GEOMETRY OF LINEAR ILL-POSED PROBLEMS IN VARIABLE HILBERT SCALES

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1 GEOMETRY OF LINEAR ILL-POSED PROBLEMS IN VARIABLE HILBERT SCALES PETER MATHÉ AND SERGEI V. PEREVERZEV Abstract. The authors study the best possible accuracy of recovering the unknown solution from linear ill-posed problems in variable Hilbert scales. A priori smoothness of the unknown solution is expressed in terms of general source conditions, given through index functions. Emphasis is on geometric concepts. The notion of regularization is appropriately generalized, and the interplay between qualification of regularization and index function becomes visible. A general adaptation strategy is presented and its optimality properties are studied. 1. Introduction In the present paper we consider numerical solution of operator equations Ax = y under presence of noise, which means we are given (1) y δ = Ax + δξ, where the operator A acts between Hilbert spaces X and Y and the noise ξ is assumed to be bounded ξ 1. A numerical method S for approximation x, based on observations y δ is given as an arbitrary mapping S : Y X. Its error at any problem instance x X is then given by (2) e(x, S, δ) := sup x S(y δ ). ξ 1 The worst-case error over a class F of problem instances is determined as (3) e(f, S, δ) := sup e(x, S, δ). x F The best possible accuracy is defined by minimization over all numerical methods, i.e., (4) e(f, δ) := inf e(f, S, δ). S:Y X Date: March 28,

2 2 PETER MATHÉ AND SERGEI V. PEREVERZEV In the present context we are interested in the asymptotic behavior of e(f, δ) as δ 0, when the class of problem instances A ϕ (R) is given through index functions ϕ as (5) A ϕ (R) := {x X, x = ϕ(a A)v, v R}, where A ϕ (R) is called source condition. For regularization we shall later on assume more specifically, that the index function is continuous, increasing and satisfies ϕ(0) = 0. The problem under consideration is the following. Suppose we are given an index function ϕ. Which qualification of a chosen regularization guarantees the optimal order of approximation, uniformly over A ϕ (R), after appropriate regularization? When the source conditions are given in terms of powers ϕ(t) := t µ, the answer to this question is known: We should use regularization of qualification p µ/2, see [28, Thm. 1.2]. Our aim is to answer the above question in the context of general source conditions. Moreover, we are going to show, how conditions on ϕ can be given, which are easy to verify, and which allow classical regularization, in particular Tikhonov s, to yield the best order. The presentation of the material is as follows. We use this overview to point at relevant references. Within the body of the presentation only a few further references are given, if necessary. We first recall the concept of variable Hilbert scales. Major contributions were made by Hegland in [6, 7] and Tautenhahn [25]. There are also other papers, where implicitly such concept can be found, we mention the recent paper on regularization of non-linear problems by Bakushinskiĭ [1] and the study by Mair [11]. Having the basic parameters, which drive a variable Hilbert scale, we may describe the degree of ill-posedness. This is in accordance with the notion as originally introduced by Wahba [29] in statistical context. More recently we mention [21]. The present authors tried to capture the main idea in [14]. This approach is generalized here. The best possible accuracy in the general context, without making the notion of variable Hilbert scales explicit, was given in [9]. Here we rely on this description. Hegland [7] also relied upon it, while Tautenhahn gave another proof in [25]. Actually, a corresponding problem was studied by Melkmann and Micchelli [17], we refer also to Micchelli and Rivlin [18]. Turning form the original problem (1) to the symmetrized one is common in many studies, we mention [28, Chapt. 2]. Within the present framework, the original problem and its symmetrized version, see (15)

