Regularization of ill-posed problems involving constant-coefficient pseudo-differential operators

This paper deals with the wavelet regularization for ill-posed problems involving linear constant-coefficient pseudo-differential operators. We concentrate on solving ill-posed equations involving these operators, which are behaving badly in theory and practice. Since a wide range of ill-posed and inverse problems in mathematical physics can be described and rewritten by the language of these operators, it has gathered significant attention in the literature. Based on a general framework, we classify ill-posed problems in terms of their degree of ill-posedness into mildly, moderately, and severely ill-posed problems in a certain Sobolev scale. Using wavelet multi-resolution approximations, it is shown that wavelet regularizers can achieve order-optimal rates of convergence for pseudo-differential operators in special Sobolev space both for the a priori and the a posteriori choice rules. Our strategy, however, turns out that both schemes yield comparable convergence rates. In this setting, ultimately, we provided some prototype examples for which our theoretical results correctly predict improved rates of convergence.


Introduction
The theory of pseudo-differential operators (ΨDOs) was launched by Kohn and Nirenberg around 1964. This theory has its roots in the theory of singular integrals and Fourier analysis. * Author to whom any correspondence should be addressed.
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It was subsequently clarified and extended by many prominent mathematicians, notably Hörmander. In connection with the modern theory of differential equations, the theory of ΨDOs plays a vital role and proposes a powerful and flexible way of applying Fourier analysis to the study of constant and variable-coefficient operators and singularities of distributions [12]. It turns out that the process of computing the ΨDOs leads to an ill-posed problem and thereby provides a general framework for studying a much wider class of inverse and ill-posed problems. For instance, many classical and non-classical ill-posed problems can be classified by the language of ΨDOs, including numerical differentiation [13], the Cauchy problems associated with the Laplace and Helmholtz equations [20,32], ill-posed analytic continuation problem [10,24], inverse and backward problems [7,[21][22][23][33][34][35] and so on. During the last forty years, much progress has been made on numerical ΨDOs including wavelet approximation methods, Fourier method and Galerkin methods. A suitable method that has turned out to be particularly successful in numerical experiments is the wavelet method. One central aim of this manuscript is to point out some of the underlying mathematical frameworks for this method.
Wavelet theory came out in the early 1980s as a new rigorous theory of mathematics for analyzing the non-stationary behavior of signals. Wavelets are just fashionable new waves that will soon come to rest. What has made the recent explosion of mathematical activities centered around wavelet theory are certainly a bandwagon effect, describing local features of physical phenomenon, and of course mathematical beauty. The potential of the wavelets will be one of the central themes of our discussions. Wavelet techniques are in some ways much more powerful tools than other conventional or classical techniques. It will be pointed out that the wavelet-based regularization scheme facilitates our analysis. During our analysis, we will take advantage of a certain kinds of band-limited wavelets called Meyer wavelets. They are easy to compute with fast decreasing property in the time domain and suitable for representing band-limited signals with sharp cut-off. These wavelets have compact support in the frequency domain as well as a closed form representation therein. In fact, they have good localization in the frequency domain but relatively poor localization in the time domain [5,28]. This property enables us to filter high-frequency noise injected into the signal from destructing the true solution. Such features can be employed to study some ill-posed problems that are well-understood in the frequency domain. In contrast to Daubechies wavelets, the Meyer wavelets are of special importance, mainly because they have the so-called oversampling property for which the coefficients can be calculated by sampling procedures [41,42]. Meyer wavelets are frequently used for studying a wide range of inverse and ill-posed problems including numerical analytic continuation problem [10,24], Cauchy problem for the Helmholtz and Laplace equations [20,32], backward problem in heat propagation [7,[21][22][23][33][34][35], some inverse problems involving parabolic PDEs [26] and so on.
