Mathematical Problems in Engineering

Volume 2018, Article ID 4790950, 16 pages

https://doi.org/10.1155/2018/4790950

## A Novel Fractional Tikhonov Regularization Coupled with an Improved Super-Memory Gradient Method and Application to Dynamic Force Identification Problems

^{1}College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin 150001, China^{2}School of Mechanical Engineering, Heilongjiang University of Science and Technology, Harbin 150022, China

Correspondence should be addressed to Chunping Ren; moc.anis@nipnuhcner

Received 8 January 2018; Revised 27 March 2018; Accepted 15 April 2018; Published 2 July 2018

Academic Editor: Kishin Sadarangani

Copyright © 2018 Nengjian Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a novel inverse technique to provide a stable optimal solution for the ill-posed dynamic force identification problems. Due to ill-posedness of the inverse problems, conventional numerical approach for solutions results in arbitrarily large errors in solution. However, in the field of engineering mathematics, there are famous mathematical algorithms to tackle the ill-posed problem, which are known as regularization techniques. In the current study, a novel fractional Tikhonov regularization (NFTR) method is proposed to perform an effective inverse identification, then the smoothing functional of the ill-posed problem processed by the proposed method is regarded as an optimization problem, and finally a stable optimal solution is obtained by using an improved super-memory gradient (ISMG) method. The result of the present method is compared with that of the standard TR method and FTR method; new findings can be obtained; that is, the present method can improve accuracy and stability of the inverse identification problem, robustness is stronger, and the rate of convergence is faster. The applicability and efficiency of the present method in this paper are demonstrated through a mathematical example and an engineering example.

#### 1. Introduction

In this paper, we present a novel regularization technique to solve the following linear operator equations of the form:where , and represent the identified vector, the measured vector, and a compact linear operator, respectively;

Under these circumstances, a generalized solution of (1) can be obtainedwhere is the Moore–Penrose pseudoinverse of which is usually an ill-conditioned operator, as, for example, it could be if , which causes (2) to be a typical ill-posed problem; see [1, 2].

Ill-posed problem exists widely in many fields of science and has important application in engineering [3]. However, in many cases of linear equation which appears in practical engineering problem, generally, it is clear that the measured vector is rarely obtained exactly. Conversely, noises are contained in owing to the measurement error. Consequently, when is severely ill-conditioned and is disturbed through noise, we will be confronted with a problem in which the generalized solution of an ill-posed linear equation may be seriously deviated from the exact solution [4–6].

In mathematical theory, there are several typical regularization methods which can solve the ill-posed problem in practical engineering applications. For example, the standard Tikhonov regularization (TR) method, as a traditional technique, has been widely adopted for overcoming the ill-posed nature (or numerical instability), such as system parameter identification, load source identification problem, and so on [7–10]. T. S. Jang et al. [11] applied the standard TR method to identify the loading source of infinite beams on an elastic foundation from given information of vertical deflection of infinite beams. J. Gao et al. [12] utilized the standard TR method for identifying the unknown load source. V. Melicher et al. [13] identified the piecewise constant parameter using the standard TR method. In addition, many works are mainly concerned with the identification of load position and load number in nonlinear system using the standard TR method [14–16]. However, the standard TR method is not completely perfect; there are some unavoidable limitations and disadvantages: (i) the standard TR method may determine approximate solutions that are too smooth; (ii) the approximate solution may lack many details that the desired exact solution might possess; therefore, it is very necessary to discuss other regularization methods to improve the limitations and disadvantages of the regularization method mentioned above.

In the past few years, fractional Tikhonov regularization (FTR) method was gradually introduced to deal with the ill-posed problems mentioned above [17]. Furthermore, the FTR method has been applied to many fields and obtained some remarkable achievements [18–20]. M. E. Hochstenbach et al. [21] used the FTR method to obtain approximate solution of higher quality which is better than the standard TR method in solving linear system equation with a severe ill-posedness. In [22], we can see that the FTR method is significantly better than the standard TR method, which was proved by several numerical examples. It can be found from [23] that the FTR method can obtain more approximation solutions than the standard TR method, which was discussed through several computed examples. J. Zhang et al. [24] illustrated that the FTR method was more suitable for tackling partly textured images problems. Hence, the FTR method mentioned above has many advantages, to a certain extent, which may not only solve the shortcomings of the standard TR method, but also be able to provide a useful help for obtaining the effective solution of the ill-posed problem. However, FTR method is also not very perfect, which has great limitations for solving the ill-posed problem. First, the solutions of the larger singular values can not be retained well. Then it is not completely effective for the large-scale ill-posed problem. Thus, it is very necessary to discuss an effective technique to tackle the limitations of the problem encountered.