3 ILL-POSED PROBLEMS IN VARIABLE HILBERT SCALES 3 below, may be studied equivalently. Note however, that this equivalence does not extend to discretizations, since there are discretizations for (1), which are not obtained from ones of the symmetrized problem. The study of discretizations is deferred to a separate paper, see [15]. The observation, that under certain conditions one can switch from the original problem to a diagonal one in l 2 was exploited in [12] and also in [14] by the present authors. This is particularly useful, when the noise is Gaussian white noise, thus orthogonally invariant. The important notion of the qualification of a regularization method was first introduced in [28, Chapt. 2, 3]. Here this concept is generalized to variable Hilbert scales, where its properties become transparent. The importance of appropriate qualification of regularization methods in variable Hilbert scales was recently emphasized in [20, 4.7]. Bakushinskiĭ [1] tried to capture the quality of the source condition within classical regularization. We mention the paper by Deuflhard, Engl and Scherzer [2], where the importance of determining the required qualification for general source condition became evident (Remark 3.2, there). Data-based adaptation to find the optimal regularization parameter, known as discrepancy principle dates back to Phillips [23], predating even Tikhonov s original paper [26]. It has then been reinvented by Morozov [19] and Marti [13]. However, it is known, that this principle does not provide the best order of approximation for all type of source conditions, for which Tikhonov regularization is optimal, see e.g. [5]. So, the question arises, whether there are strategies, which adapt to unknown smoothness, uniformly for all such source conditions. This has been studied by Gfrerer [3] and more recently by Tautenhahn and H amarik [24]. The proposed strategies are still not satisfactory, since additional approximate solutions, e.g. by iterated Tikhonov regularization have to be computed. The procedure of adapting to unknown smoothness, which our proposal is based on, was first studied in the context of statistics by Lepskiĭ [10]. Since then many authors have adopted this approach towards various applications, we mention [27] and [4]. The a posteriori principle proposed in the present paper is free from the above mentioned drawback of discrepancy principle. Namely, for the first time one has an adaptive principle that allows to reach the best order of accuracy for all linear ill-posed problems that in principle can be treated in optimal way by the regularization method with fixed qualification. In a final example we treat classes of source conditions, where classical regularization works. This class covers all source conditions, studied do far, in particular by Hohage [8]. This also sheds light on the

4 4 PETER MATHÉ AND SERGEI V. PEREVERZEV discussion in [2, Remark 3.2]. Under certain concavity assumptions, classical regularization of qualification, capturing these, is suited for regularization. 2. Variable Hilbert scales and the degree of ill-posedness Here we briefly summarize the concept of variable Hilbert scales, as introduced in Hegland [6, 7]. The basic ingredient is a non-negative compact self-adjoint operator T : X X, acting in a given initial Hilbert space X. Moreover T is assumed to be injective. Its singular numbers are denoted by (s k ) k=1, arranged in non-increasing order. In particular a := s 1 = T. Since T is compact, the only limit point is 0. T admits a (monotonic) Schmidt representation for an orthonormal system u 1, u 1,..., given by T x = s j x, u j u j, x X. j=1 Any function ϕ : (0, a] (0, ) is called an index function. We denote I(0, a] the set of all index functions on (0, a]. As pointed out in [6], this forms a multiplicative group. Each such function can be assigned a pre-hilbert space in the following way. Let { } n F := x, x = x, u j u j, n <, j=1 be the linear space of finite expansions in u 1, u 2,.... Given ϕ I(0, a] we can endow F with scalar product x, u j y, u j x, y :=, x, y F. ϕ 2 (s j ) j=1 The completion of F is denoted by X ϕ. The family {X ϕ, ϕ I(0, a]}, is called a variable Hilbert scale. For the understanding of this concept the following facts are important. (1) There is an embedding X ϕ X ψ, iff there is a constant C <, such that ϕ(s k ) Cψ(s k ). (2) The above embedding is compact if and only if lim ϕ(s k)/ψ(s k ) = 0. k (3) The adjoint space X ϕ to X ϕ is isometric to X 1/ϕ.