To the best of our knowledge, the first progress in connection with ΨDO equations has been achieved by Dahmen et al [3,4] using wavelet approximation methods. In these papers, they comprehensively discussed and characterized stability and convergence of their method for the general case of variable symbols in terms of simple conditions on the Fourier transform of the generating refinable function. But in these papers, the estimates were established for the exactly given right-hand side of the operator equation, and the influence of the data errors was not investigated. This is, however, the substantial part when someone dealing with ill-posed problems. An introductory investigation on the ill-posed problems and ΨDOs has been achieved in [16]. In the year 2009, Fu et al [14] have proposed a Fourier regularization scheme for recovering the stability of numerical ΨDOs. In the year 2016, Cheng et al [2] have presented a wavelet regularization method to solve this problem. However, both methods were applied for one-dimensional ΨDOs and took advantage of an a priori approach.
All estimates were worked out in Hilbert space L 2 (R) which in some sense do not produce rigid qualitative analysis for ill-posed problems. Moreover, in the year 2017, Feng et al [11] have introduced an a posteriori wavelet regularization scheme to attack the problem. The most recent article in connection with ΨDOs is due to Hofmann et al [27], where they studied the regularization analysis of linear ill-posed equations involving certain kinds of operators called multiplication operators in the case in which the multiplier function is positive almost everywhere and zero is an accumulation point of the range of this function. Our ultimate goal is a systematic analysis of multiscale techniques, the so-called wavelet techniques for the numerical ΨDOs, which in particular includes rates of convergence for the solutions of corresponding problems.
During our investigation, it will be seen that the a priori wavelet regularization takes advantage of the regularization parameter depending on the solution smoothness and the level of noise. Although our strategy provides an order optimal rates of convergence, it asymptotically takes advantage of the minimal amount of information need to achieve the best possible error estimates. However dealing with practical problems one cannot use the a priori regularization methods, mainly because the choice of regularization parameter actually depends upon the a priori information and also the level of noise. Unfortunately, the a priori choice of information is in practice challenging, simply because the degree of smoothness is in general not precisely known. This fact motivates us to search for an a posteriori choice rule in which the regularization parameter no longer depends on the a priori information and is constructed during the algorithm. To derive some a posteriori rates of convergence we will link our strategy with the well-known Morozov's discrepancy principle.
The main goal of this manuscript is to contribute to the general mathematical analysis of the wavelet method. Some prototype examples are also provided. We give mainly a general framework to obtain stability estimates. The greater part of numerical experiments can be found in the corresponding references.
The layout of this paper is as follows. In section 2, we set off our mathematical problem as a general framework for unifying some different classes of ill-posed problems into one package by means of pseudo-differential operators. Indeed, depending on how much is the degree of ill-posedness, we classify ill-posed problems into mildly, moderately, and severely ill-posed problems. In section 3, we describe Meyer wavelet systems and their general multiresolution analysis as powerful tool to study our mathematical problem. Moreover, we introduce some auxiliary lemmas which will be used for figuring out some theoretical consequences. In sections 4 and 5, we separately discuss the wavelet regularization analysis for classified illposed problems, deriving some order optimal rates of convergence both for the a priori and a posteriori parameter choice rules. Finally, we will represent some applicable prime examples to support our qualitative analysis in section 6.

