1 Introduction

Nowadays, inverse analysis is often used to solve the problems of mechanics [1–4]. In the framework of this paper, one considers the linear operator equation Ax=u where A : \(\mathbb{E}\rightarrow \) \(\mathbb{F}\) describes an injective and bounded linear operator from the Hilbert space \(\mathbb{E}\) into another Hilbert space \(\mathbb{F}\) with non-closed range. In many applications, Ax=u turns out to be ill-posed [5, 6]. In the literature, many strategies have been developed to solve this equation. For this purpose, regularization methods will be reviewed including Singular Values Decomposition (SVD) [7], the Tikhonov regularization [5, 8] and the Landweber iteration [9]. One of the main drawbacks of the SVD is that when it is applied to practical problems, the approximation is cursed by the instability (which is a consequence from the decay to zero of the singular values of the kernel). In addition, the cost of the computing time is great enough. Thus, it is useful to consider techniques that do not depend on the SVD. One of the most prominent classes of these strategies is the method of regularization. In regularization, original ill-posed problem is replaced with the nearby well-posed problem which is numerically stable. The Tikhonov method of regularization consists of minimizing the regularized Tikhonov functional; one notes that this method is often used. The disadvantages of this method are that on the one hand it requires inverting the regularization of the normal operator γ+A ∗ A and this inversion may be very costly in practice in terms of computing; on other hand, the choice of the regularization parameter γ is not evident; it influences significantly the approximate solution. Indeed a choice of γ too small, leads to the approximate solution which is affected by the instability of the original ill-posed problem. While a choice of γ too large, tends to over-smooth the approximate solution with consequent loss of information.

The Landweber iteration method is an iterative technique in which no inversion is necessary. It is defined to solve the equation Ax=u as follows:

in which, r>0 and n∈ℕ. This iteration can be made optimal by choosing the appropriate number of iteration. One notes that the standard method of Landweber is failed to solve the equation Ax=u when u is the result of measurements, in the sense that an approximate solution of minimal norm can not be obtained. Hence, the equation Ax=u does not have a solution in the classical sense. In this context, Ivanov [10] and Arcangeli [11] introduced respectively the concepts of quasi-solution and pseudo-solution in order to solve the problem. However, in general the quasi-solution is not unique, thus a choice of the best quasi-solution was necessary to obtain an optimal approximate solution of the problem. The aim of this paper is to develop a relationship between the Landweber method and L-pseudo solution. In many applications, a quasi solution of minimal norm can not be obtained in \(\mathbb{E}\). Hence, we determine the quasi solution of minimal norm in another Hilbert space \(\mathbb{G}\) in which the minimization of the norm can be obtained. This approximate solution of minimal norm is the L-pseudo solution. In this perspective, the Landweber iteration becomes:

x 0 being arbitrary, (AL −1)∗ the adjoint operator of AL −1, and (a n ) n a sequence of real positive numbers such that na n converges to a constant when n tends to infinity. This new procedure can be understood as the image by the bijective bounded linear operator \(L:\mathbb{E}\rightarrow \mathbb{G}\) of the standard Landweber iteration in the space \(\mathbb{G}\) where the minimization can be obtained. We point out that this new procedure generalizes the classical Landweber iteration. Indeed, taking \(\mathbb{G}=\mathbb{E}\) and L=Id, we obtain the pseudo-solution see [12].

The paper is organized as follows: in Sect. 2 the statement of the problem is proposed, in Sect. 3, one introduces the quasi-solution, pseudo-solution and L-pseudo-solution, in Sect. 4 some new results were established by using stochastic methods. Therefore, a stable solution is obtained. In Sect. 5, the validity of our approach is checked by numerical example which is standard in Rheology. We finish by some remarks and conclusion.

2 Statement of the problem

Let \(\mathbb{E}\), \(\mathbb{F},\mathbb{G}\) be separable Hilbert spaces, \(( \varOmega ,\mathcal{F},\mathbb{P} ) \) be a probability space and \(A:\mathbb{E}\rightarrow \mathbb{F}\) describes an injective and bounded linear operator from \(\mathbb{E}\) into the space \(\mathbb{F}\) with non-closed range R(A). Let L be a bijective bounded linear operator from \(\mathbb{E}\) into \(\mathbb{G}\) without lost of generality, we assume that ∥AL −1∥≤1.

Let us consider the following operator equation

(1)

where u∈ \(\mathbb{F}\) and x is the unknown solution in \(\mathbb{E}\). In practice, the second member of (1) is the result of measurements, and is usually known just approximately [13–17]. A natural way to deal with the treatment of the measurement errors is to consider the probabilistic framework. Instead of considering deterministic equation (1), we shall think of it as realization of a stochastic identity:

where \(\xi :\varOmega \rightarrow \mathbb{F}\) is the noise.