5 ILL-POSED PROBLEMS IN VARIABLE HILBERT SCALES 5 (4) The following interpolation inequality holds true for any two increasing functions ϕ, ψ, such that the composition ϕ 2 (ψ 2 ) 1 is convex. ) ) (ϕ 2 ) 1 ( x 2 θ/ϕ x 2 θ (ψ 2 ) 1 ( x 2 θ/ψ x 2 θ, x X max{ϕ,ψ} 1 θ \ 0. for any index function θ. (5) The following transfer relation is true. Let {X ϕ } be a scale generated by T and {Y ψ } be generated by the operator θ(t ), for some increasing function θ. Then an easy calculation shows, that X ϕ is isometric to Y ϕ θ 1. (6) Let A: X Y be any injective compact operator. The scales {X ϕ, ϕ I(0, a]}, generated by A A and {Y ϕ, ϕ I(0, a]}, generated by AA are isometrically isomorphic. As a consequence, any Hilbert scale in X, generated by some operator with singular values s 1, s 2,... is isometrically isomorphic to some scale in l 2, generated by a corresponding diagonal operator with diagonal entries s 1, s 2,.... From now on we shall assume, that the scale is generated by T := A A. We mention, that then A ϕ (R) is the ball of radius R in X ϕ. With this preliminary discussion we may turn to the description of the degree of ill-posedness, following previous discussion in [14]. We need to define the degree of ill-posedness of the operator A as well as the effective smoothness of the solution. The latter is certainly represented by ϕ, if the error is measured in X. It is represented by the ϕ/ν, the distance between the spaces X ϕ and X ν, if we agree to measure the error in the latter space. The degree of ill-posedness, say λ of the operator A is defined by (6) λ := inf { ψ, A 1 : X X ψ 1 }. Note, that in contrast to [14, 2], we need to fix the constant, since otherwise any multiple of λ would be admissible. An easy computation shows, that we arrive at λ(t) = 1/ t. As it will appear in Theorem 1 below, the error can be expressed by the two indices, the effective smoothness, and the distance between X ϕ and X λ, which is (7) Θ(t) := tϕ(t), t > 0, precisely as ϕ Θ 1, if Θ was increasing. This function will play a crucial role in choosing the regularization parameter as well as for representing the error, see Section 3.

6 6 PETER MATHÉ AND SERGEI V. PEREVERZEV 3. Best possible accuracy Within the context of variable Hilbert scales the best possible accuracy is known for a variety of index functions ϕ and for deterministic noise. For later use we briefly recall the approach. As usual, the best possible accuracy is represented by e(a ϕ (R), δ) = sup { x, x ϕ R, Ax δ}. Introducing v with x = ϕ(a A)v, and rescaling, we may rewrite this as (8) e 2 (A ϕ (R), δ) = R 2 sup { ϕ(a A)v 2, v 2 1, Aϕ(A A)v 2 δ 2 /R 2}. In many cases the optimization problem (8) has an explicit solution, we refer to [9] and also [6, 4], [25, Thm. 2.1]. It is immediate, that the function Θ from (7) is strictly increasing if ϕ was. Theorem 1. Let ϕ be any index function, for which Θ is strictly increasing, Θ(t) 0 as t 0. Then (9) e(a ϕ (R), RΘ(s j )) Rϕ(s j ), j = 1, 2,.... Moreover, if the function t ϕ 2 ((Θ 2 ) 1 (t)) is concave, then (10) e 2 (A ϕ (R), δ) = R 2 s(δ 2 /R 2 ), δ R A, where s is a piece-wise linear spline, interpolating (11) s(θ 2 (s j )) = ϕ 2 (s j ), j = 1, 2,.... As a consequence, (12) e(a ϕ (R), δ) Rϕ(Θ 1 (δ/r)), δ R A. Proof. The first statement is trivial. Representation (10) with (11) follows from [9]. It remains to prove (12). Let δ/r A be given. Then there are some k and convex combination with α + β = 1 for which (δ/r) 2 = αθ 2 (s k ) + βθ 2 (s k+1 ). By (10) we may write e 2 (A ϕ (R), δ) = R 2 αϕ 2 (s k ) + R 2 βϕ 2 (s k+1 ) (13) = R 2 αϕ 2 ((Θ 2 ) 1 (Θ 2 (s k ))) + R 2 βϕ 2 ((Θ 2 ) 1 (Θ 2 (s k+1 ))) R 2 ϕ 2 ((Θ 2 ) 1 ((δ/r) 2 )) = R 2 ϕ 2 (Θ 1 (δ/r)),, where we used the concavity assumption to derive (13). The proof is complete.