Mathematical setting
Let M : X → Y be a linear injective bounded operator between Hilbert spaces X and Y with non-closed range R(M). We study the numerical solution of inverse problems formulated as linear ill-posed operator equations where f † ∈ D(M) ⊂ X is the unknown true solution and u † ∈ R(M) ⊂ Y is the exact data which is observed approximately by u obs ∈ Y and subsequently is controlled through the following deterministic noise model Here, δ > 0 is a constant and stands for the level of noise. In order to give a qualitative picture, we will restrict our attention to constant coefficient ΨDO operators between Sobolev spaces. Basically, we assume that the forward operator M has a bounded inverse as a mapping between which means M is a smoothing operator of the order ν in the scale of Sobolev spaces. By · H q we denote the norm of Sobolev space H q (R N ) with smoothness order q ∈ R as where F : S(R N ) → S(R N ) is called the unitary Fourier operator on Schwartz space S(R N ). For some cross connections between the wavelets and general function spaces including ordinary Sobolev spaces, Besov spaces, Bessel potential spaces etc one can refer to [38,39]. There are many well-known examples of type (3) including the Radon transform (with ν = N−1 2 ), Symm's operator (with ν = 1), and more popular ΨDOs with the symbol m(·). In this paper, we consider the constant-coefficient ΨDOs described by the following form where D stands for differential operator and the complex-valued function m(·) ∈ C ∞ (R N \{0}) is called the symbol of order r ∈ R satisfying the so-called Hörmander's growth condition [3] |∂ a ξ m(ξ)| C a (1 + ξ 2 ) r−|a| , for all multi-index a ∈ N N 0 and some constant C a . In section 4, we shall confine our attention to functions m(ξ) that behave qualitatively like polynomials or homogeneous functions of ξ as ξ → ∞-that is, they grow or decay like powers of ξ , and differentiation in ξ lowers the order of growth. From (1) and (5) we have where for the sake of simplicity we usedf (ξ) := (F f )(ξ). Obviously, when the function (m(·)) −1 is unbounded the equation (7) behaves badly and the process of calculating the solution f being very complicated, mainly because the high frequency components in the given dataû(·) are uncontrollably amplified through the multiplication function m(·) and thereby it causes the blow-up in the solution. However, the equation (7) is ill-posed and naturally suffers from the so-called Hadamard instability. It will be pointed out that wide ranges of ill-posed problems can be formulated as equation (7). Indeed, different symbols, produce different kinds of ill-posed problems. It is well-known that if A : X → Y is a linear compact operator between Hilbert spaces X and Y, then there is a non-increasing infinite sequence (σ j (A)) j∈N of positive singular values which is infinite and approaches to zero when the range R(A) of A is infinite dimensional. Then the first kind linear operator equation is ill-posed and one can associate the nonnegative number ν as the degree of ill-posedness for such problems in the following sense [18] ν := sup{μ : Due to Wahba [40], the linear problem (8) is distinguished as the mildly ill-posed problem if 0 < ν < 1, the moderately ill-posed problem if 1 ν < ∞, and the severely ill-posed problem if ν = ∞ (i.e., no such μ exists such that σ j (A) = O( j −μ ) as j → ∞). We know that the linear operator equation (8)  However, due to Nashed [30], not all ill-posed problems are related to compact operators and that the distinction between the ill-posedness of linear operator (8) is of type I and type II. In this sense, the linear ill-posed operator equation (8) is called of type I if A is not compact and of type II if A is compact. Note that the difference between the type I and type II of ill-posed problems does not reflect the fact that how much difficult it is to solve an ill-posed problem in a realistic situation. As the operator M in equation (7) is not compact, the ill-posedness of its corresponding operator equation is of type I. Unfortunately, in this case it is not quite easy to characterize the degree of ill-posedness based on the approach (9) by a single constant ν. However, we make a cross connection between these traditional approaches and our idea concerning the degree of ill-posedness.
A similar approach is mentioned in [6] where a pair of characteristics including smoothing order and smoothing potential are introduced to quantify the smoothing properties of a convolution operator A. In this framework, a further approach based on both the smoothing properties of the operator A and the solution smoothness for classifying linear illposed problems is used. In the sequel, motivated by the framework defined in [6] we introduce a quantitative measure of the degree of ill-posedness to classify some different kinds of ill-posed problems which come out from the operator equation (7) in the following sense: Note that this quantity can also be measured as a lower limit as It can be inferred from this definition that the smoother the forward operator, the higher the degree of ill-posedness of associated problem. Thus, the smoother the symbol m(·), the greater losses of information in the solution process and errors in the solution.

