Generalized Qualification and Qualification Levels for Spectral Regularization Methods
GGeneralized Qualification and Qualification Levelsfor Spectral Regularization Methods
T. Herdman, R. D. Spies and K. G. Temperini
Abstract.
The concept of qualification for spectral regularization methods (SRM)for inverse ill-posed problems is strongly associated to the optimal order of convergenceof the regularization error ([2], [5], [6], [11]). In this article, the definition ofqualification is extended and three different levels are introduced: weak, strong andoptimal. It is shown that the weak qualification extends the definition introducedby Math´e and Pereverzev ([6]), mainly in the sense that the functions associatedto orders of convergence and source sets need not be the same. It is shown thatcertain methods possessing infinite classical qualification, e.g. truncated singularvalue decomposition (TSVD), Landweber’s method and Showalter’s method, alsohave generalized qualification leading to an optimal order of convergence of theregularization error. Sufficient conditions for a SRM to have weak qualification areprovided and necessary and sufficient conditions for a given order of convergence tobe strong or optimal qualification are found. Examples of all three qualification levelsare provided and the relationships between them as well as with the classical conceptof qualification and the qualification introduced in [6] are shown. In particular, SRMshaving extended qualification in each one of the three levels and having zero or infiniteclassical qualification are presented. Finally several implications of this theory in thecontext of orders of convergence, converse results and maximal source sets for inverseill-posed problems, are shown.
Keywords.
Qualification, Regularization method, Inverse ill-posed problem.
This work was supported by DARPA/SPO, NASA LaRC and the National Institute of Aerospaceunder grant VT-03-1, 2535, and in part by AFOSR Grants F49620-03-1-0243 and FA9550-07-1-0273, byConsejo Nacional de Investigaciones Cient´ıficas y T´ecnicas, CONICET, and by Universidad Nacionaldel Litoral, U.N.L., Argentina, through project CAI+D 2006, P.E. 236.T. Herdman, Interdisciplinary Center for Applied Mathematics, ICAM, Virginia Tech, Blacksburg,VA 24061, USA. E-mail: [email protected]. D. Spies, Instituto de Matem´atica Aplicada del Litoral, IMAL, CONICET-UNL, G¨uemes 3450,S3000GLN, Santa Fe, Argentina. Departamento de Matem´atica, Facultad de Ingenier´ıa Qu´ımica,UNL, Santa Fe, Argentina. E-mail: [email protected]. G. Temperini, IMAL, CONICET-UNL, G¨uemes 3450, S3000GLN, Santa Fe, Argentina.Departamento de Matem´atica, Facultad de Humanidades y Ciencias, UNL, Santa Fe, Argentina.E-mail: [email protected] a r X i v : . [ m a t h . NA ] J u l eneralized Qualification and Qualification Levels for SRM
1. Introduction and preliminaries
Let
X, Y be infinite dimensional Hilbert spaces and T : X → Y a bounded linearoperator. If R ( T ), the range of T , is not closed it is well known that the linear operatorequation T x = y (1)is ill-posed, in the sense that T † , the Moore-Penrose generalized inverse of T , is notbounded [2]. The Moore-Penrose generalized inverse is strongly related to the least-squares (LS) solutions of (1). In fact equation (1) has a LS solution if and only if y belongs to D ( T † ), the domain of T † , which is defined as D ( T † ) . = R ( T ) ⊕ R ( T ) ⊥ . Inthat case, x † . = T † y is the best approximate solution (i.e. the LS solution of minimumnorm) and the set of all LS solutions of (1) is given by x † + N ( T ). If the problem isill-posed, then x † does not depend continuously on the data y . Hence if instead of theexact data y , only an approximation y δ is available, with (cid:13)(cid:13) y − y δ (cid:13)(cid:13) ≤ δ , where δ > T † y δ does not exist or, if itexists, then it will not necessarily be a good approximation of x † , even if δ is very small.This instability becomes evident when trying to approximate x † by standard numericalmethods and procedures. Thus, for instance, except under rather restrictive conditions([4], [12]), the application of the standard LS approximations procedure on a sequence { X n } of finite dimensional subspaces of X , whose union is dense in X , will result in asequence { x n } of LS approximating solutions which does not converge to x † (see [9]).Moreover, this divergence can occur with arbitrarily large speed (see [10]).Ill-posed problems must be regularized before pretending to successfully attackthe problem of numerically approximating their solutions. Regularizing an ill-posedproblem such as (1) essentially means approximating the operator T † by a parametricfamily of continuous operators { R α } , where α is called the regularization parameter.More precisely, for α ∈ (0 , α ) with α ∈ (0 , + ∞ ], let R α : Y → X be a continuous (notnecessarily linear) operator. The set { R α } α ∈ (0 ,α ) is said to be a “family of regularizationoperators” (FRO) for T † , if for every y ∈ D ( T † ), there exists a parameter choice rule α = α ( δ, y δ ) such thatlim δ → + sup y δ ∈ Y (cid:107) yδ − y (cid:107) ≤ δ (cid:13)(cid:13) R α ( δ,y δ ) y δ − T † y (cid:13)(cid:13) = 0 . Here the parameter choice rule α : IR + × Y → (0 , α ) is such thatlim δ → + sup y δ ∈ Y (cid:107) yδ − y (cid:107) ≤ δ α ( δ, y δ ) = 0 . If y ∈ D ( T † ), then x † satisfies the normal equation ( T ∗ T ) x † = T ∗ y and x † can bewritten as x † . = T † y = (cid:90) (cid:107) T (cid:107) +0 λ dE λ T ∗ y, (2) eneralized Qualification and Qualification Levels for SRM { E λ } λ ∈ IR is the spectral family associated to the self-adjoint operator T ∗ T (see [1],[2]). However, since we are assuming that R ( T ) is not closed (and therefore D ( T † ) (cid:40) Y ),if y / ∈ D ( T † ) then the integral in (2) does not exist since in that case 0 ∈ σ ( T ∗ T ) and λ has a pole at 0. Moreover in this case, the operator T † defined in (2) for y ∈ D ( T † ), is notbounded. For that reason, many regularization methods are based on spectral theoryand consist on defining R α . = (cid:82) (cid:107) T (cid:107) +0 g α ( λ ) dE λ T ∗ where { g α } is a family of functionsappropriately defined such that for every λ ∈ (0 , (cid:107) T (cid:107) ] there holds lim α → + g α ( λ ) = λ .Let { g α } α ∈ (0 ,α ) be a parametric family of functions g α : [0 , + ∞ ) → IR defined forall α ∈ (0 , α ). We shall say that { g α } α ∈ (0 ,α ) is a “spectral regularization method”(SRM), if it satisfies the following hypotheses: H1 . For every fixed α ∈ (0 , α ) , g α ( λ ) is piecewise continuous with respect to λ , for λ ∈ [0 , + ∞ ); H2 . There exists a constant C > α ) such that | λg α ( λ ) | ≤ C forevery λ ∈ [0 , + ∞ ); H3 . For every λ ∈ (0 , + ∞ ), lim α → + g α ( λ ) = λ . It can be shown that if { g α } α ∈ (0 ,α ) is a SRM then the family of operators { R α } α ∈ (0 ,α ) defined by R α . = (cid:90) g α ( λ ) dE λ T ∗ = g α ( T ∗ T ) T ∗ , is a FRO for T † ([2], Theorem 4.1). In this case we shall say that { R α } α ∈ (0 ,α ) is a“spectral regularization family” for T † . The use of this terminology has to do with thefact that each one of its elements is defined in terms of an integral with respect to thespectral family { E λ } λ ∈ IR associated to the operator T ∗ T . Note that given the operator T , it is sufficient that g α ( λ ) be defined for λ ∈ [0 , (cid:107) T (cid:107) ], since E λ is “constant” outsidethat interval.It is well known that for ill-posed problems it is not possible to reconstruct theexact solution x † with any degree of accuracy unless additional a-priori informationabout x † is available ([10], [2] Proposition 3.11). On the other hand, given certain a-priori information about x † , it could be desirable to know the best order of convergence(of the regularization error (cid:13)(cid:13) R α y − x † (cid:13)(cid:13) as a function of the regularization parameter α ,or of the total error (cid:13)(cid:13) R α y δ − x † (cid:13)(cid:13) as a function of the noise level δ ), that can be achievedwith a regularization method under those a-priori assumptions. Conversely, given anorder of convergence, one could be interested in determining the possible existence of“source sets” on which a certain regularization method reaches that order of convergence.In this case it could further be of interest to determine “maximal source sets”. Allthese problems are strongly related to the concepts of qualification and saturation of aregularization method ([2], [3], [5], [6], [7], [8]).In [11] the notion of qualification of a regularization method was introduced for thefirst time and the decisive role of this concept in relation to the order of convergence ofthe regularization error was shown. In the sequel, we shall simply denote with { g α } theSRM { g α } α ∈ (0 ,α ) . We now recall the definition of classical qualification for SRMs (see eneralized Qualification and Qualification Levels for SRM Definition 1.1.
Let { g α } be a SRM and denote with I ( g α ) the set I ( g α ) . = { µ ≥ ∀ λ ∈ [0 , + ∞ ) , ∃ k > such that λ µ | − λg α ( λ ) | ≤ k α µ , ∀ α ∈ (0 , α ) } and let µ . = sup µ ∈I ( g α ) µ . If < µ < + ∞ , we say that { g α } has classical qualification andin that case the number µ is called “order” of the classical qualification. Remark 1.1.
Note that ∈ I ( g α ) by virtue of H2 and therefore I ( g α ) is alwaysnonempty. In [6] Math´e and Pereverzev first introduced the following definition of qualificationfor a spectral regularization method, formalizing and extending the classical notion ofthe concept.
Definition 1.2.
Let ρ : (0 , a ] → (0 , ∞ ) be an increasing function. It is said that theregularization method { g α } has qualification ρ if there exists a constant γ ∈ (0 , ∞ ) suchthat sup λ ∈ (0 ,a ] | − λg α ( λ ) | ρ ( λ ) ≤ γ ρ ( α ) ∀ α ∈ (0 , a ] . (3)In this article we generalize the previous concept, mainly by allowing the function ρ ( λ ) appearing in the left hand side of (3) to be substituted by a general function s ( λ )with similar properties. Remark 1.2.
It is important to point out that in [2] the “classical qualification”of a method was defined to be the number µ in Definition 1.1 (even in the case µ = ∞ ). However, from our point of view the “generalized qualification” of a methodwill not be a number but rather a function of the regularization parameter α as anorder of convergence in the sense of Definition 1.2. In the case of SRMs with classicalqualification of positive finite order µ , the corresponding generalized qualification will beshown to be the function ρ ( α ) = α µ , coinciding with the classical approach. Since in theextreme cases µ = 0 and µ = ∞ that function does not define an order of convergence,we have preferred to exclude them from the definition of classical qualification (Definition1.1) and, accordingly, we shall say that the method does not have classical qualification. The organization of this article is as follows. In Section 2 the concepts of weakand strong source-order pair and of order-source pair are defined and three qualificationlevels for SRM are introduced: weak, strong and optimal. A sufficient condition forthe existence of weak qualification is provided and necessary and sufficient conditionsfor an order of convergence to be strong or optimal qualification are given. In Section3, examples of all qualification levels are provided and the relationships between themand with the classical qualification and the qualification introduced in [6] are shown.In particular, SRMs having qualification in each one of the three levels and not havingclassical qualification are presented. Finally several implications of this theory in thecontext of orders of convergence, converse results and maximal source sets for inverseill-posed problems are shown in Section 4. eneralized Qualification and Qualification Levels for SRM
2. Source-order and order-source pairs. Generalized qualification andqualification levels.
It is well known that there exist SRMs for which the corresponding µ given in Definition1.1 is infinity, e.g. truncated singular value decomposition (TSVD), Landweber’smethod and Showalter’s method. However, a careful analysis leads to observe thatthe concept of qualification as optimal order of convergence of the regularization errorremains alive underlying most of these and many other methods. In this section wegeneralize the definition of qualification introduced by Math´e-Pereverzev in [6] andthereby the notion of classical qualification of a SRM. Also three different levels ofqualification are introduced: weak, strong and optimal. These levels introduce naturalhierarchical categories for the SRMs and we show that the generalized qualificationcorresponds to the lowest of these levels. Moreover, a sufficient condition whichguarantees that a SRM possesses qualification in the sense of this generalization isprovided and necessary and sufficient conditions for a given order of convergence to bestrong or optimal qualification are found.We denote with O the set of all non decreasing functions ρ : IR + → IR + such thatlim α → + 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 ) > λ > . If moreover s is increasing, then it isan index function in the sense of Math´e-Pereverzev ([6]). Definition 2.1.
Let ρ, ˜ ρ ∈ O . We say that “ ρ precedes ˜ ρ at the origin” and we denoteit with ρ (cid:22) ˜ ρ , if there exist positive constants c and ε such that ρ ( α ) ≤ c ˜ ρ ( α ) for every α ∈ (0 , ε ) . Definition 2.2.