7 ILL-POSED PROBLEMS IN VARIABLE HILBERT SCALES 7 Remark 1. We first make the important observation, that the error tends to 0 only, if ϕ tends to 0, which means, if X ϕ is compactly embedded in X. We further note, that the concavity assumption on ϕ 2 ((Θ 2 ) 1 ) is equivalent to convexity of ρ(t) := Θ 2 (ϕ 2 (t)), which is just the assumption made in Tautenhahn [25, 1.1(iii)], thus Theorem 1 summarizes previous results. However, the representation of the error in terms of the function ϕ Θ 1 turns out to be useful. Under fairly general assumptions the function δ ϕ(θ 1 (δ/r)) also provides a lower bound. Precisely we state Corollary 1. Assume that ϕ is increasing and obeys a 2 -condition, i.e., there is C < for which ϕ(2t) Cϕ(t), 0 < t a. Assume furthermore, that t ϕ 2 ((Θ 2 ) 1 ) is concave. If moreover, the singular numbers of A A obey s j+1 /s j γ > 0, then there is a constant c γ > 0, such that (14) e(a ϕ, δ) c γ Rϕ(Θ 1 (δ/r)), 0 < δ Ra. Proof. First, iterating the 2 -condition, if necessary, we can find c γ, such that ϕ(γt) c γ ϕ(t), 0 < t a. Secondly, by monotonicity of Θ, given δ Ra, there is an index i, for which Θ 2 (s i+1 ) δ 2 /R 2 Θ 2 (s i ). Recall, that by (11), the exact error is provided through the piece-wise linear spline s. Taking all above facts into account we can estimate s(δ 2 /R 2 ) s(θ 2 (s i+1 )) = ϕ 2 (s i+1 ) ϕ 2 (γs i ) c 2 γϕ 2 (s i ) = c 2 γϕ 2 ((Θ 2 ) 1 (Θ 2 (s i ))) which allows to complete the proof. c 2 γϕ 2 ((Θ 2 ) 1 (δ 2 /R 2 )) = c 2 γϕ 2 (Θ 1 (δ/r)), A sufficient condition for ϕ 2 ((Θ 2 ) 1 ) to be concave is given in Proposition 1. Suppose ϕ is non-decreasing and twice differentiable on (0, a). Then ϕ 2 ((Θ 2 ) 1 ) is concave, provided t log ϕ(t) was concave. Sketch of the proof. Let f(t) := ϕ 2 ((Θ 2 ) 1 (t)). Re-parametrization yields the implicit formula f(tϕ 2 (t)) = ϕ 2 (t). Differentiating this twice results in the following representation in terms of s := Θ 2 (t). f (s) = ((ϕ)2 (s)) ((Θ) 2 (s)) + (log ϕ2 (s)) ϕ 4 (s)/((θ) 2 (s)),

8 8 PETER MATHÉ AND SERGEI V. PEREVERZEV which is negative, if (log ϕ 2 (s)) The general linear ill-posed problem Let us recall the original equation y δ = Ax + δξ. It is interesting to relate this to the symmetrized equation A y δ = A Ax + δa ξ, or after letting z δ := A y δ and ζ := A ξ we arrive at (15) z δ = A Ax + δζ. The advantage of (15) is the following: It is entirely defined within the Hilbert scale X ϕ, ϕ I(0, a]. But, the error ζ is now bounded in X t, as shows ζ t = A ξ t = ξ, the latter norm is the one in Y. Since A is injective, its adjoint has dense range, such that the noise is is not degenerate. Moreover, any method z δ S(z δ ) for solving equation (15) corresponds a method for solving (1), letting y δ S(A y δ ), which is seen to have the same error. On the other hand, the original problem is not simpler, since the best possible accuracy of problem (15) is the same as for the original one, which can be seen by an easy calculation. In this sense the problems as given by equations (1) and (15) are equivalent. Returning to (15), it is natural, and it has applications when studying random noise, a problem, which will be treated elsewhere in [16], to extend to more general assumptions on the noise, precisely we assume, that ζ ψ 1, for some index function ψ. The description of the problem is complete, after fixing the space X ν, in which the error will be measured. As a shorthand for the problem under consideration we will agree to write (A A: X ϕ X ν, A ϕ (R), A ψ, δ), showing all parameters involved. The following two relations between the parameters are basic. (1) The embedding X ϕ X ν is compact. (2) The limit lim t 0 tϕ(t)/ψ(t) = 0. As can be seen from the reasoning below, if the first assumption is violated, then the error does not tend to 0. Moreover, we note, that A A has a bounded inverse from X ψ X ψ/t. The second relation means that the embedding X ϕ X ψ/t is compact, and our problem is really ill-posed. Thus below we shall assume, that the above relations are satisfied.