Remark 1.
In Hilbert scales there is another way to describe the quantitative measure of ill-posedness [8]. To be more precise, the operator equation A f = u with bounded operator A between Hilbert spaces X and Y is ill-posed with a degree ν > 0 in the sense that there exists two constants 0 < c C < ∞ such that for all q ∈ R, we have the following In this way, the corresponding operator equation is called mildly ill-posed if 0 < ν < 1, moderately ill-posed if 1 ν < ∞, and severely ill-posed if ν = ∞.
Our mathematical tool to treat these different ill-posed problems in one mathematical framework is the method based on the Meyer wavelet systems. The mathematical introductory to the Meyer wavelet systems is discussed in the next section.

Meyer wavelet systems
In late 1986, Meyer and Mallat recognized that the construction of different wavelet systems can elegantly be described by the so-called multi-resolution analysis (MRA) [28]. This is a general framework in which functions f ∈ L 2 (R N ) can be traced as a limit of successive approximations where each one is a smoother version of the function f . A natural way to construct N-dimensional MRA is the tensor product [5,25,28,43]. To work out this construction, we first introduce one-dimensional Meyer's wavelet and scaling functions. The Meyer scaling function is described by the dilation equation as It is also used to construct the mother wavelet of Meyer where d n = γ 1−n (−1) n . The most natural way to pass from the one-dimensional setting to the multi-dimensional setting is to use the tensor product. Using this general concept, the N−dimensional Meyer scaling function can be defined as Similarly, the corresponding Meyer wavelet function is defined by Let V J be the closed linear span defined by This space is also called the scale space. Similarly, we denote by W J the closed linear span, called the detail space, and is defined by In frequency space, the Meyer scaling function Φ(·) has compact support in the following form The orthogonal projection operators P J : tively defined by the following relations, and where ·, · indicates the L 2 -inner product. The basic tenet of MRA is that whenever a set of nested closed subspaces like V J fulfills the principles of MRA, then there exists an orthonormal wavelet basis for L 2 (R N ). To be more precise, the family of parametrized func- For the convenient purposes, we set Utilizing (19), for J ∈ N, we get where Γ J := R N \Λ J . Also, we have Another important operator to develop our approach is M J : where χ J denotes the characteristic function of the cube Λ J . We also recall some useful properties as Moreover, it can be seen that from equation (19) where the multi-index κ satisfies |κ| m and C N depends only on N.
In what follows, we recall the well-known Jackson and Bernstein inequalities.
Theorem 4 (Jackson's inequality) ( [17,28]). Let {V j } j∈Z be an m-regular MRA. Then for J ∈ N and ϕ ∈ H q (R N ), the following inequality holds: where C = C(p, q) is a positive constant depending on real constants p, q satisfying −m < q < p < m.
Theorem 5 (Bernstein's inequalities) ( [17,28]). Let {V j } j∈Z be an m-regular Meyer's MRA, and suppose −m < q < q + r < m. Then where D x i is the differential operator and C 0 is a positive constant.
where C 1 is a positive constant.
Then there exists a positive constant C such that for all q ∈ R, J ∈ N and ϕ ∈ V J , the following inequality is satisfied whereC is a positive constant.
In the next section, we introduce our regularization strategy based on the a priori and the a posteriori choice rules. These rules are multi-level approaches in terms of wavelet systems.

Wavelet regularization analysis: mildly and moderately cases
In this section, we first analyze the cases mildly and moderately ill-posed problems in one scenario. Here, we give some theoretical consequences based on the a priori and a posteriori choice rules.

The a priori choice rule
Consider the pseudo-differential operator M : where ν > 0 is the degree of smoothness for the ΨDO M. We assume that the following conditions are hold: where M is a positive non-dimensional a priori bound. This condition sometimes is called the smoothness condition. C3 Assume that the symbol of the operator M is continuous and there exist two positive constants α and β such that or where ν > 0 and ξ = 0.
These conditions describe the standard setup for studying ill-posed problems. According to definition 1, μ(M) = ν for which 0 < ν < 1 yields the mildly and for 1 ν < +∞ the moderately ill-posed case. Based on equation (7), we define the operator R := M −1 by Ru † := f † . Then the following lemma shows that the solution of the pseudo-differential operator equation (35) in the case of mildly and moderately ill-posedness is stable in the sense that it depends continuously on the data within the Meyer MRA.