Let ρ, ˜ ρ ∈ O . We say that “ ρ and ˜ ρ are equivalent at the origin” andwe denote it with ρ ≈ ˜ ρ , if they precede each other at the origin, that is, if there existconstants ε, c , c , ε > , < c < c < ∞ such that c ρ ( α ) ≤ ˜ ρ ( α ) ≤ c ρ ( α ) for every α ∈ (0 , ε ) . Clearly, “ ≈ ” introduces an order of equivalence in O . Analogous definitions andnotation will be used for s, ˜ s ∈ S . Definition 2.3.
Let { g α } be a SRM, r α ( λ ) . = 1 − λg α ( λ ) , ρ ∈ O and s ∈ S . i) We say that ( s, ρ ) is a “weak source-order pair for { g α } ” if it satisfies s ( λ ) | r α ( λ ) | ρ ( α ) = O (1) for α → + , ∀ λ > . (4) ii) We say that ( s, ρ ) is a “strong source-order pair for { g α } ” if it is a weak source-order pair and there is no λ > for which O (1) in (4) can be replaced by o (1) . That is,if (4) holds and also lim sup α → + s ( λ ) | r α ( λ ) | ρ ( α ) > ∀ λ > . (5) eneralized Qualification and Qualification Levels for SRM iii) We say that ( ρ, s ) is an “order-source pair for { g α } ” if there exist a constant γ > and a function h : (0 , α ) → IR + with lim α → + h ( α ) = 0 , such that s ( λ ) | r α ( λ ) | ρ ( α ) ≥ γ ∀ λ ∈ [ h ( α ) , + ∞ ) . (6)In the previous definitions we shall refer to the function ρ as the “order ofconvergence” and to s as the “source function”. The reason for using this terminologywill become clear in Section 4 when we shall see applications of these concepts in thecontext of direct and converse results for regularization methods.The following observations follow immediately from the definitions.(i) If ( s, ρ ) is a weak source-order pair for { g α } which is not a strong source-order pair,then there exists λ > α → + s ( λ ) | r α ( λ ) | ρ ( α ) = 0 and therefore ( ρ, s ) cannotbe an order-source pair for { g α } . Thus if ( ρ, s ) is an order-source pair and ( s, ρ )is a weak source-order pair, then ( s, ρ ) is further a strong source-order pair in thesense of ii ) .(ii) Let ρ, ˜ ρ ∈ O .(a) If ( s, ρ ) is a weak source-order pair for { g α } and ρ (cid:22) ˜ ρ then ( s, ˜ ρ ) is also aweak source-order pair for { g α } .(b) If ( s, ρ ) is a weak source-order pair for { g α } and ˜ s ∈ S is such that thereexists c > s ( λ ) ≤ c s ( λ ) for every λ >
0, then (˜ s, ρ ) is also a weaksource-order pair for { g α } .In the following definition we introduce the concept of generalized qualification andthree different levels of it. Definition 2.4.
Let { g α } be a SRM. i) We say that ρ is “weak or generalized qualification of { g α } ” if there exists afunction s such that ( s, ρ ) is a weak source-order pair for { g α } . ii) We say that ρ is “strong qualification of { g α } ” if there exists a function s suchthat ( 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 } (it is sufficient that ( s, ρ ) be aweak source-order pair) and ( ρ, s ) is an order-source pair for { g α } . It is important to observe that weak qualification generalizes the concept ofqualification introduced by Math´e and Pereverzev in [6] and therefore, the notion ofclassical qualification. In fact, if { g α } has continuous qualification ρ ( α ) in the sense ofDefinition 1.2 and lim α → + ρ ( α ) = 0, then the function˜ ρ ( α ) . = , si α = 0; ρ ( α ) , si 0 < α ≤ a ; ρ ( a ) , si α > a . (7) eneralized Qualification and Qualification Levels for SRM { g α } . However, these two notions are not equivalent. We shallsee later on that it is possible for a function to be weak qualification of a SRM and notbe qualification according to Definition 1.2 (see comments at the end of Section 3).It is timely to note here that if { g α } has classical qualification of order µ , then ρ ( α ) = α µ is weak qualification of { g α } and moreover ( λ µ , α µ ) is a weak source-order pairfor { g α } for every µ ∈ (0 , µ ]. Conversely, if for µ >
0, ( λ µ , α µ ) is a weak source-orderpair for { g α } , then this method has classical qualification (of order µ ≥ µ ) providedthat µ . = sup { µ : ( λ µ , α µ ) is a weak source-order pair for { g α } } < + ∞ .The following result provides a sufficient condition for the existence of weakqualification of a SRM. Theorem 2.1.
Let { g α } be a SRM such that for every fixed λ > , g α ( λ ) is decreasingin α , for α ∈ (0 , α ) . a) If there exist an increasing function h : (0 , α ) → IR + with lim α → + h ( α ) = 0 , ρ ∗ ∈ O and ε > such that for every α ∈ (0 , ε ) , sup λ ∈ [ h ( α ) , + ∞ ) | r α ( λ ) | ≤ ρ ∗ ( α ) , (8) then { g α } has weak qualification and in that case ρ ∗ is weak qualification of the method. b) If for every α ∈ (0 , α ) , r α ( λ ) is positive and monotone decreasing for λ ∈ (0 , + ∞ ) , then it is always possible to find h and ρ ∗ as in a) satisfying (8) for all α ∈ (0 , α ) .Proof. a) Let h : (0 , α ) → IR + be an increasing function with lim α → + h ( α ) = 0, ρ ∗ ∈ O and ε > α ∈ (0 , ε ) condition (8) holds.Case I: there exists ˜ α ∈ (0 , ε ) such that sup λ ∈ [ h (˜ α ) , + ∞ ) | r ˜ α ( λ ) | > h ( α ) is increasing, it follows that sup λ ∈ [ h ( α ) , + ∞ ) | r α ( λ ) | > α ∈ (0 , ˜ α ].Let λ >