9 ILL-POSED PROBLEMS IN VARIABLE HILBERT SCALES 9 The respective degree of ill-posedness of the operator, corresponding to (6) in Section 2, is now given through ψ/t, which is the minimal index function, for which the operator A A is invertible. The same reasoning as in the proof of Theorem 1 yields the following Corollary 2. Let us make the following assumptions, in addition to the relations above. (1) The function ϕ/ν is increasing. (2) The function tϕ/ψ is increasing. (3) The function t (ϕ/ν) 2 ((t 2 ϕ 2 /ψ 2 ) 1 (t)) is concave. Then the following error bound holds true. (16) e(a A: X ϕ X ν, A ϕ (R), A ψ, δ) R(ϕ/ν) ( (tϕ/ψ) 1 (δ/r) ). Remark 2. Theorem 1 is of course a special case, with ν 1 and ψ(t) = t. For the important case ν 1, we may provide the following sufficient condition for concavity. Proposition 2. Suppose, ϕ and ψ ate twice differentiable on (0, a). Under (1) and (2) the function t ϕ 2 ((t 2 ϕ 2 /ψ 2 ) 1 (t)) is concave, provided log ϕ was concave and t t 2 /ψ 2 (t) was convex. Sketch of the proof. Let f denote the function under consideration. If we furthermore abbreviate g := t 2 /ψ 2 and h := ϕ 2, then we obtain the following implicit representation. f(g(t)h(t)) = h(t). Differentiating this twice yields (suppressing the variable t), f (gh)[(gh) ] 2 = 1 (gh) [ g h 2 (log ϕ) h g (h ) 2 g ], such that under the above assumptions f is non-negative. Remark 3. Note, that for the function ψ(t) = t, the function t t 2 /ψ 2 (t) is just t t, which is convex in a trivial manner. We end this section with the following discussion. If two problems (A A: X ϕ X ν, A ϕ (R), A ψ, δ) and (B B : X ϕ X ν, A ϕ (R), A ψ, δ) are related via a monotone transformation θ, for which A A = θ(b B), ϕ = ϕ θ and ψ = ψ θ, then the best possible accuracies coincide. It is thus interesting to know, whether a given ill-posed problem with operator, say B : X Z, mapping X into some other Hilbert space Z, corresponds to some problem within the Hilbert scale, generated by

10 10 PETER MATHÉ AND SERGEI V. PEREVERZEV A A. This is is the case, if B admits a (monotonic) Schmidt representation B = j=1 β ju j z j, with the same orthonormal system u 1, u 2,..., as A A. Such condition turned out to be useful by Mair and Ruymgaart [12]. As mentioned in Section 2, this allows to reduce the original ill-posed problem to a diagonal one in l 2. If this is fulfilled, then we can assign θ(β j ) = s j, j = 1, 2,.... If this correspondence is increasing, then the above calculus applies. 5. Regularization The above upper bounds are obtained by some abstract argument and are not supported by an explicit method. Hegland [6] and Tautenhahn [25] have indicated that certain specific methods, involving the index function ϕ, may be used, to achieve the best possible accuracy as given in (12). It is however easy to design a method, say S, which realizes e(a ϕ (R), S, δ) 2Rϕ(Θ 1 (δ/r)). The considerations in Section 4 allow to restrict our attention to selfadjoint problems (15), with noise in X t. To this end, given δ Ra, let α satisfy Θ(α) = δ/r. Furthermore, let N = N(α) := max {j, s j α}. It is then straight forward to see, that (17) S N (y δ ) := N j=1 1 s j y δ, u j u j, realizes e(a ϕ (R), S, δ) Rϕ(α) + δ/ α 2Rϕ(α), from which the assertion follows. This reveals two things. First, up to a factor 2, the upper bound can be achieved without additional concavity assumptions. Second, the above method (17) requires complete knowledge of the operator, which is numerically infeasible. For this reason regularization methods, which replace the above spectral cut-off by feasible functions of the operator A A have been developed. The study of such regularizations within the framework of variable Hilbert scales is the scope of this section. Specifically, we are interested in regularization methods given by some operator function α g α (A A), 0 < α a, i.e., the approximation to x A ϕ is given by choosing some α = α(δ) and letting x α,δ := g α (A A)z δ (= g α (A A)A y δ ).