When carrying out n independent experiments, we obtain a sample {u 1,u 2,…,u n } of noisy data which is given by:

where u ex represents the unknown exact value of the second member of (1) and the noise is represented by a sequence \(( \xi_{i} )_{i\in \mathbb{N} ^{\ast }}\) of independent and identically distributed functional random variables with zero mean defined on (Ω,ℙ) with values into the Hilbert space \(\mathbb{F}\).

The main objective here is to estimate the solution of (1). The proposed approach has two main steps. The first step is concerned by the estimation of the second member. The strong law of large numbers gives the empirical mean \(\bar{u}\) as a natural exhaustive estimate. Therefore the problem becomes, solving the following equation:

(2)

In addition given that \(A ( \mathbb{E} ) \) is not closed, we can consider that \(\bar{u}\notin A ( \mathbb{E} ) \) almost surely. Consequently, the problem (2) is an ill-posed problem [5, 6] and does not have a solution in the classical sense. This will lead to the introduction of quasi-solution [10], pseudo-solution [11] and L-pseudo-solution [18]. In the second step, we use stochastic methods in order to build stable solution estimate.

3 Quasi-solution and L-pseudo-solution

3.1 Quasi-solution [10]

Let M be a non-empty subset of \(\mathbb{E}\). The quasi-solution on M of Eq. (2) is the element x ∗ of M satisfying almost surely the following equality

(3)

Denote by \(Q_{\bar{u}}\) the set of all quasi-solutions on M, i.e.

Remark 1

If M is compact, the quasi-solution exists for all \(u\in \mathbb{F}\) [6]. Moreover, if \(\bar{u}\in AM\), there exists a unique quasi-solution which coincides with the exact solution of Eq. (2).

Proposition 2

If M is convex, then \(Q_{\bar{u}}\) is convex.

Proof

The subset of minimizers of the convex functional M∋x↦∥Ax−u∥ is convex. □

3.2 Pseudo-solution [11, 18]

The pseudo-solution x ∗ of Eq. (2) is the quasi-solution of almost surely minimal norm.

3.3 L-pseudo-solution [11, 18]

The L-pseudo-solution \(\bar{x}_{\ast }\) of Eq. (2) is a quasi-solution satisfying almost surely

4 Results

In what follows, A and L are operators defined previously in Sect. 2. We first show the existence and unicity of the L-pseudo-solution \(\bar{x}_{\ast }\). Then we estimate it by the following recurrent procedure

(4)

x 0 being an arbitrary random variable, B ∗ the adjoint operator of B=AL −1 and (a n ) n a sequence of real positive numbers such that na n converges to a constant when n tends to infinity. (η n ) n is a sequence of independent and identically distributed functional random variables modeling the noise related to the algorithm, satisfying E∥η i ∥2<+∞ and Cramer’s condition:

(5)

where \(\emph{E}\) design the mathematical expectation and H a positive constant.

The last term of the right side of equality (4) is added in order to improve the iteration; so that η n models a noise on the algorithm.

Following this recurrent procedure, we subsequently establish exponential inequalities of Bernstein-Frechet type which enable us to build a confidence interval for the L-pseudo-solution of Eq. (2) and deduce the almost complete and the quadratic mean convergences of the procedure (4) to \(L\bar{x}_{\ast }\).

Theorem 3

Let P be the orthogonal projection operator on \(\overline{R(A)}\). If \(P\bar{u}\in \) \(\overline{R(A)}\) almost surely, then there exists a unique L-pseudo-solution in the convex subset M of \(\mathbb{E}\) of Eq. (2)

Proof

Since, \(\bar{u}-P\bar{u}\) is orthogonal to \(Ax-P\bar{u}\) then, we have

(6)

If x ∗ is a quasi-solution, then it satisfies the relation (3).

Applying (6) to both members of (3), we obtain

then

(7)

Inversely, if the relation (7) is true, then

Consequently x ∗ is a quasi-solution.