Lemma 7 (Stability).
Let {V j } j∈Z be an m-regular Meyer MRA. Then for all ϕ ∈ V J , with J ∈ N the following stability estimate holds where α andC are the same constant appeared in condition C3 and lemma 6, respectively.
Proof. For all ϕ ∈ V J , we have where we have used the lemma 6.
In what follows, we introduce a regularizer based on the Meyer wavelet as where J stands for the regularization parameter. This parameter plays an essential role in recovering the regularized solution. It controls the compromise between stability and approximation. The next theorem provides an asymptotic order optimal error estimate for the solution of pseudo-differential operator equation (1).

Theorem 8 (Convergence rate of the a priori rule).
Suppose that the conditions C1, C2 and C3 hold true. Then with where u † ∈ V J opt and [[a]] denotes the largest integer not exceeding a.
Proof. The fundamental or total error can be estimated as follows where A1 := Ru † − R J u † H q is called the approximation error and B1 := R J u † − R J u obs H q is called the propagated error. For quantity A1 we have where A2 := Ru † − P J Ru † H q and B2 := P J Ru † − P J RP J u † H q . Applying (28) and Jackson's inequality, we have For quantity B2 we have where A3 := Ru † H q (Γ J+1 ) and B3 := RQ J u † H q (Λ J+1 ) . Simple calculations reveal that Also, for quantity B3 we get Combining A3 and B3 with B2, we arrive at Adding up quantities A1 and B1, one can estimate quantity A1 as Now, we turn back to quantity B1. Using condition C1 and lemma 6, we have Consequently, the fundamental error can be estimated as where C † := max{ N αC , 1 + C + NC β α }. Now, it is necessary to minimize the right-hand side of inequality (52). To this end, we first suppose that J ∈ R and setting It is clear that J opt := J min = log 2 (p−q)M νδ 1 p−q+ν is a minimizer for the function ϑ(·). As J ∈ N we take It follows, Therefore, where C(p, q, ν) : Eventually, setting C ‡ := C † C(p, q, ν) we get the desired result.
As mentioned in condition C1, in the case q > 0 it is not consistent to demand u † − u obs H q δ, simply because the measured data u obs are belong to L 2 (R N ). Therefore, from theorem 8 we have the following consequence we have where u † ∈ V J opt and [[a]] denotes the largest integer not exceeding a.
This corollary states that when q = 0, then p must be positive which in turn is equivalent to the raise of demand for the smoothness degree of the exact solution f † . In fact, the larger p is, the more restrictive is the condition C3.

Remark 2.
It is important to estimate Ru † − R J opt u obs Hq in the Hq-scale forq q. But in this case we obtain nothing new as u † − u obs Remark 3. If 0 < q < N 2 and q N 2 − N p for 2 < p < ∞, then it follows from the continuity of the embedding H q (R N ) → L p (R N ) that under the assumption of theorem 8 we also have an error bound with respect to the L p −norm (59)

The a posteriori choice rule
As is known from the previous subsection, the regularization parameter generally depends upon the noise level and the a priori information. But precisely determining the a priori information is generally impossible in practice. Thus processing data with wrong a priori information will result in an unwanted regularized solution. In this subsection, we will introduce the a posteriori approach where the regularization parameter no longer depends on the a priori information and the solution smoothness but in turn is certainly constructed during the algorithm. The structure of this approach is based on the Morozov discrepancy principle. Based on this principle, we first introduce the following consequence.

Lemma 10.
Suppose that the conditions C1, C2, and C3 hold true. Moreover, suppose that the regularization parameter, J, is selected such that where θ > 1 is a constant. Then there holds the following inequality Proof. It follows from the triangle inequality where N1 := u † H q (Γ J−1 ) and N2 := Q J−1 u † H q (Λ J−1 ) . These quantities can separately be evaluated as From (27), it follows Therefore, On the other hand, Combining (65) with (66), the desired consequence is derived.
Note that according to [8] in the a posteriori choice rule, the a priori information bound M is no longer required for the choice of the regularization parameter. It is just required for the theoretical analysis of the convergence rates of the regularized solution. The next theorem is the main result and provides the a posteriori convergence rate of the Hölder-type.

