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On the existence of global saturation for spectral regularization methods with optimal qualification

A family of real functions {g_\alpha} defining a spectral regularization method with optimal qualification is considered. Sufficient condition on the family and on the optimal qualification guaranteeing the existence of saturation are established. Appropriate characterizations of both the saturation function and the saturation set are found and some examples are provided.


Introduction
Let X, Y be infinite dimensional Hilbert spaces and T : X → Y a bounded linear operator with nonclosed range R(T ). It is well known that under these conditions T † , the Moore-Penrose generalized inverse of T , is unbounded ( [1]) and therefore the linear operator equation is ill-posed. The Moore-Penrose generalized inverse can be used to define the least squares solutions of (1). In fact equation (1) has a least squares solution if and only if y ∈ D(T † ) . = R(T ) ⊕ R(T ) ⊥ and in that case, x † .
= T † y is the least squares solution of minimum norm and the set of all least-squares solutions of (1) is given by x † + N (T ). Since T † is unbounded, x † does not depend continuously on the data y. Therefore, if instead of the exact data y, a noisy observation y δ is available, y δ = T x + δξ, where the noise ξ is assumed to be bounded, ξ ≤ 1, then it is possible that T † y δ does not even exist and if it does, it will not necessarily be a good approximation of x † ( [9], [10]). This instability becomes evident when trying to approximate x † by traditional numerical methods and procedures.
Ill-posed problems must be first regularized if one wants to successfully attack the task of numerically approximating their solutions. Regularizing an ill-posed problem such as (1) essentially means approximating the operator T † by a parametric family of bounded operators {R α }, where α is the so called "regularization parameter". If y ∈ D(T † ), then the best approximate solution x † of (1) can be written as x † = T 2 + 0 1 λ dE λ T * y where {E λ } is the spectral family associated to the operator T * T (see [1]). This is mainly why many regularization methods are based on spectral theory and consist of defining R α . = T 2 + 0 g α (λ) dE λ T * where {g α } is a family of functions appropriately chosen such that for every λ ∈ (0, T 2 ] there holds lim α→0 + g α (λ) = 1 λ . It is important to emphasize however that no mathematical trick can make stable a problem that is intrinsically unstable. Whatever the case, there is always loss of information. All a regularization method can do is to recover the largest possible amount of information about the solution of the problem, maintaining stability. It is often said that the art of applying regularization methods consist always in maintaining an adequate balance between accuracy and stability. Usually accuracy can be improved with increasing assumptions (or information) on the regularity of the exact solution. In 1994, however, Neubauer ([7]) showed that certain spectral regularization methods "saturate", that is, they become unable to continue extracting additional information about the exact solution even upon increasing regularity assumptions on it. In his article, Neubauer introduced for the first time the idea of the concept of "saturation" of regularization methods. Saturation is however a rather subtle and complex issue in the study of regularization methods for inverse ill-posed problems and the concept has, for many years, escaped rigorous formalization in a general context. Neubauer's idea referred to the best order of convergence that a method can achieve independently of the smoothness assumptions on the exact solution and on the selection of the parameter choice rule. In 1997, Neubauer ([8]) showed that this saturation phenomenon occurs for instance in the classical Tikhonov-Phillips method. Later on, in 2004, Mathé ([5]) proposed a general definition of the concept of saturation for spectral regularization methods. However, the concept of saturation defined by Mathé is not applicable to general regularization methods and it is not fully compatible with the original idea of saturation proposed by Neubauer in [7]. In particular, for instance, the definition of saturation given in [5] does not imply uniqueness and therefore, neither a best global order of convergence. More recently, in 2011, Herdman, Spies and Temperini (see [3]) developed a general theory of global saturation for arbitrary regularization methods, formalizing the original and intuitive idea first outlined by Neubauer in 1994 ([7]).