11 ILL-POSED PROBLEMS IN VARIABLE HILBERT SCALES 11 By the spectral calculus, each function defined on (0, a) taking real values can be assigned a respective function taking non-negative operators to self-adjoint ones. Therefore we may and do identify g α with its real valued function. Our aim is to discuss the interplay between certain qualification of the regularization and the index function. A look at the error for solving (15) using g α shows, that for any x X we have (18) x g α (A A)z δ = x g α (A A)A Ax δg α (A A)ζ. Therefore it is natural to assume that I g α (A A)A A: X X C, 0 < α a, even convergence to 0 as α 0 will be required later on, and g α (A A)A A: X t X C α, 0 < α a. The latter requirement is somewhat arbitrary. We want however, that the optimal parameter α corresponds to the spectral cut-off, i.e., it should solve Θ(α) = δ/r. Under this side restriction, the asymptotics g α (A A)A A: X C t X α is necessary. In terms of real functions g α these requirements can be expressed as follows. Definition 1. A family g α, 0 < α a is called regularization, if there are constants γ and γ for which and sup 1 λg α (λ) γ, 0 < α a. 0<λ a sup λ gα (λ) γ, 0 < α a, 0<λ a α The regularization g α is said to have qualification ρ, for an increasing function ρ : (0, a) R +, if (19) sup 1 λg α (λ) ρ(λ) γρ(α), 0 < α a. 0<λ a Remark 4. It is worthwhile to note, that for any given ρ respective regularization of qualification ρ can be constructed. Actually, the spectral cut-off (17), which corresponds to { 1/λ, α λ a, g α (λ) := 0, 0 < λ < α, has arbitrary qualification.

12 12 PETER MATHÉ AND SERGEI V. PEREVERZEV The classical regularizations are special cases of the general definition by using monomials of prescribed degree, thus sup λ q 1 λg α (λ) γ q α q, for every 0 q p. 0<λ a In this case, we call this classical qualification of order p. We now turn to the study of the interplay between qualification, say ρ and properties of the index function, say ϕ. Definition 2. We say, that the qualification ρ covers ϕ, if there is c > 0 such that (20) c ρ(α) ϕ(α) inf ρ(λ) α λ a ϕ(λ), 0 < α a. If this is the case, then we shall say, that ρ covers ϕ with constant c. Formally, we need the slightly weaker assumption. Lemma 1. The qualification ρ covers ϕ, if there are c > 0 and 0 < t 0 a, such that (20) is fulfilled with inf restricted to α λ t 0 and constant c. Proof. If α t 0 λ a, then by monotonicity of ρ and of ϕ, we conclude ρ(λ) ϕ(λ) ρ(t 0) ϕ(a) cϕ(t 0) ϕ(a) ρ(α) ϕ(α), making use of (20) for α λ t 0, only, to obtain the last estimate. Remark 5. If the function λ ρ(λ)/ϕ(λ) is increasing, then (20) is certainly satisfied with c = 1. Since the infimum in (20) is certainly less then or equal its value at a, necessarily X ρ X ϕ. The importance of this notion is expressed in the following Proposition 3. Let ϕ be any non-decreasing index function and let g α be a regularization of qualification ρ that covers ϕ. Then (21) sup 1 λg α (λ) ϕ(λ) γ ϕ(α), α a, 0<λ a c where γ is from (19) and c from (20). Proof. We introduce the function λ 1 λg α (λ) ϕ(λ). We need to show, that it is uniformly bounded by the right hand side in (21), for any value of α. We distinguish two cases. First, if λ α, then (21) is