0. Then for every α ∈ (0 , ˜ α ], | r α ( λ ) | ρ ∗ ( α ) ≤ | r α ( λ ) | sup λ ∈ [ h ( α ) , + ∞ ) | r α ( λ ) | . (9)Since lim α → + h ( α ) = 0, there exists α ∗ ∈ (0 , ˜ α ) such that λ ∈ [ h ( α ) , + ∞ ) for every α ∈ (0 , α ∗ ] , from which it follows that for every α ∈ (0 , α ∗ ], | r α ( λ ) | sup λ ∈ [ h ( α ) , + ∞ ) | r α ( λ ) | ≤ . (10)¿From (9) and (10) it follows that for every λ > α → + | r α ( λ ) | ρ ∗ ( α ) ≤ . Then, for any bounded s ∈ S the pair ( s, ρ ∗ ) satisfies (4), i.e., it is a weak source-orderpair for { g α } . Thus we have proved that ρ ∗ is weak qualification of { g α } . eneralized Qualification and Qualification Levels for SRM λ ∈ [ h ( α ) , + ∞ ) | r α ( λ ) | = 0 for every α ∈ (0 , ε ).Let λ >
0. Since lim α → + h ( α ) = 0, there exists α ∗ ∈ (0 , ε ) such that λ ∈ [ h ( α ) , + ∞ )for every α ∈ (0 , α ∗ ] . Then | r α ( λ ) | ≤ sup λ ∈ [ h ( α ) , + ∞ ) | r α ( λ ) | = 0 for every α ∈ (0 , α ∗ ), fromwhat it follows that r α ( λ ) = 0. Then, for any s ∈ S , s ( λ ) r α ( λ ) ρ ∗ ( α ) = 0 for all α ∈ (0 , α ∗ ) . Therefore, ( s, ρ ∗ ) is a weak source-order pair for { g α } , which implies that ρ ∗ is weakqualification of { g α } . (Note that in this case any ρ ∗ ∈ O is weak qualification of { g α } .) b) Let { g α } be a SRM such that for every α ∈ (0 , α ), r α ( λ ) is positive andmonotone decreasing for λ ∈ (0 , + ∞ ). For λ > f ( λ ) . = (1 − e − λ ) θ ( λ ), where θ ( λ ) . = sup { γ ∈ (0 , α ) : r α ( λ ) ≤ λ ∀ α ∈ (0 , γ ) } . Since for every λ >
0, lim α → + r α ( λ ) = 0, it follows that given λ > γ = γ ( λ ) > r α ( λ ) ≤ λ for every α ∈ (0 , γ ). Then θ ( λ ) (cid:54) = −∞ , moreover θ ( λ ) ∈ (0 , α ] for every λ > f ( λ ) ∈ (0 , α ) for every λ >
0. On the otherhand, since for every α ∈ (0 , α ), r α ( λ ) is decreasing for λ >
0, it follows immediatelythat f is strictly increasing. Furthermore, since f is bounded, it has countably manyjump discontinuity points. Therefore, it is possible to assume, without loss of generality,that f is continuous (since, if it is not, we can redefine it in such a way that it becontinuous, by subtracting the jumps at the discontinuity points).Thus f : IR + → (0 , α ) is continuous, strictly increasing with lim λ → + f ( λ ) = 0.Therefore, its inverse function f − exists over the range of f and it is strictly increasingand continuous with lim α → + f − ( α ) = 0. It is possible to extend f − to (0 , α ) in such away that it preserves all these properties. We shall denote with h this extension.For α ∈ (0 , α ), we define z ( α ) . = sup λ ∈ [ h ( α ) , + ∞ ) | r α ( λ ) | = r α ( h ( α )) . Since for every α ∈ (0 , α ), r α ( λ ) is positive for all λ >
0, it follows that z ( α ) is also positive. Sincefor every λ > f ( λ ) < θ ( λ ), the definition of θ ( λ ) implies that r f ( λ ) ( λ ) ≤ λ for every λ >
0, or equivalently, r α ( h ( α )) ≤ h ( α ) for every α ∈ (0 , α ). Then 0 < z ( α ) ≤ h ( α )for every α ∈ (0 , α ) and the fact that lim α → + h ( α ) = 0 implies that lim α → + z ( α ) = 0. Iffurther z is a non decreasing function, then z ∈ O and it suffices to define ρ ∗ . = z . Onthe contrary, since z is bounded and positive with lim α → + z ( α ) = 0, there always exists afunction ρ ∗ ∈ O such that z ( α ) ≤ ρ ∗ ( α ) for every α ∈ (0 , α ), as we wanted to show. (cid:4) ¿From the previous Theorem, it follows that the SRMs { g α } such that for every λ > g α ( λ ) is decreasing for α ∈ (0 , α ) and for every α ∈ (0 , α ), r α ( λ ) is positiveand decreasing for λ >
0, do possess weak qualification. It is important to observe thatmost of the usual SRMs do in fact satisfy these conditions. In particular this is so forLandweber’s and Showalter’s methods. eneralized Qualification and Qualification Levels for SRM { g α } and ρ ∈ O , we define s ρ ( λ ) . = lim inf α → + ρ ( α ) | r α ( λ ) | for λ ≥ . (11)Note that s ρ (0) = 0 . In the next three results we will see that the characteristics of a given function ρ ∈ O , as a possible strong or optimal qualification of a SRM, can be determined fromproperties of that function s ρ . Proposition 2.1. (Necessary and sufficient condition for strong qualification.) Afunction ρ ∈ O such that s ρ ∈ S is strong qualification of { g α } if and only if < s ρ ( λ ) < + ∞ for every λ > . (12) Proof.
Suppose that ρ is strong qualification of { g α } . Then there exists a function s ∈ S such that ( s, ρ ) is a strong source-order pair for { g α } . Then, for every λ > s ρ ( λ ) = lim inf α → + ρ ( α ) | r α ( λ ) | = 1lim sup α → + | r α ( λ ) | ρ ( α ) = s ( λ )lim sup α → + s ( λ ) | r α ( λ ) | ρ ( α ) . Thus (12) follows from (4) and (5).Conversely, suppose now that 0 < s ρ ( λ ) < + ∞ for every λ >
0. We will show that ρ is strong qualification of { g α } . For that let us see that ( s ρ , ρ ) is a strong source-orderpair for { g α } . Since 0 < s ρ ( λ ) < + ∞ for every λ >
0, it follows thatlim sup α → + s ρ ( λ ) | r α ( λ ) | ρ ( α ) = s ρ ( λ ) lim sup α → + | r α ( λ ) | ρ ( α ) = 1 ∀ λ > . Then, s ρ verifies (4) and (5), which, together with the fact that s ρ ∈ S , implies that( s ρ , ρ ) is a strong source-order pair and thus ρ is strong qualification of { g α } . (cid:4) Proposition 2.2.