Related in a dual way to the concept of saturation is the concept of qualification of a spectral regularization method, introduced by Mathé and Pereverzev in 2003 ( [6]). This concept is strongly related to the optimal order of convergence of the regularization error, under certain "a-priori" assumptions on the exact solution. In 2009 Herdman, Spies and Temperini ( [2]) generalized the concept of qualification and introduced three hierarchical levels of it: weak, strong and optimal qualification. There, it was shown that the weak qualification generalizes the definition introduced in [6].
In this work, some light on the existence of saturation for spectral regularization methods with optimal qualification is shed. In particular, sufficient conditions on the family of real functions {g α } defining the method and on the optimal qualification ρ, which guarantee the existence of saturation, are established. Moreover, in those cases, appropriate characterizations of both the saturation function and the saturation set are provided.

Preliminaries
In this section we shall recall some basic concepts on global saturation of regularization methods for inverse ill-posed problems theory (for more details see [3]). In the sequel, T : X → Y will be a bounded linear operator with non-closed range between two Hilbert spaces X and Y . Without loss of generality we will assume that the operator T is invertible (in the context of inverse problems it is customary to work with the Moore-Penrose generalized inverse of T since one seeks least squares solutions of the problems; therefore the lack of injectivity of T is never a relevant issue). Also, for simplicity of notation and unless otherwise specified, we shall assume that all subsets of the Hilbert space X under consideration are not empty and they do not contain x = 0.
Let M ⊂ X. We shall say that a function ψ : X × IR → IR belongs to the class F M if there exists a = a(ψ) > 0 such that ψ is defined in M × (0, a), with values in (0, ∞) and it satisfies the following conditions: ψ(x, δ) = 0 for all x ∈ M , and 2. ψ is continuous and non-decreasing as a function of δ in (0, a) for each fixed x ∈ M .
One may think of F M as the collection of all possible δ-"orders of convergence" on the set M .
be a family of regularization operators for the problem T x = y. The "total error of {R α } α∈(0,α 0 ) at x ∈ X for a noise level δ" is defined as Note that E tot {Rα} is the error in the sense of the largest possible discrepancy that can be obtained for an observation of y within noise level δ, with an appropriate choice of the regularization parameter α.
for all x ∈ M and for every δ ∈ (0, r).
be a family of regularization operators for the problem T x = y, M ⊂ X and ψ ∈ F M . We say that ψ is an "upper bound of convergence for the total error of With U M (E tot {Rα} ) we shall denote the set of all functions ψ ∈ F M that are upper bounds of convergence for the total error of {R α } α∈(0,α 0 ) on M .
The following two definitions formalize certain comparisons of bounds of convergence on different sets of X, which will be needed later to introduce the concept of global saturation. Next we recall the concept of global saturation introduced in [3].
. It is said that ψ S is a "global saturation function of {R α } over M S " if ψ S satisfies the following three conditions: S3. There is no upper bound of convergence for the total error of {R α } that is a proper extension of ψ S (in the variable x) and satisfies S1 and S2, that is, there exist noM M S andψ ∈ UM (E tot {Rα} ) such thatψ satisfies S1 and S2 with M S replaced byM and ψ S replaced byψ.
The function ψ S and the set M S are refer to as the saturation function and the saturation set, respectively.
This conception of global saturation essentially establishes that in no point x * ∈ X, x * = 0, can exist an upper bound of convergence for the total error of the regularization method that is "strictly better" than the saturation function ψ S at any point of the saturation set M S .
The following definitions will be needed both to recall the concept of qualification as introduced in [2], as well as in the rest of the article.
We denote with O the set of all non-decreasing functions ρ : IR + → IR + such that lim α→0 + ρ(α) = 0 and with S the set of all continuous functions s : IR + 0 → IR + 0 satisfying s(0) = 0 and such that s(λ) > 0 for every λ > 0. Note that if s ∈ S is non-decreasing, then s is an index function in the sense of Mathé-Pereverzev ( [6]).