Considering the limitations of TR method and FTR method in a certain case, we propose a novel fractional Tikhonov regularization (NFTR) method to offer a stable solution for the ill-posed problems in practical engineering applications. In our work, the ill-posed problem processed by the proposed method is redefined as a class of unconstrained optimization problems to solve the following minimum problem: In which : is a continuously differentiable function, and denotes the smoothing functional using the NFTR method.

In fact, there are many optimization algorithms that can improve the efficiency of solving (3), like conjugate gradient (CG) method, memory gradient (MG) method, super-memory gradient (SMG) method, and so on [25–27]. It is generally known that CG method does not achieve global convergence for common functions under inexact line searches [28], whereas MG method requires dealing with the trust region subproblem [29]. Although SMG method is superior to CG and MG method, in some cases, it is also not globally convergent [30]. Therefore, it is very meaningful to find a global convergence algorithm based on the SMG method. Hence, in this paper, an improved super-memory gradient (ISMG) method is introduced to deal with unconstrained optimization problems of (3).

This article is focused on a novel stable inverse technique developed to provide an optimal solution for the ill-posed problems in practical engineering. Compared with our previous works, some highlights and advances of the proposed technique are derived in this article.

First of all, the article is different from our previous studies, and the highlights of this article are described as follows: (i) a novel fractional Tikhonov regularization (NFTR) coupled with an improved super-memory gradient (ISMG) method is proposed; (ii) solving the ill-posed problem procedure is regarded as a class of unconstrained optimization problems; (iii) the detailed information of the solution is clearly enhanced; (iv) the technique we propose can effectively overcome the smoothness of the solution of the ill-posed problem; (v) global convergence and stability of the solution are proved.

Then, some advances of the proposed technique are described as follows: (i) a stable optimal solution for the ill-posed dynamic force identification problems can be obtained; (ii) the proposed method yields approximate solutions of higher quality than the standard TR method and FTR method; (iii) the technique proposed does not need any a priori information on the model of the inverse identification; only the measured response is sufficient for the present method; (iv) the applicability of the present technique is examined through a mathematical example and an engineering example.

In this paper we propose a novel fractional Tikhonov regularization (NFTR) coupled with an improved super-memory gradient (ISMG) method to solve the ill-posed dynamic force identification problems in engineering applications. The remainder of this paper is formulated as follows. The inverse problem model is described in Section 2. In Section 3, we present the NFTR method for solving the ill-posed problem. In Section 4, ISMG method is introduced to deal with an optimization problem. The convergence properties of the present method are proved in Section 5. In Section 6, we discuss the application of a mathematical and an engineering example. Section 7 shows some meaningful findings and new conclusions.

#### 2. Description of Inverse Problem Model

The model of inverse problems encountered in science and engineering can be uniformly expressed by a Fredholm integral equation of the first kind [31–33]: where , , and represent the identified object, the kernel function, and the measured response, respectively.

The discretization form of (4) is processed based on the rectangular formula in which is time interval; we consider thatThen we haveHence, (5) can be simply written as The vector represents available data, which is contaminated by an error . The error may stem from measurement inaccuracies or discretization. Thus, where is the unknown error-free vector associated with . We will assume the unavailable error-free systemto be consistent and denote its solution of minimal Euclidean norm by . We would like to determine an approximation of by computing a suitable approximate solution of (10). Due to the ill-conditioning of the matrix and the error in , the solution of the least-squares problem (10) of minimal Euclidean norm is typically a poor approximation of , and a small perturbation in may give rise to an arbitrarily large perturbation in , or even make the problem unsolvable. Moreover, the right-hand side function that is available in applications represents data that is contaminated by noise. Thus, instead of , the error-contaminated function is available [34, 35]. Therefore, it is very necessary to find an effective method to solve the ill-posed problem.

#### 3. The Regularization Method for Solving The Ill-Posed Problem

In some instances, we all know that some traditional numerical methods are not able to solve (8) effectively. However, the regularization method, as a way to deal with the ill-posed problem, has been used by mathematical workers and engineering technicians [36–38]. In the past few years, the standard Tikhonov regularization (TR) and fractional Tikhonov regularization (FTR) method have been successfully applied to many engineering fields. Although many meaningful achievements have been made, there are still many limitations. Hence, in this work, considering the disadvantages of the standard TR method and FTR method, a novel fractional Tikhonov regularization (NFTR) method is discussed.

##### 3.1. The Standard Tikhonov Regularization (TR) Method

The standard Tikhonov regularization (TR) method is a popular approach to determine an approximation of x; see, e.g., [39–41] for properties and applications. This method replaces the minimization problem (8) by a penalized least-squares problem of the following form [42]:in which , and , ,, and represent the regularization matrix, the regularization parameter, and the unit matrix, respectively.