Let ρ ∈ O be strong qualification of { g α } and s ∈ S . Then ( s, ρ ) is astrong source-order pair for { g α } if and only if there exists k > such that s ( λ ) ≤ k s ρ ( λ ) for every λ > .Proof. Since ρ is strong qualification, by Proposition 2.1 it follows that s ρ ( λ ) > λ > s, ρ ) is a strong source-order pair for { g α } . Then thereexist positive constants k and ε such that s ( λ ) | r α ( λ ) | ρ ( α ) ≤ k for every λ > α ∈ (0 , ε ).Then, for every λ > s ( λ ) s ρ ( λ ) = s ( λ ) lim sup α → + | r α ( λ ) | ρ ( α ) = lim sup α → + s ( λ ) | r α ( λ ) | ρ ( α ) ≤ k, and therefore s ( λ ) ≤ k s ρ ( λ ) for every λ > k > s ( λ ) ≤ k s ρ ( λ ) for every λ >
0. Since s ρ ( λ ) >
0, it then follows that k ≥ s ( λ ) s ρ ( λ ) = lim sup α → + s ( λ ) | r α ( λ ) | ρ ( α ) ∀ λ > , eneralized Qualification and Qualification Levels for SRM s, ρ ) is a weak source-order pair for { g α } . Moreover since s ( λ ) and s ρ ( λ ) arepositive for all λ >
0, it follows that s ( λ ) verifies (5) and therefore ( s, ρ ) is, furthermore,a strong source-order pair for { g α } . (cid:4) Theorem 2.2. (Necessary and sufficient condition for optimal qualification.) Afunction ρ ∈ O such that s ρ ∈ S is optimal qualification of { g α } if and only if s ρ verifies (6) and (12) .Proof. Suppose that ρ is optimal qualification. Then ρ is strong qualification andit follows from Proposition 2.1 that s ρ verifies (12). Moreover since ρ is optimalqualification, there exists s ∈ S such that ( s, ρ ) is a strong source-order pair and ( ρ, s )is an order-source pair. From the latter it follows that there exist a constant γ > h : (0 , α ) → IR + with lim α → + h ( α ) = 0, such that s ( λ ) | r α ( λ ) | ρ ( α ) ≥ γ ∀ λ ∈ [ h ( α ) , + ∞ ) . (13)On the other hand, since ( s, ρ ) is a strong source-order pair for { g α } , it follows fromProposition 2.2 that there exists k > s ( λ ) ≤ k s ρ ( λ ) for every λ > . (14)¿From (13) and (14) it follows that s ρ ( λ ) | r α ( λ ) | ρ ( α ) ≥ γk ∀ λ ∈ [ h ( α ) , + ∞ ) , that is, s ρ satisfies (6) as we wanted to show.Conversely, suppose that s ρ ∈ S verifies (6) and (12). By Proposition 2.1 we havethat ( s ρ , ρ ) is a strong source-order pair for { g α } and (6) implies that ( ρ, s ρ ) is anorder-source pair. Then, ρ is optimal qualification of { g α } . (cid:4) Next we will show the uniqueness of the source function.
Theorem 2.3. If ρ is optimal qualification of { g α } then there exists at most one function s (in the sense of the equivalence classes induced by Definition 2.2) such that ( s, ρ ) is astrong source-order pair and ( ρ, s ) is an order-source pair for { g α } . Moreover if s ρ ∈ S ,then s ρ is such a unique function.Proof. Given that ρ is optimal qualification of { g α } , there exists at least one function s such that ( s, ρ ) is a strong source-order pair and ( ρ, s ) is an order-source pair for { g α } .Suppose now that there exist s and s such that ( s , ρ ) and ( s , ρ ) are strong source-order pairs and ( ρ, s ) and ( ρ, s ) are order-source pairs for { g α } . Then there exist γ > h : (0 , α ) → IR + with lim α → + h ( α ) = 0, such that s ( λ ) | r α ( λ ) | ρ ( α ) ≥ γ for every λ ∈ [ h ( α ) , + ∞ ). Then, s ( λ ) = s ( λ ) s ( λ ) | r α ( λ ) | ρ ( α ) s ( λ ) | r α ( λ ) | ρ ( α ) ≤ s ( λ ) γ s ( λ ) | r α ( λ ) | ρ ( α ) ∀ λ ∈ [ h ( α ) , + ∞ ) , ∀ α ∈ (0 , α ) . (15) eneralized Qualification and Qualification Levels for SRM s , ρ ) is a strong source-order pair, there exist positiveconstants k and ε such that s ( λ ) | r α ( λ ) | ρ ( α ) ≤ k ∀ λ > , ∀ α ∈ (0 , ε ) . (16)From (15) and (16) it follows that s ( λ ) ≤ kγ s ( λ ) ∀ λ ∈ [ h ( α ) , + ∞ ) , ∀ α ∈ (0 , ε ) . Since lim α → + h ( α ) = 0 we have that s ( λ ) ≤ kγ s ( λ ) for every λ >
0. Analogously, byinterchanging s and s it follows that there exists ˜ k > s ( λ ) ≤ ˜ k s ( λ ) forevery λ > s ≈ s .Suppose now that s ρ ∈ S . Since ρ is optimal qualification of { g α } it follows fromTheorem 2.2 that s ρ verifies (6) and (12). Then, s ρ is the unique function such that( s ρ , ρ ) is a strong source-order pair and ( ρ, s ρ ) is an order-source pair for { g α } . (cid:4) The following is a result about the uniqueness of the order.
Theorem 2.4. If ( s, ρ ) and ( s, ρ ) are strong source-order pairs for { g α } and thereexists lim α → + ρ ( α ) ρ ( α ) , then ρ ≈ ρ .Proof. Suppose that ( s, ρ ) and ( s, ρ ) are strong source-order pairs for { g α } . We willfirst show that lim sup α → + ρ ( α ) ρ ( α ) > . Suppose thatlim sup α → + ρ ( α ) ρ ( α ) = 0 . (17)Since ( s, ρ ) is a strong source-order pair we have that s ( λ ) | r α ( λ ) | ρ ( α ) = O (1) for α → + , ∀ λ > < lim sup α → + s ( λ ) | r α ( λ ) | ρ ( α ) = lim sup α → + s ( λ ) | r α ( λ ) | ρ ( α ) ρ ( α ) ρ ( α ) . It follows from (17) and (18) that the lim sup on the right-hand side of the previousexpression must be equal to zero, which is a contradiction. Then, lim sup α → + ρ ( α ) ρ ( α ) > . Similarly, it is shown that lim sup α → + ρ ( α ) ρ ( α ) > . Since there exists lim α → + ρ ( α ) ρ ( α ) , we then havethat 0 < lim α → + ρ ( α ) ρ ( α ) < + ∞ and 0 < lim α → + ρ ( α ) ρ ( α ) < + ∞ . Then, ρ (cid:22) ρ and ρ (cid:22) ρ , thatis, ρ ≈ ρ , as we wanted to show. (cid:4) eneralized Qualification and Qualification Levels for SRM
3. Examples
In this section we present several examples which illustrate the different qualificationlevels previously introduced as well as the relationships between them and with theconcept of classical qualification and the qualification introduced in [6]. Although someof these examples are only of academic interest and nature, they do serve to show theexistence of regularization methods possessing qualification in each one of the levelsintroduced in this article.