Now, let us show the existence of the L-pseudo-solution. As \(Q_{\bar{u }}\) is not empty, \(\{ \Vert Lx_{\ast }\Vert ,x_{\ast }\in Q_{ \bar{u}} \} \) is also a non-empty subset of ℝ+ which is bounded from below and hence admits a lower bound α. The lower bound property in ℝ implies

(8)

Note that (∥Lx ∗n ∥) n converges almost surely towards α. The parallelogram identity applied to Lx ∗n+p and Lx ∗n , then to \(Ax_{\ast n+p}-\bar{u}\) and \(Ax_{\ast n}-\bar{u}\) implies, for any natural integer p

(9)

and

(10)

Set

Since (x ∗n ) n is in \(Q_{\bar{u}}\), it follows from (9) and (10)

and

Hence, (Lx ∗n ) n and (Ax ∗n ) n are almost surely Cauchy sequences respectively in the Hilbert spaces \(\mathbb{G}\) and \(\mathbb{F}\). Set

Since

and A is continuous, thus

consequently \(\bar{x}_{\ast }\) is a quasi-solution of \(Ax=\bar{u}\). In addition, taking the limit in (8), we obtain \(\Vert L\bar{x}_{\ast }\Vert =\inf_{x\in Q_{\bar{u}}} \Vert Lx_{\ast }\Vert \), which means that \(\bar{x}_{\ast }\) is a L-pseudo-solution. Its unicity results from the parallelogram law and the convexity of the subset \(Q_{\bar{u}}\). □

Lemma 4

Set \(Lz_{n+1}=Lx_{n+1}-L\bar{x}_{\ast }\). If \(P\bar{u}\in R(A)\) almost surely then, the procedure (4) is equivalent to

(11)

where

(12)

Proof

Subtracting \(L\bar{x}_{\ast }\) from both sides of (4) and from Theorem 3, \(P\bar{u}\in R(A)\) almost surely implies

(13)

By iterating (13) we obtain (11). □

Lemma 5

For every \(z\in \mathbb{G}\),

where D is given by (12).

Proof

First of all, D is bounded in \(\mathbb{G}\). Show that

Assume that there exists α≥1 such that

The upper bound property implies that ∥B∥2=2, which is absurd with the hypothesis ∥B∥≤1. Thus, ∥D∥<1.

On the other hand, since

then

and

(14)

because D n is a continuous operator. Finally, we obtain the result by taking the limit in (14). □

Lemma 6

Let (a n ) n∈ℕ be a sequence of real positive number, if (a n ) n∈ℕ converges to zero, then

Proof

By expansion

we obtain the result from the convergence of (a n ) n and (∥D∥n) n to zero. Thus,

(15)

 □

Lemma 7

There exists a sequence of random variables (ζ k ) k∈ℕ such that

Proof

Let ς be a discrete random variable independent of η n such that

Moreover, we assume that random variable ς satisfies

Then, it is sufficient to take

 □

Corollary 8

(16)

Proof

This inequality follows from the expansion of the hyperbolic function into series. □

Theorem 9

Let (η n ) n be a sequence of independent and identically distributed functional random variables with zero mean on \(( \varOmega ,\mathcal{F} ) \) with values into the Hilbert space \(\mathbb{F}\), satisfying E∥η i ∥2<+∞ and Cramer’s condition (5). Let (a n ) n be a sequence of real positive numbers such that na n converges to a constant when n tends to infinity. Then the L-pseudo-solution denoted \(\bar{x}_{\ast }\) satisfies the following exponential inequality

where D is given by (12).

Proof

According to Lemma 4, we have

Triangular inequality and probability properties give

From Lemma 5 and for n large enough, we obtain

and hence,

Chernoff inequality gives

(17)

Using the independence of random variables ζ k ,k=0,1,…,n and (16), (17), we obtain

(18)

Since Eζ k =0, the expansion of the exponential function around zero gives

Besides,

Combining Lemma 7 and (5), we obtain

It is sufficient to choose

in the above inequality, to obtain

Therefore, (18) is equivalent to

(19)

The right side of (19) is minimal at

By substituting t ∗ in (19), we obtain

(20)

 □

Remark 10

From (20), we obtain

According to (15), we have

which implies the existence of a natural integer n γ such that

Or furthermore

so that the L-pseudo-solution belongs to the closed ball of center \(x_{n_{\gamma }+1}\) and radius ε with a probability greater than or equal to 1−γ.

Theorem 11

Let (η n ) n be a sequence of independent and identically distributed functional random variables with zero mean on \(( \varOmega ,\mathcal{F} ) \) with values into the Hilbert space \(\mathbb{F}\), satisfying E∥η i ∥2<+∞ and Cramer’s condition (5). Let (a n ) n be a sequence of real positive numbers such that na n converges to a constant when n tends to infinity. Let \(\bar{x}_{\ast }\) be L-pseudo-solution, then the procedure (4) converges almost completely and in quadratic means to \(L\bar{x}_{\ast }\).