##### 3.2. Fractional Tikhonov Regularization (FTR) Method

It is well known that Tikhonov regularization in standard form typically determines a regularized solution that is too smooth; i.e., many details of the desired solution are not represented by the regularized solution. This shortcoming led Klann and Ramlau to introduce the fractional Tikhonov regularization method [17]. Subsequently another approach, also referred to as fractional Tikhonov regularization (FTR), was investigated in [21], the model of which is described as follows:Then we know In which is defined with the aid of the Moore–Penrose pseudoinverse of . where denotes fractional order and . denotes the transpose of . It is obvious that FTR method is different than that of TR method; it mainly considers the influence of fractional order on the identified objects, which thus may be more suitable for solving the ill-posed identification problem. However, FTR method is not perfect, which has great limitations for solving the ill-posed problem. First, the solutions of the larger singular values can not be retained well. Then it is not completely effective for solving the large-scale ill-posed problem. Thus, it is very necessary to find an effective method to solve the limitations of problem we encountered.

##### 3.3. A Novel Fractional Tikhonov Regularization (NFTR) Method

From Sections 3.1 and 3.2, on the one hand, we can find that the regular solution is obtained by using the standard TR method that may be too smooth and may lose some details of the solution we expected. On the other hand, the solutions of the larger singular values can not be retained perfectly using the FTR method for dealing with some practical complex engineering applications. Hence, in order to solve the limitations of TR method and FTR method in a certain case, in this paper, a novel fractional Tikhonov regularization (NFTR) method is proposed. The model of the proposed method is described as follows: in which where , , , , and represent the regularization matrix, the diagonal matrix, right column orthogonal matrix of singular value decomposition of , adjustment coefficient matrix, and symmetric positive semidefinite matrix, respectively. In this work, Γ is arranged as the identity matrix; that is, , in which can be described as follows:Hence, the NFTR method corresponding to the normal equation can be expressed in the following form:where denotes the transpose of .

Introduce the singular value decomposition (SVD) [43]:in which and

are orthogonal matrices, andThe singular values are ordered according toin which the index is the rank of ; see, e.g., [43] for discussions on properties and the computation of the SVD.

Combining (18) and (19), then we haveHence, we can obtain the regular solution of (22):Meanwhile, the filter factor of TR, FTR, and NFTR method can be expressed as follows:

##### 3.4. Stability Analysis

In the paper, we may determine a suitable regular parameter by the discrepancy principle [44, 45]; we choose > 0 so that where is equivalent to the noise level , which satisfies , is the noise vector. is a user-supplied constant independent of , and .

Using (23) and (24), we haveFinding the partial derivative of and in (26), respectively, then

We can find from (27) that is a monotonous decreasing function. When , then , so we can obtain , also a monotonous decreasing function.

However, it is clearly seen from (27) that the regularization parameter becomes large and the norm of regular solution becomes small. Due to being a monotonous decreasing function, so reducing can increase the norm of regular solution according to .

Theorem 1. *For , the regularization parameter meets (25), and then one has *

*Proof. *The following form can be found:Then, the following form is obtained by substituting (24) and (25) into (29).Offering the derivative of We can see from the previous formula that ; then Hence, the conclusion is proved.

In this paper, the NFTR method is of order optimal when the regularization parameter is chosen according to the discrepancy principle, and the proof process is shown as follows.

Let , ; then (18) can be described as a normal equation:where and denote the transpose of and .

For our analysis, we assume a Hölder-type smoothness assumption, i.e., that the minimal norm solution of the error-free problem (1) satisfies a smoothness condition of the form, for some constant ,With .

Here is the singular system of the operator . is a parameter which controls the a priori assumptions on the regularity of the domain set.

Theorem 2 ([17, Theorem 3.1]). *Let and . is a regularizing filter and assume that for ,and where and , are constants independent of . The definition of is the same as that of . Then with the a priori parameter choice The NFTR method induced by the filter is order optimal for all .*

Lemma 3. *The regularizing filter from (24) with fulfills Theorem 2 with and .*

*Proof. *The filter is continuous on . The regularizing properties of the filter are easily verified. One sees thatHence, as long as , the suprema in Theorem 2 are attained as local maximal, which can be derived by simple calculus. One obtainsandConsequently, by Theorem 2, the NFTR method (15) is order optimal for and .

#### 4. The Algorithm for Solving Unconstrained Optimization Problem

In general, it has always been known that the regular solution of (15) may give rise to an arbitrarily large perturbation in actual solution. However, one of the highlights of this work is expressed as follows: (15) is deemed considered as an unconstrained optimization problem, and the optimal solution of (15) can be obtained by using some optimization algorithms. In this paper, an improved super-memory gradient (ISMG) method was introduced to solve objective function of (15). Most of the well-known iterative algorithms for solving (15) take the following form [46–49]:where is the search direction.