Example 1.
Tikhonov-Phillips regularization method { g α } , where g α ( λ ) . = λ + α has classical qualification of order µ = 1 ([2]). We will see that ρ ( α ) = α is optimalqualification in the sense of Definition 2.4 iii) . In fact, for λ > r α ( λ ) = αα + λ and if ρ ( α ) = α then s ρ ( λ ) = lim inf α → + ρ ( α ) | r α ( λ ) | = lim α → ( λ + α ) = λ >
0, that is, s ρ verifies (12). Alsosince s ρ ( λ ) | r α ( λ ) | ρ ( α ) = λλ + α ≥ ∀ λ ∈ [ α, + ∞ ) , we have that s ρ verifies (6). From Theorem 2.2 it then follows that ρ ( α ) = α is optimalqualification of { g α } . Example 2.
Let { g α } be the family of functions associated to the truncatedsingular value decomposition (TSVD), g α ( λ ) . = (cid:40) λ , if λ ∈ [ α, + ∞ )0 , if λ ∈ [0 , α ) . It follows that µ = + ∞ , where µ is as in Definition 1.1. Therefore, TSVD does nothave classical qualification. In this case we have that r α ( λ ) = (cid:40) , if λ ∈ [ α, + ∞ )1 , if λ ∈ [0 , α ) . Let h ( α ) = α and ρ ∈ O . Thensup λ ∈ [ h ( α ) , + ∞ ) | r α ( λ ) | = sup λ ≥ α | r α ( λ ) | = 0 ≤ ρ ( α ) for every α ∈ (0 , α ) . Then, it follows from Theorem 2.1.a) that any function ρ ∈ O is weak qualificationof the method. However, TSVD does not have strong qualification. In fact, for anyfunction ρ ∈ O we have that s ρ ( λ ) = lim inf α → + ρ ( α ) | r α ( λ ) | = + ∞ for every λ >
0. Proposition2.1 implies then that ρ is not strong qualification of the method. In [6] it was observedthat TSVD has arbitrary qualification in the sense of Definition 1.2. Example 3.
For α ∈ (0 , α ) we define g α ( λ ) . = 1 − e − α λ + e − α , for every λ ∈ [0 , + ∞ ) . eneralized Qualification and Qualification Levels for SRM { g α } satisfies the hypotheses H1-H3 and thereforeis a SRM. Since r α ( λ ) = λ λ e α for all λ ∈ [0 , + ∞ ), it follows that for every µ > | r α ( λ ) | λ µ α µ = (1 + λ ) λ µ λ e α α µ + α µ = o (1) for α → + for every λ ∈ [0 , + ∞ ) . Then, { g α } does not have classical qualification (more precisely µ = + ∞ , where µ isas in Definition 1.1).We will now show that ρ ( α ) = e − α is optimal qualification of { g α } . Since s ρ ( λ ) = lim inf α → + ρ ( α ) | r α ( λ ) | = λ λ ∈ (0 , + ∞ ) for every λ >
0, it follows from Proposition2.1 that ρ is strong qualification of { g α } . Moreover since s ρ ( λ ) | r α ( λ ) | ρ ( α ) = λλ + e − α ≥ ∀ λ ∈ [ e − α , + ∞ ) , it follows that s ρ verifies (6). Theorem 2.2 then implies that ρ ( α ) = e − α is optimalqualification of { g α } . Example 4.
For α ∈ (0 , α ) with α < e − , define g α ( λ ) . = 1 + (ln α ) − λ − (ln α ) − , for every λ ∈ [0 , + ∞ ) . Clearly, { g α } satisfies hypotheses H1-H3 and therefore is a SRM. Since r α ( λ ) = λ − λ ln α for all λ ∈ [0 , + ∞ ), it follows that for every µ > | r α ( λ ) | λ µ α µ = (1 + λ ) λ µ α µ − λ α µ ln α → + ∞ for α → + for every λ ∈ [0 , + ∞ ) . Then, µ = 0 and therefore { g α } does not have classical qualification.However, we will show that ρ ( α ) = − (ln α ) − is optimal qualification of { g α } . Infact, since s ρ ( λ ) = lim inf α → + ρ ( α ) | r α ( λ ) | = lim α → λ − (ln α ) − λ = λ λ ∈ (0 , + ∞ ) for every λ > s ρ ( λ ) | r α ( λ ) | ρ ( α ) = λλ − (ln α ) − ≥ ∀ λ ∈ [ − (ln α ) − , + ∞ ) , it follows from Theorem 2.2 that ρ ( α ) = − (ln α ) − is optimal qualification of { g α } . Example 5.
Let { g α } be the Tikhonov-Phillips regularization method, which, aspreviously mentioned, it has classical qualification of order µ = 1. In Example 1 wesaw that ρ ( α ) = α is optimal qualification of this method and therefore it is also weakqualification of it. Since α (cid:22) α it follows from Definition 2.4.i) and Observation 2.a)that ρ ∗ ( α ) = α is also weak qualification. However, ρ ∗ is not strong qualification ofthe method. In fact, for any s ∈ S , we have thatlim sup α → + s ( λ ) | r α ( λ ) | ρ ∗ ( α ) = lim sup α → + s ( λ ) α α + λ = 0 ∀ λ > . Example 6.
Let { g α } be the SRM defined in Example 4. This method doesnot have classical qualification since µ = 0. We proved that − (ln α ) − is optimalqualification and therefore, it is also weak qualification. Since − (ln α ) − (cid:22) ( − ln α ) − , eneralized Qualification and Qualification Levels for SRM ρ ( α ) = ( − ln α ) − is weakqualification. Let us show now that ρ is not strong qualification of the method. For any s ∈ S , we have thatlim sup α → + s ( λ ) | r α ( λ ) | ρ ( α ) = lim sup α → + s ( λ ) (1 + λ )(1 − λ ln α ) ( − ln α ) − = 0 ∀ λ > . It is important to observe that if ρ ( α ) = α µ is strong qualification of a SRM thenit follows immediately from the definition of strong source-order pair that the methodhas classical qualification of order µ . The converse, however, is not true as the nextexample shows. Hence it is the weak and not the strong qualification what generalizesthe classical notion of this concept. Example 7.