Theorem 11 (Convergence rate of the a posteriori rule).
Suppose that the conditions C1, C2, and C3 are hold. Moreover, suppose that the regularization parameter, J, is selected such that where θ > 1 is a constant. Then there holds (68)
The equation (I − P J )u obs H q = θδ for θ > 1 is uniquely solvable, provided u − u obs H q δ < u obs H q with q 0. In practice, the condition u obs H q > δ certainly makes sense since otherwise the right-hand side would be less than the noise level δ, and R J u obs = 0 would be an acceptable approximation to Ru † [8]. However, if the equation has multiple solutions, we will understand J as its minimal solution.

Wavelet regularization analysis: severely ill-posed case
In this section, we present both the a priori and a posteriori wavelet regularization analysis for solving the pseudo-differential operator equation in the severely ill-posed case.

The a priori choice rule
For σ > 0, consider the problem of solving pseudo-differential operator equation associated with the parameter y ∈ [0, σ] by where M y : H q (R N ) → H q (R N ) and in terms of the Fourier transform is described by Here, D stands for differential operator, and m y (·) : R N −→ C is called the symbol of M y . This symbol will behave like an exponential function as ξ → ∞. However, the situations where the severely and exponentially ill-posed problems can play a role are not the same and do not describe a similar concept. For more information about the exponentially ill-posed problems one can see [19]. We assume that the following conditions hold true: Since the observed data u obs are generally in L 2 (R N ). H2 There exists p > q such that −m < q < p < m and f † σ H p M, where M is a positive non-dimensional a priori bound. H3 There exists two positive constants α and β such that where γ and ν are two positive parameters.
Note that in the case of severely ill-posed problems the symbol m y (·) cannot belong to Hörmander's symbol class (see e.g., [3]), mainly because the symbol m y (·) does not satisfy the Hörmander's growth condition (6). In this case the corresponding operator cannot be considered as a constant-coefficient ΨDO but rather a Fourier multiplier operator. The operator equation (81) can be formulated in the frequency space as According to definition 1, the degree of ill-posedness μ(M y ) = +∞. Because no finite power ν exists. So the problem (81) is severely ill-posed and the smoother the symbol of the linear pseudo-differential operator M y (D), the higher the degree of ill-posedness of the associated problem (81). Therefore, the smoother the symbol, the greater losses of information in the process of finding the solution and creeping errors into the solution. However, the drawbacks coming up here can be treated within the language of wavelet analysis. It turns out that for this general class of ill-posed problems there are many order optimal convergence rates. Similar to previous section and to dig these results, we consider the operator equation R y u † = f † y , where R y := M −1 y . Then the following stability estimate holds true within the Meyer MRA. Lemma 12 (Stability). Let {V j } j∈Z be an m-regular Meyer MRA in L 2 (R N ). Then for all ϕ ∈ V J , with J ∈ N the following stability estimate holds where α andC are the same constant appeared in condition H3 and lemma 6, respectively.
Remark 5. Lemma 12 shows that for the general class of severely ill-posed problems formulated in terms of pseudo-differential operators there exists a certain type of stability within the Meyer MRA. That means the solution of the problem continuously depends upon the data.
To establish some order optimal convergence rates, we introduce the wavelet regularizer as follows: for all J ∈ N and y ∈ [0, σ]. To work out some order optimal convergence rates, we state the following auxiliary lemma.