Proof

The almost complete convergence follows from Theorem 9. Indeed, we have

From Cauchy series rule, follows that

is convergent. It leads that

which ensures the almost complete convergence.

It remains now to show the quadratic means convergence. From the Hilbert structure of \(\mathbb{G}\), we obtain

(21)

Taking the means (expectation) of both sides of (21), we show that its right side converges to zero.

Indeed, since the first term of (21) is deterministic, then Lemma 5 implies

D k B ∗ is linear and Eη k =0, hence

which converges to zero.

According to (15) and Cramer inequality, we obtain convergence of the third term in the right side of (21) to zero. Applying Hölder inequality to its last term gives

Since

then Lemma 6 implies

Consequently

Finally, the convergence in mean square of the procedure (4) to \(L\bar{x}_{\ast }\) results from the convergence to zero of the expectation of each term in (21) and from (11). □

5 Numerical example

The method developed in this paper is applied to a standard inverse problem of Rheology [19]. Strictly speaking, starting from the relaxation function, we will estimate the creep function. To this end, the Volterra equation of the first kind is discretized. Theoretically, the relaxation function can be determined by applying a step-strain. In experiments, the step-strain test cannot be performed due to a finite short ramp time. To overcome this difficulty, the relaxation loading is performed by imposing a constant rate in the interval times [0,t 0] where t 0 is the time ramp, followed by a constant strain. We point out that the ramp test tends to a “perfect” step-strain loading for t 0→0 Lee and Knauss [20] derived a procedure for the determination of relaxation function from ramp test. They showed that the relaxation function can be determined exactly for times t≥10t 0. This is known as the factor-of-ten rule [21]. The method is recursive and contains numerical differentiation of stress that is leading to unstable solutions. Sorvari and Malinen [22] improved this method by using numerical arguments.

Hence, the relaxation function is unknown in the interval times [0,t 0]. In addition if t 0 is relatively large then oscillations can be observed in the beginning of relaxation due to the inertia effects, inducing severe errors on the values of the relaxation function. We note that the disturbed experimental data can cause large errors in numerical interconversion and in mathematical sense the problem is said to be ill-posed [23]. Indeed, if the relaxation function is determined from experimental data and used in the procedure of interconversion to obtain the creep function, small errors in the relaxation function can produces large errors on the creep function. This is a typical feature of ill-posed problems. To obtain the creep function, Sorvari and Malinen [24] used the Tikhonov regularization method [5]. The drawback of this approach is the choice of the regularization parameter which influences significantly the solution. Many strategies have been developed in the literature to choose this parameter [4].

We recall the Volterra equation of the first kind which connects the relaxation and creep functions [25]:

(22)

G and J are the relaxation function and creep compliance respectively.

By setting t=0, an initial condition can be found between the relaxation function and creep function

To test the powerful of our method, the non-dimensional relaxation function of the Maxwell model is considered, i.e.

(23)

where \(\xi =\frac{G_{R}}{G_{1}}\), G R is the relaxed modulus, G R +G 1 is the instantaneous modulus and τ R is the relaxation time in seconds.

Substituting Eq. (23) into Eq. (22), the equation which results from it, is discretized and rearranged in the form of Eq. (1) [26], one obtains:

(24)

with

The analytical solution of the non dimensional creep function is:

(25)

In order to apply Eq. (11), one needs to check that the norm ∥D∥≤1 where D is given in Eq. (12). To this end, Eq. (24) is multiplied by a factor α which is computed so that the norm ∥I−(αA)∗(αA)∥≤1. Hence, the algorithm of Eq. (4) is implemented in commercial Package MATLAB (see the program in Appendix in the supplementary online material) and the non-dimensional creep function is computed. The obtained results are compared with the prediction of the analytic solution of Eq. (25). The plots of the non dimensional analytical creep function and results of simulation are shown on Figs. 1 to 8 (see the supplementary online material) for different values of the material parameters.

6 Conclusion

The Tikhonov regularization method is standard to solve inverse problems. The major drawback of this method is the choice of the regularization parameter. To overcome this difficulty, the Landweber iterative method is may be a solution. To this end, one takes the L-pseudo-solution as an optimal approximate solution in terms of minimal norm. The characteristic feature of iterative methods (deterministic or stochastic) is may be their numerical stability because no inversion is necessary. In a future work it is interesting to consider the case where the random variables are dependent i.e. mixing case and to study the behavior of the L-pseudo-solution in terms of consistence. This work is an attractive subject in Banach spaces. One can be interested by a possible use of these results to solve inverse problem within the non linear framework.