In this paper, a new search direction of ISMG method is determined as follows:where denotes the gradient of at the point , and . is defined as follows:

of (41) is the step length, which can be computed using modified nonmonotone line search technique, , and satisfies the following inequalities:where , and .

*Algorithm 4. **Step 1*. Given an initial point , , , and ; is a given positive integer, , and , set .*Step 2*. Compute ; if , then terminate; else go to the next step. *Step 3*. Compute based on (42).*Step 4*. Compute based on (44).*Step 5*. Set .*Step 6*. Set and go to Step 2.

Lemma 5. *For any line search, the search direction is defined by (42); then one has *

*Proof. *If , so we haveand , so it has , .

Hence, we obtainIf , according to (42) and (43), we can obtain the following:Then, we have Hence, (45) is proved.

Lemma 6. *For any , then one has*

*Proof. *If , so we haveIf , according to (42) and (43), we can obtain the following:So we have Hence, (50) is proved.

It is clear that we can draw the following conclusion according to (45) and (50):

#### 5. The Convergence Properties Analysis for Optimal Algorithm

In this section, we propose an algorithm and then we study the global convergence property of this algorithm. Firstly, we make the following two assumptions, which have been widely used in the literature to analyze the global convergence of the presented method.

*Assumption 7. *The objective function is considered to be bounded in the following circumstances:

*Assumption 8. *The gradient is Lipschitz continuous; there exists a constant , for any ; we have

Lemma 9. *Suppose that Assumptions 7 and 8 hold; then Algorithm 4 is well-posed; that is, there exists the step length , which makes (44) hold.*

*Proof. *According to (44), we have It indicates that the following formula is set up for when there exists , and .Hence, (44) is proved.

Lemma 10. *Suppose that Assumptions 7 and 8 hold, , and there is a constant ; then*

*Proof. * is defined; we just need proof that . First of all, we prove it with a contradiction; suppose that holds, and there are a series of subfamily ; then Set , and is the maximum in , which meets (44). Thus, can not meet (44). Then Further, we have Based on the mean value theorem, then in which .

Thus, it follows from (62) and (63) that From Assumption 8 and Cauchy Schwartz inequality, we can obtainFrom (64) and (65), it follows thatThen, we haveCombining (67) and (54), we obtainObviously, we clearly see that (68) and (60) are contradictory. Hence, we haveThen, combining (44) and (45), we can obtainin which, set .

Hence, (59) is proved.

According to (see [50, 51]), it is clear that the sequence is contained in , and there exists a constant , such that

, suppose a<x<y<b; then it can satisfy the following conditions: , for small enough. Because is a convex function, so we obtainLet ; we haveSet , then Hence, we get

Therefore, according to the above discussion, we can draw a conclusion that the objective function defined in (15) is differentiable and Lipschitz continuous.

#### 6. Examples and Discussions

In this section, a mathematical example and an engineering example are investigated to clarify how to solve the solution of the ill-posed problem quickly and accurately by using a novel fractional Tikhonov regularization (NFTR) coupled with an improved super-memory gradient (ISMG) method.

##### 6.1. A Mathematical Example

In this section, our first example is a severely ill-posed Fredholm integral equation of the first kind given bySet , ; then in which the perturbation of the right-hand side is set as follows:where , in which , , and denote the standard deviation of , a random number within , and the noise level, respectively.

Then (77) was discretized, which can be obtained:in which , .

The problem of solving (76) is an ill-posed problem; we solve the approximate solution of (76) with the right-hand side given a perturbation. The true solution of (76) is ; then we need to calculate the identified solution . Here, TR method, FTR method, and the present method in this paper are applied to deal with the problem.

In order to further discuss the quality of the different methods used and assess the degree of the approximation of the identified result to the true result, we provide the following evaluation metrics.

Relative error (RE):Correlation coefficient (CC):

The program processing is executed in the MATLAB regularization tools package, and the condition number of the matrix is calculated by the MATLAB function cond; it can be found that , which means that the matrix is singular.

In our experiments, the noise vector is selected as a normal random distribution vector with zero mean and normalized to an ideal noise level:In this experiment, the regularization parameter is selected by using the discrepancy principle, in which and . Other parameters are set to the following: , , , , and The computer stops the iteration processing when is met.

denotes the optimal value of the fractional order that gives the most accurate identified result of (76) when is selected using the discrepancy principle ( and ).

In this section, we employ the standard TR, FTR, and the present method in the paper to identify the results of , respectively.

First of all, when and the noise level is set to 10%, 5%, and 1%, then RE and CC and iterative steps of the identified result are listed in Table 1.