For α ∈ (0 , α ) with α < / h α ( λ ) . = αα + ln( αα + λ )and g α ( λ ) . = − h α ( λ ) λ + h α ( λ ) , if λ ∈ [2 α, + ∞ ) − h α (2 α )2 α + h α (2 α ) = (cid:16) α − α +2 α ln 3 (cid:17) − , if λ ∈ [0 , α ) . In this case, r α ( λ ) . = α (1+ λ ) λ ln( αα + λ )+ α (1+ λ ) , if λ ∈ [2 α, + ∞ )1 − λ (cid:16) α − α +2 α ln 3 (cid:17) − , if λ ∈ [0 , α ) . One can immediately show that { g α } is a SRM with classical qualification of order µ = 1. However, ρ ( α ) = α is not strong qualification of the method. In fact, for any s ∈ S , we can see that s ( λ ) | r α ( λ ) | α = o (1) for α → + , ∀ λ ≥ ρ is strong qualification which is notoptimal, then ∀ λ >
0, the function s ρ ( λ ) | r α ( λ ) | ρ ( α ) it is not of bounded variation as a functionof α in any neighborhood of α = 0. Even so, the following three examples show theexistence of SRM having strong but not optimal qualification and they show that strongqualification in no case implies optimal qualification. Example 8.
Given k ∈ IR + , for α, λ > g kα ( λ ) . = λ − (1 − e − λα ) − α k λ − / (cid:12)(cid:12)(cid:12) sin( λ /α ) (cid:12)(cid:12)(cid:12) , eneralized Qualification and Qualification Levels for SRM r kα ( λ ) = e − λα + α k λ − / (cid:12)(cid:12)(cid:12) sin( λ /α ) (cid:12)(cid:12)(cid:12) . It can be immediately checked that { g kα } isa SRM with classical qualification of order k . With ρ ( α ) = α k we have that ∀ λ > s ρ ( λ ) = lim inf α → + α k e − λα + α k λ − / (cid:12)(cid:12)(cid:12) sin( λ /α ) (cid:12)(cid:12)(cid:12) = 1lim sup α → + (cid:16) α − k e − λα + λ − / (cid:12)(cid:12)(cid:12) sin( λ /α ) (cid:12)(cid:12)(cid:12)(cid:17) = λ / . Since s ρ ( λ ) = λ / ∈ S , from Proposition 2.1 it follows that ( s ρ , ρ ) is a strong source-order pair and ρ ( α ) = α k is strong qualification of the method. However, for every λ >
0, lim inf α → + s ρ ( λ ) | r α ( λ ) | ρ ( α ) = lim inf α → + (cid:34) λ / α − k e − λα + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin (cid:32) λ α (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:35) = 0 . Therefore equation (6) does not hold and ρ ( α ) = α k is not optimal qualification of themethod. Example 9.
For α, λ > g α ( λ ) as follows: g α ( λ ) . = λ − (1 − e − λα ) − e − √ α λ − / (cid:12)(cid:12)(cid:12) sin( λ /α ) (cid:12)(cid:12)(cid:12) , so that r α ( λ ) = e − λα + e − √ α λ − / (cid:12)(cid:12)(cid:12) sin( λ /α ) (cid:12)(cid:12)(cid:12) . It can be immediately verified that { g α } is a SRM which does not have classicalqualification ( µ = ∞ ). However, with ρ ( α ) . = e − √ α we have that s ρ ( λ ) = lim inf α → + ρ ( α ) r α ( λ )= 1lim sup α → + (cid:104) e − λα + √ α + λ − / | sin( λ /α ) | (cid:105) = λ . Since s ρ ( λ ) = λ / ∈ S , by Proposition 2.1 ( s ρ , ρ ) is a strong source-order pair and ρ ( α ) = e − / √ α is strong qualification of the method. However, ∀ λ > α → + s ρ ( λ ) | r α ( λ ) | ρ ( α ) = lim inf α → + (cid:16) λ / e √ α − λα + | sin( λ /α ) | (cid:17) = 0 , and therefore (6) does not hold and ρ ( α ) = e − √ α is not optimal qualification of themethod. Example 10.
For 0 < α < λ > g α ( λ ) . = λ − (1 − e − λα ) + (ln α ) − λ − / (cid:12)(cid:12)(cid:12) sin( λ /α ) (cid:12)(cid:12)(cid:12) , eneralized Qualification and Qualification Levels for SRM r α ( λ ) = e − λα − (ln α ) − λ − / (cid:12)(cid:12)(cid:12) sin( λ /α ) (cid:12)(cid:12)(cid:12) . Just like in Examples 8 and 9 it can be easily checked that { g α } is a SRM whichdoes not have classical qualification ( µ = 0), that ρ ( α ) = − α is strong but not optimalqualification of the method and that ( s ρ , ρ ) is a strong source-order pair with s ρ ( λ ) = λ .Note that examples 2, 3, 4, 6, 9 and 10 correspond to SRMs which do not haveclassical qualification but, however, they do have generalized qualification, falling insome of its three different levels. Also Landweber’s method and Showalter’s method,which as previously pointed out do not have classical qualification (in both cases µ = ∞ ), are SRMs defined by g α ( λ ) . = λ (1 − (1 − µλ ) α ) (where α ≤ , µ < (cid:107) T (cid:107) ) and g α ( λ ) . = λ (1 − e − λα ), respectively. It can be easily proved, by using Theorem 2.1, that ρ ( α ) = (1 − µα ) α is weak qualification of Landweber’s method and ρ ( α ) = e − √ α is weakqualification of Showalter’s method. However, in this last case it can be easily shownthat ρ ( α ) = e − √ α does not satisfy condition (3) and therefore ρ ( α ) is not qualificationin the sense of Definition 1.2.The different qualification levels introduced in this article and the relationshipsbetween them are visualized in Figure 1.
4. Orders of convergence, converse results and maximal source sets
The generalization of the concept of qualification of a SRM introduced in the previoussections is strongly related with and it has a broad spectrum of applications in thecontext of orders of convergence, converse results and maximal source sets for inverseill-posed problems. We present next some results in this direction. However, we pointout that this is not the main objective of the present article. For that reason, some ofthis results will be stated without proof. More detailed results in this regard will appearin a forthcoming article.Let
X, Y be infinite dimensional Hilbert spaces and T : X → Y a bounded, linearinvertible operator such that R ( T ) is not closed. For s ∈ S , the set R ( s ( T ∗ T )), willbe referred to as the “source set associated to the function s and the operator T ”. Inall that follows, the hypothesis s ∈ S can be replaced by s continuous on σ ( T ∗ T ) and s ∈ M , where M is the set of all functions f : IR → IR +0 which are measurable withrespect to the measures d (cid:107) E λ x (cid:107) for every x ∈ X .The following direct result, whose proof follows immediately from the concept ofweak source-order pair, states that if the exact solution x † of the problem T x = y belongs to the source set R ( s ( T ∗ T )) and ( s, ρ ) is a weak source-order pair for { g α } ,then the regularization error (cid:13)(cid:13) R α y − x † (cid:13)(cid:13) has order of convergence ρ ( α ). For brevityreasons we do not give the proof here. eneralized Qualification and Qualification Levels for SRM Figure 1.