Lemma 13 ([37]). Let ρ : [0, a] −→ R be defined by
where 0 < a < 1, b, d ∈ R + and c ∈ R. Then the function ρ(·) is invertible and has the following inverse form In detail, we will show that how the Meyer wavelet theory can recover the exact solution in such away that the stability property being preserved. The next theorem will show this argument. rate of the a priori rule). Let u † ∈ V J with J ∈ N. Suppose also the conditions H1, H2 and H3 hold true. Then with

Theorem 14 (Convergence
the following order optimal convergence rate is satisfied where [[a]] denotes the largest integer not exceeding a.
Proof. By the triangle inequality the fundamental error can be estimated as follows where A1 := R y u † − R y,J u † H q and B1 := R y,J u † − R y,J u obs H q . For quantity A1, we have where A2 := R y u † − P J R y u † H q and B2 := P J R y u † − P J R y P J u † H q . Applying Jackson's inequality and (28), we get To evaluate B2, we use (24) and obtain where A3 := R y Q J u † H q (Λ J+1 ) and B3 := R y u † H q (Γ J+1 ) . First we estimate quantity A3 by applying lemma 6 as where we have used the fact that Q J u † ∈ V J+1 . On the other hand, we have for quantity B3 Combining now quantities A3 and B3 with B2, we arrive at Also, inserting quantities A2 and B2 into A1, we get Finally, quantity B1 can simply be estimated through lemma 6 as follows Consequently, the fundamental error is worked out as where C † := max{ N αC , (1 + C + NC) β α }. Setting κ(J) := exp(γy2 Jν )δ + exp γ(y − σ)2 Jν 2 −J(p−q) M and η := e −γ2 Jν such that η ∈ (0, 1), we can find a minimizer for the function κ(J). In fact, by minimizing κ(J), we have Therefore, δ M = η σ 1 γ ln 1 η − p−q ν . According to lemma 13 we conclude b = σ, d = 1 γ and c = p−q ν . Therefore, we have for M δ → 0. Eventually, the regularization parameter J min is worked out as Since J = J min ∈ N, we set J opt := [[J min ]] as Therefore, the optimal value for the function κ(·) is worked out as for δ → 0. The fundamental error is ultimately estimated as for δ → 0.
According to condition H1, it is not appropriate to ask u − u obs H q δ with q > 0 since generally the measured data u obs live in Hilbert space L 2 (R N ). Hence if we replace u − u obs H q δ by u − u obs L 2 δ, we have the following result Corollary 15. Let u † ∈ V J with J ∈ N and the conditions H1, H2 and H3 be fulfilled for 0 < p < m. Then with the following asymptotic order optimal convergence rate holds true where [[a]] denotes the largest integer not exceeding a.
Especially, at y = 0 the fundamental error, R y u † − R y,J opt u obs L 2 , is of order optimal O(δ). Remark 6. In practice, the a priori information M is not known exactly. Hence if one take then there holds

The a posteriori choice rule
As is known from the previous section, the regularization parameter generally depends upon the noise level and the a priori information. But precisely determining the a priori information is generally impossible in practice. Thus a special strategy involving the a posteriori rule is necessary to achieve a better error estimates in this case. However, in this section, we will introduce an a posteriori approach where the regularization parameter no longer depends upon the a priori information and the solution smoothness. The structure of this approach is based on the Morozov discrepancy principle.
Lemma 16. Suppose that the conditions H1, H2, and H3 hold true. Moreover, suppose that the regularization parameter, J, is selected such that where θ > 1 is a constant. Then there holds the following inequality Proof. It follows from the triangle inequality where N1 := u † H q (Γ J−1 ) and N2 := Q J−1 u † H q (Λ J−1 ) . These quantities can separately be evaluated as From (27), it follows Therefore, On the other hand, Finally, relations (117) and (118) will give us the main consequence.
The next theorem is the main result and provides the a posteriori convergence rate of the Hölder-logarithmic type.
Theorem 17 (Convergence rate of the a posteriori rule). Let the conditions H1, H2, and H3 be fulfilled. Moreover, suppose that the regularization parameter, J, is selected such that where θ > 1 is a constant. Then there holds (120) Proof. From triangle inequality, we have where Q1 := R y u † − R y,J u † H q and Q2 := R y,J u † − R y,J u obs H q . From lemma 6 and (113), we have If we take J = 1 ν log 2 ln 2βM δ(θ−1) , then after some simple calculations quantity Q2 can be estimated as For quantity Q1, we arrive at where Q3 := (I − P J )R y u † H q and Q4 = R y (I − P J )u † H q . Using (28) and the Jackson inequality, we obtain Applying J and doing some calculations, we get Employing the Hölder inequality, quantity Q4 is estimated as It can readily be seen that Inserting J into (127) and then simplifying, we arrive at Plugging (126) and (129) into (124), we obtain Combining now quantities (123) and (130) with (121), we ultimately get where This completes the proof.