We say that "ρ andρ are equivalent at the origin" and we denote it with ρ ≈ρ, if they precede each other at the origin, that is, if there exist constants ε > 0, Clearly "≈" is an equivalence relation and it introduces in O a partial ordering. Analogous definitions and notation will be used for s,s ∈ S.
iii) We say that (ρ, s) is an "order-source pair for {g α }" if there exist a constant γ > 0 and a function h In the context of the previous definitions we refer to the function ρ as the "order of convergence" and to s as the "source function".
We are now ready to define the concept of qualification in its three different levels as it was introduced in [2].
ii) We say that ρ is "strong qualification of {g α }" if there exists a function s such that (s, ρ) is a strong source-order pair for {g α }.
iii) We say that ρ is "optimal qualification of {g α }" if there exists a function s such that (s, ρ) is a strong source-order pair for {g α } and (ρ, s) is an order-source pair for {g α }.
Note that since condition (4) implies condition (3), in the definition of optimal qualification above the requirement that (s, ρ) be strong source-order pair can be replaced by the one that (s, ρ) be a weak source-order pair. Now given the SRM {g α } and ρ ∈ O, we define Note that s ρ (0) = 0 and if s ρ is continuous, s ρ ∈ S. The next theorem provides necessary and sufficient condition, in terms of s ρ , for an order of convergence ρ ∈ O to be optimal qualification. Theorem 2.11. ( [2]) Let {g α } be a SRM and ρ ∈ O such that s ρ ∈ S. Then ρ is optimal qualification of {g α } if and only if s ρ verifies (4) and The next theorem shows the uniqueness of the source function. The following converse result, where regularity properties of the exact solution are derived in terms of the rate of convergence of the regularization error, will be needed later. This result states that if the regularization error has order of convergence ρ(α) and (ρ, s) is an order-source pair, then the exact solution belongs to the source set given by the range of the operator s(T * T ).

Saturation of spectral regularization methods with optimal qualification
The purpose of this section is to shed some light on the saturation of SRM with optimal qualification. More precisely, we will establish sufficient conditions on the family of functions {g α } and on the optimal qualification ρ guaranteeing the existence of saturation. Moreover, for those methods we will provide appropriate characterizations of both the saturation function and the saturation set. Then, let {g α } α∈(0,α 0 ) be a SRM and consider the following hypothesis: x ∈ X s and δ ∈ (0, Θ(α 0 )). Then ψ ∈ F X s and, moreover, ψ is an upper bound of convergence for the total error of {R α } α∈(0,α 0 ) on X s , that is, Proof. Since ρ is continuous and non-decreasing and ρ(0 + ) = 0 it follows that Θ(t) is continuous and strictly increasing on (0, +∞) with Θ(0 + ) = 0. Therefore Θ −1 exists and has the same properties. It then follows that ψ is continuous and non-decreasing as a function of δ in (0, Θ(α 0 )) for each fixed x ∈ X s , and ψ(x, 0 + ) = 0 for all x ∈ X s . Hence ψ ∈ F X s . On the other hand, since (s, ρ) is a weak source-order pair for {g α } α∈(0,α 0 ) , there exist positive constants c andα such that s(λ) |r α (λ)| ≤ cρ(α) for every α ∈ (0,α), λ ∈ (0, T 2 ]. Moreover, from hypothesis H2 and the fact that ρ is non-decreasing it follows that the previous inequality holds (perhaps with a different positive constant c) for every α ∈ (0, α 0 ), that is, Now, for every p ≥ 0 we define the source sets X s,p . = {x ∈ X : x = s(T * T )ζ, ζ ≤ p}. Then for each x ∈ X s there exists p x > 1 such that x ∈ X s,px . On the other hand, since Θ is continuous and strictly increasing in (0, α 0 ), there exists a uniqueα x ∈ (0, α 0 ) such that x ∈ X s,px and Θ(α x ) = δ px . Therefore, Now, since y δ = T x + δξ, ξ ≤ 1 and x = s(T * T )ζ with ζ ≤ p x , it follows immediately that Since Θ(α . Hence by virtue of (9) one has that where the last inequality follows from the fact that p x > 1 and both ρ and Θ −1 are non-decreasing functions. From (8) and (10) it follows that for every δ ∈ (0, Θ(α 0 )), This proves that ψ ∈ U X s (E tot {Rα} ). (Saturation for FSRO with optimal qualification.) Let {g α } α∈(0,α 0 ) be a SRM satisfying hypothesis H4 and having optimal qualification ρ, r α (λ) = 1 − λg α (λ), R α = g α (T * T )T * and suppose that s ρ ∈ S (where s ρ is as defined in (5) ). Assume further that the following hypotheses hold: a) The function ρ is of local upper type β, for some β ≥ 0. b) There exist positive constants γ 1 , γ 2 , λ * , c 1 , with λ * ≤ T 2 and c 1 > 1 such that where h is as in (4). (Note that by virtue of Theorem 2.12 and the fact that s ρ ∈ S, there exists only one function s ∈ S satisfying (4), that is, s = s ρ .) iii) |r α (λ)| is non-decreasing with respect to α for each λ ∈ (0, To prove this theorem we will need three previous lemmas. The first one is a somewhat technical result, the second one deals with the existence of an a-priori parameter choice rule leading to a worst total error having an appropriate order of convergence, while the third one is a converse result. Proof. Let {E λ } be the spectral family of T * T . It suffices to show that for every α ∈ (0, α 0 ), x ∈ X, the function r −2 α (λ) is integrable with respect to the measure d E λ x 2 . Let α ∈ (0, α 0 ) fixed. Since (ρ, s) is an order-source pair for {g α }, there exist a constant γ > 0 and a function h : (0, α 0 ) → IR + with lim Now, since α ∈ (0, α 0 ), it follows from hypothesis b.ii) of Theorem 3.3 that r α (λ) ≥ γ 1 > 0 for every λ ∈ [0, h(α)). Then From (11) and (12) it follows that Lemma 3.5. Let {g α } be a SRM, R α = g α (T * T )T * , (ρ, s) an order-source pair for {g α } and assume that hypotheses b.ii) and b.iii) of Theorem 3.3 hold. Let ϕ : (0, +∞) → IR + be a continuous, nondecreasing function satisfying lim δ→0 + ϕ(δ) = 0 and x * ∈ X, x * = 0.
for t > 0, and suppose that: a) The function ρ is of local upper type β, for some β ≥ 0. b) There exist positive constants γ 1 , γ 2 , λ * , c 1 , with λ * ≤ T 2 and c 1 > 1 such that and c ≥ 1 such that λ n ↓ 0 and λn λ n+1 ≤ c for every n ∈ IN. If for some x ∈ X we have that then x ∈ R(s(T * T )). In particular, if ρ is optimal qualification of {g α } and s ρ ∈ S, then x ∈ R(s ρ (T * T )).
Proof. Without loss of generality we may assume that α 0 ≤ λ * c 1 and x = 0 (this is so because hypotheses a) and c) are independent of α 0 and if b) holds for α ∈ (0, α 0 ) then it holds for α ∈ (0,α 0 ) for everyα 0 < α 0 with the same constants, while if x = 0 the result of the Lemma is trivial).
where Gᾱ . = F c 1ᾱ − Fᾱ with {F λ } being the spectral family associated to T T * and if GᾱT x = 0, arbitrary with Gᾱz = 1, in other case.
Remark 3.7. Note that since sup Having stated and proved the previous three lemmas, we are now ready to prove Theorem 3.3.
Also, since ψ is trivially invariant over X sρ , ψ does not depend on x. Thus, it satisfies condition S2.

Examples
Although the main results of this article are very theoretical in nature, we provide below two examples of regularization methods with optimal qualification which do possess saturation. In both cases the saturation function and saturation set are found.

Conclusions
In this article families of real functions {g α } defining a spectral regularization methods with optimal qualification were considered. Sufficient conditions on the family and on the optimal qualification guaranteeing the existence of saturation were found. Appropriate characterizations of both the saturation function and the saturation set were given and two examples were provided.