Relationships between the different qualification levels, the classicalqualification and the qualification defined in [6].
Theorem 4.1.
Let ρ ∈ O be weak qualification of { g α } and s ∈ S such that ( s, ρ ) isa weak source-order pair for { g α } . If x † . = T † y ∈ R ( s ( T ∗ T )) then (cid:13)(cid:13) ( R α − T † ) y (cid:13)(cid:13) = O ( ρ ( α )) for α → + . It is important to note here that the previous result can be viewed as ageneralization of Theorem 4.3 in [2], to the case of SRM with weak qualification andgeneral source sets. In fact, that result corresponds to the particular case in which { g α } has classical qualification of order µ . eneralized Qualification and Qualification Levels for SRM ρ ( α ) and ( ρ, s ) is an order-source pair, then the exact solution belongs tothe source set given by the range of the operator s ( T ∗ T ). Theorem 4.2. If ( ρ, s ) is an order-source pair for { g α } and (cid:13)(cid:13) ( R α − T † ) y (cid:13)(cid:13) = O ( ρ ( α )) for α → + , then x † ∈ R ( s ( T ∗ T )) . Proof.
The proof follows immediately from the definition of order-source pair for theSRM { g α } . (cid:4) It is interesting to note that Theorem 4.2 can also be viewed as a generalization ofTheorem 4.11 in [2]. In fact, this corresponds to the particular case in which s ( λ ) . = λ µ y ρ ( α ) . = α µ . If moreover ρ is optimal qualification then the reciprocal of Theorem 4.2also holds. This is proved in the following theorem. Theorem 4.3. If ρ is optimal qualification of { g α } and s ρ ∈ S , then (cid:13)(cid:13) ( R α − T † ) y (cid:13)(cid:13) = O ( ρ ( α )) for α → + if and only if x † ∈ R ( s ρ ( T ∗ T )) . Proof.
Let ρ be optimal qualification of { g α } and s ρ ∈ S . Then by Theorem 2.3, ( ρ, s ρ )is an order-source pair for { g α } and since (cid:13)(cid:13) ( R α − T † ) y (cid:13)(cid:13) = O ( ρ ( α )) for α → + , itfollows from Theorem 4.2 that x † ∈ R ( s ρ ( T ∗ T )) . Conversely, if x † ∈ R ( s ρ ( T ∗ T )), since by virtue of Theorem 2.3 ( s ρ , ρ ) is a strongsource-order pair, Theorem 4.1 implies that (cid:13)(cid:13) ( R α − T † ) y (cid:13)(cid:13) = O ( ρ ( α )) for α → + . (cid:4) An important result regarding existence and maximality of source sets is thefollowing: if ρ is strong qualification of a SRM and s ρ ∈ S it follows from Proposition2.2 that R ( s ρ ( T ∗ T )) is a maximal source set where ρ is order of convergence of theregularization error. More precisely we have the following result. Theorem 4.4.
Let ρ ∈ O be strong qualification of { g α } such that s ρ ∈ S and s ∈ S .If ( s, ρ ) is a strong source-order pair for { g α } and R ( s ( T ∗ T )) ⊃ R ( s ρ ( T ∗ T )) then R ( s ( T ∗ T )) = R ( s ρ ( T ∗ T )) .Proof. Under the hypotheses of the Proposition 2.2, there exists k > s ( λ ) ≤ k s ρ ( λ ) for every λ >
0, which implies that R ( s ( T ∗ T )) ⊂ R ( s ρ ( T ∗ T )). (cid:4) If moreover ρ is optimal qualification the following stronger result is obtained. Theorem 4.5. If ρ ∈ O is optimal qualification of { g α } and s ρ ∈ S , then R ( s ρ ( T ∗ T )) is the only source set where ρ is order of convergence of the regularization error of { g α } .Proof. This result follows immediately from Theorem 2.3. (cid:4)
Examples:
1. For the Tikhonov-Phillips regularization method the only source set where ρ ( α ) = α is optimal qualification is R ( s ρ ( T ∗ T )) = R ( T ∗ T ), since in this case s ρ ( λ ) = λ . eneralized Qualification and Qualification Levels for SRM ρ ( α ) = e − α is optimal qualification of { g α } and s ρ ( λ ) = λ λ . Since λ λ ≈ λ it follows that R ( s ρ ( T ∗ T )) = R ( T ∗ T ) is the onlysource set where ρ is order of convergence of the regularization error.3. In Example 8 of the previous section, for ρ ( α ) = α k we have that s ρ ( λ ) = λ / .Since ρ is strong qualification of this SRM, it follows that R ( s ρ ( T ∗ T )) = R ( T ∗ T ) / isa maximal source set where ρ ( α ) is order of convergence of the regularization error.4. As pointed out at the end of Section 3, ρ ( α ) = e − √ α is weak qualification ofShowalter’s method. It can be easily shown that for every s ∈ S , ( s, ρ ) is a weak source-order pair for the method. Therefore, it follows from Theorem 4.1 that the regularizationerror (cid:13)(cid:13) R α y − x † (cid:13)(cid:13) has order of convergence ρ ( α ) = e − √ α whenever x † ∈ (cid:83) s ∈ S R ( s ( T ∗ T )).5. Same as 4. happens with Landweber’s method and ρ ( α ) = (1 − µ √ α ) α .
5. Conclusions
In this article we have extended the definition of qualification for spectral regularizationmethods introduced by Math´e and Pereverzev in [6]. This extension was constructedbearing in mind the concept of qualification as the optimal order of convergence of theregularization error that a method can achieve ([2], [5], [6], [11]). Three different levelsof generalized qualification were introduced: weak, strong and optimal. In particular,the first of these levels extends the definition introduced in [6] and a SRM havingweak qualification which is not qualification in the sense of Definition 1.2 was shown.Sufficient conditions for a SRM to have weak qualification were provided, as well asnecessary and sufficient conditions for a given order of convergence to be strong oroptimal qualification. Examples of all three qualification levels were provided andthe relationships between them as well as with the classical concept of qualificationand the qualification introduced in [6] were shown. Several SRMs having generalizedqualification in each one of the three levels and not having classical qualification werepresented. In particular, it was shown that the well known TSVD, Showalter’s andLandweber’s methods do have weak qualification. Finally several implications of thistheory in the context of orders of convergence, converse results and maximal sourcesets for inverse ill-posed problems, were briefly shown. More detailed results on theseimplications will appear in a forthcoming article.
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