Remark 7.
On comparing the convergence rates for all type of ill-posedness, one is thus come to the conclusion that all rates of convergence derived by the a priori and the a posteriori regularization approaches are generally of comparable accuracy and no meaningful discrepancy is observed. But it should be pointed out that evaluating and working with the a posteriori regularization approach is more economical in compared to the a priori approach; mainly because the regularized solution is independent of the a priori information, M, and the degree of smoothness, p.

Applications
In this section we discuss the capability of the presented method to solve inverse and ill-posed partial differential equation problems containing parabolic and elliptic problems, numerical fractional differentiation, the analytic continuation problem, and one source problem.
As m(ξ) = O( ξ −2 ), the degree of ill-posedness is μ(S) = 2 which means that the problem of source identification is moderately ill-posed. To establish the rate of convergence, we suppose that for some q 0 the deterministic noise model, φ † − φ obs H q δ, and the smoothness condition, f (·) H p M hold true for some q + 2 < p. We consider the regularization operator as If one take the following wavelet regularization parameter then the following a priori error estimate is satisfied . Also, the following rate of convergence by the a posteriori choice rule is established as follows where C * := C2 p−q + 2NC θ−1 ( θ−1 4 ) p−q p−q+2 + ( θ+1 2 ) p−q p−q+2 .

Applications for severely ill-posed problems
Example 3 (Analytic continuation problem) (see [10,24]). Let f (z) = f (x + iy) be a complex-valued analytic function on the high-dimensional infinite complex domain Ω defined by Ω := {x + iy ∈ C N : x ∈ R N , y y 0 , y, y 0 ∈ R N + }, here the symbol · indicating the Euclidean norm. The central goal is to extend the function f (z) analytically from indirectly measured data of the known function Φ † (x) to the whole complex domain Ω such that f (z)| y=0 = Φ † (x). The analytic continuation problem (ACP) has attracted many research activities including many practical physical applications, for instance, inverse Laplace integral transform [1], inverse scattering problems [29], medical imaging [9,31]. Applying the Fourier transformation about the variable x in the frequency domain we get the operator equation A y f (· + iy)(ξ) = f (· + i0)(ξ), or equivalently, the following pseudo-differential operator equation A y f (· + iy)(ξ) = Φ † (ξ), A y f (· + iy)(ξ) = e y·ξ f (· + iy)(ξ), where ξ ∈ R N − . Obviously, m y (ξ) = e y ξ cos(Θ) for cos(θ) < 0. According to condition H3, y = y ∈ [0, y 0 ], γ = −cos(Θ), ν = 1, and α = β = 1. The degree of ill-posedness, in this case, is μ(A y ) = +∞ and hence the ACP is severely ill-posed. We suppose that the measured data, Φ obs , satisfies Φ † − Φ obs H q δ for some q 0. Also, consider the smoothness condition f (· + iy 0 ) H p M for some q p. We introduce the wavelet regularization operator in the following sense R AC y,J := P J R AC y P J , R AC y := A −1 y , where the regularization parameter is chosen as Then there holds the following asymptotic a priori error estimate for δ → 0. Also, there holds the following asymptotic a posteriori error estimate where k(t) ∈ C([0, T]) and g T (·) ∈ L 2 (R N ) respectively denote the positive thermal conductivity and the terminal distribution [21]. By the technique of Fourier transform we obtain in the frequency space the following operator equation B tû (ξ, t) =û(ξ, T) or equivalently the pseudo-differential operator equation