)XY, a parameter-to-observation mappings between real Hilbert spaces X and Y, is compact and directionally but not Gâteaux differentiable. Throughout this paper, it is assumed that x is the exact solution to be recovered, and the given data yδ is the an approximation of exact data y satisfyingyδyδ, where y=F(x) and δ>0 is noise level. Since the problem to estimate x is usually ill-posed in the sense that the solution does not depend continuously on data, several regularization methods have been proposed for non-smooth problem (1) to obtain a stable solution, such as non-differentiable regularization methods in Banach space [18], [16], [8] and modified iterative regularization methods in Hilbert space [9], [10].

As a classic iterative regularization, Landweber iteration has the form ofxn+1δ:=xnδtnδF(xnδ)(F(xnδ)yδ),n=0,1, where tnδ is the step size and F(xnδ) is the Fréchet derivative of F at xnδ. It is not applicable for the case that F is not Gâteaux differentiable. A potential technique is to replace F(xnδ) by a suitable linear operator Gxnδ closing to F(xnδ) in the proper sense, such as the modified Landweber method [15]. In derivative-free Landweber iteration method [9], [10], such operator Gx, satisfying the continuity of the mapping xGx, is constructed for parameter identification in certain elliptic PDEs.

In [3], a Bouligand-Landweber iteration (BL) method is proposed based on the classic Landweber iterative format (3), which uses Bouligand subdifferential in place of the non-existence F(x) for nonlinear problem (1). The BL method has the form ofxn+1δ:=xnδtnδGxnδ(F(xnδ)yδ), where tnδ is the step size, the Gxnδ is the linear operator taken from Bouligand subdifferential of F. The Bouligand subdifferential is defined as the set of limits of Fréchet derivatives in differentiable points [14, Def.2.12]. In [3], the continuity of the mapping xGx is not assumed, so a new convergence analysis for BL method is presented by introducing the concept of asymptotic stability and relaxing assumptions for classical iterative regularization methods. As first-order scheme, the BL method needs proceed large number of iterations to meet the stopping rule. The Nesterov's acceleration is introduced into the BL method to construct a fast two-point gradient Bouligand Landweber method [12]. Bouligand-Levenberg-Marquardt method considered in [4], a Newton type iteration method, also uses a family of bounded linear operators {Gx} taken from Bouligand subdifferential to replace the Fréchet derivative in the classical Levenberg-Marquardt iteration. In [5], a projected Bouligand-Landweber (PBL) method is given byxn+1δ=xnδμnδunδ,unδ=Gxnδ(F(xnδ)yδ), where μnδ=min(tnδ,M) for some fixed M and tnδ is chosen by projective strategy. The key idea of PBL method is that the new iterate is updated by projecting xnδ into the stripe which is decided by search direction unδ, the property of F and noise level. The projection technique efficiently accelerates convergence speed.

The methods by using multiple proper search directions to update the new iterate have been studied for linear and nonlinear inverse problems, which can efficiently reduce the total number of iterations as well as time consumption for obtaining satisfying approximations [17], [7], [23], [22], [20]. Inspired by this, we consider to construct a fast iterative method for solving non-smooth problem (1) by using multiply search directions.

Sequential subspace optimization (SESOP) method was proposed to update the new iterate through several search directions weighted by suitable step sizes [13]. These step sizes are chosen to project the current iterate into the intersection of several convex closed sets. The SESOP method has been introduced into the iterative regularization methods for inverse problems in Hilbert and Banach spaces [20], [17], [23], [22]. The step sizes chosen by projective method under several search directions make SESOP method have obvious advantage of reducing the numerical cost for obtaining good approximations.

In this paper, with the help of the idea of SESOP method, we consider the multiple search directions coming from J-order Homotopy perturbation (HP) iteration method. Homotopy perturbation iteration for solving nonlinear ill-posed problems was proposed in [1] with the form ofxn+1δ=xnδj=1JtnδF(xnδ)(IγnδF(xnδ)F(xnδ))(j1)(F(xnδ)yδ), where J is the order of the HP method, tnδ is the step size, γnδ is a proper parameter. The HP iterative method has more fast convergence speed in comparison with classic Landweber method.

Based on the above considerations and the non-smooth of operator F, we propose a modified homopoty perturbation iterative method based on the spirit of BL method, which utilizes a bounded linear operator Gxnδ to replace F(xnδ) in (6), i.e.,xn+1δ=xnδjInδtn,jδun,jδ,un,jδ=Gxnδ(IγnδGxnδGxnδ)(j1)(F(xnδ)yδ), where Inδ is a finite index set. In (7), the terms un,jδ,jInδ can be regarded as the search directions, then the step sizes tn,jδ can be chosen by projective strategy based on SESOP method. This method is named as SE-BHP method. In SE-BHP method, we define J stripes related to the search direction un,jδ, noise level and the structure of the operator F, then the step sizes tn,jδ are chosen to make new update xn+1δ be the projection of xnδ onto the intersection of these stripes. The PBL method in [5] can be regarded as a special case of our proposed Algorithm 3.4 with J=1, i.e., the new iterate xn+1δ is updated by one search direction.

The rest of this paper is organized as follows. In section 2, some basic notations, main assumptions and necessary preliminary results are briefly summarized. In section 3, we introduce the proposed SE-BHP method and then analyze its convergence. Numerical examples are performed to validate the efficiency of proposed methods in section 4. Finally, some conclusions are given in section 5.

Section snippets

Preliminaries

In this section, we will present some necessary concepts and auxiliary results for the following analysis. Let us begin with some results about SESOP method which have been stated in [17], [23], [22].

Definition 2.1

Let X be a real Hilbert space, we define the hyperplaneH(u,α):={xX:u,x=α} and the halfspaceH(u,α):={xX:u,xα} for uX and αR. Additionally, H<(u,α), H(u,α) and H>(u,α) can be defined similarly. We define the stripe asH(u,α,ξ):={xX:|u,xα|ξ} for uX, and α,ξR with ξ0.

Note that the

The SE-BHP method and its convergence

In this section, we will show the convergence and regularity of the SE-BHP method for the problem (1) under some suitable assumptions which have been applied in [3], [12], [5].

Let x be an arbitrary and fixed solution of (1) with exact data y. For some ρ>0, we denote by Mρ(x) the solution set of (1) contained in B¯ρ(x), i.e.Mρ(x):={xD(F)B¯ρ(x):F(x)=y}, where B¯ρ(x):={x:xxρ}.

Without the smoothness of F in (1), we use a suitable bounded linear operator Gx to replace the possibly

Numerical simulations

In this section, some numerical experiments are carried out to study the reconstruction performance of the proposed SE-BHP method.

Conclusions

A fast iterative method (SE-BHP) based on J order homotopy perturbation iteration and SESOP technique is proposed for non-smooth nonlinear problems, where the Bouligand subderivative of the forward mapping is utilized to replace the non-exist Fréchet derivative. The convergence and regularity of SE-BHP are studied by using a generalized tangential cone condition and the concept of asymptotic stability. The numerical experiments validate that the SE-BHP method with multiple search directions can

Acknowledgements

The work of Tong, S. is supported by the National Natural Science Foundation of China (No.11901373) and the Fundamental Research Funds for the Central Universities (No. GK202003007). The work of Wang, W. is supported by National Natural Science Foundation of China (No. 12071184), Zhejiang Provincial Natural Science Foundation of China (No. LY19A010009) and the Major R & D Plan of Science and Technology of Zhejiang (Project No. 2019C03099). The work of Han, B. is supported by National Natural

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