13 ILL-POSED PROBLEMS IN VARIABLE HILBERT SCALES 13 fulfilled by monotonicity of ϕ with constant γ. Otherwise, if α λ a, then we conclude 1 λg α (λ) ϕ(λ) 1 λg α (λ) ρ(λ) ϕ(λ) ρ(λ) ϕ(λ) γρ(α) sup α λ a ρ(λ) The proof is complete. γ c ρ(α)ϕ(α) ρ(α) = γ c ϕ(α). It is interesting to know, how a regularization acts, if the qualification does not cover the actual smoothness. A look at the above proof reveals the following Corollary 3. Let ϕ be any non-decreasing index function and let g α be a regularization of qualification ρ. If X ϕ X ρ, then there is a constant C < for which (22) sup 1 λg α (λ) ϕ(λ) Cρ(α), α a, 0<λ a We now state and prove the main result in this section. Theorem 2. Let ϕ be any increasing index function and let ᾱ be chosen to satisfy (23) ᾱϕ(ᾱ) = δ/r. If g α is any regularization of qualification that covers ϕ with constant c, then (24) e(a ϕ (R), gᾱ, δ) R (γ/c + γ ) ϕ(θ 1 (δ/r)), 0 < δ Ra. Proof. We analyze the error of gᾱ with the above choice of ᾱ. Recall that for any x X we have x xᾱ,δ = x gᾱ(a A)A Ax δgᾱ(a A)ζ. Therefore, and since x A ϕ (R), (25) x xᾱ,δ R (I gᾱ(a A)A A)ϕ(A A) X X + δ gᾱ(a A) X t X R sup 1 λgᾱ(λ) ϕ(λ) + δ sup λgᾱ(λ). 0<λ a 0<λ a

14 14 PETER MATHÉ AND SERGEI V. PEREVERZEV By the qualification of g α the first summand is bounded by Rγ/cϕ(ᾱ), using Proposition 3. The second summand above can be bounded by γ δ/ ᾱ. This leads to x xᾱ,δ ( Rγ/cϕ(ᾱ) + γ δ/ ᾱ ) = R (γ/c + γ ) ϕ(ᾱ). Since ᾱ = Θ 1 (δ/r), the proof of the theorem is complete. In view of Theorem 1 and Corollary 1 the above asymptotics cannot be beaten. Remark 6. It is interesting to look at the respective result when measuring the error in X ν. If ν(t) 0 as t 0, then we need regularization with less qualification! It needs to cover ϕ/ν,only. 6. Adaptation to unknown source condition To complete the picture of regularization we shall now discuss the topic of adaptation. Here the goal is to find a strategy of choosing the regularization parameter α without knowledge of ϕ. Additionally the quality of the adaptive strategy will depend on some bound of the 2 -condition. Details are given below. This strategy will work successfully uniformly for index functions, which are uniformly covered by some known qualification. It is not true for discrepancy principle. For example, Tikhonov regularization has a qualification ρ(λ) = λ, while discrepancy principle may provide order-optimal choice of regularization parameter if index function ϕ(λ) is covered by λ. We propose a scheme, based on Lepskii s original approach, which chooses the appropriate α from a finite set of possible parameters, for a given regularization, with known constant γ from (19). To this end we recall the estimate (24) e(a ϕ, gᾱ, δ) (γ/c + γ ) ϕ(θ 1 (δ)), 0 < δ a, Let us denote C γ := max {γ/c, γ }. Precisely, fix q > 1 and α 0 > δ 2, which is certainly necessary to have a nontrivial error bound. Now let α k := α 0 q k, k = 1, 2,..., n, with n such that α n 1 a α n. (26) q := {α k, k = 0, 1,..., n}. The cardinality n is of order log(a/δ 2 )/ log q log(1/δ). Now the strategy consists in computing successively α 0, α 1,... as long as (27) x αi,δ x αi 1,δ 4C γ δ αi 1.

15 It terminates with ILL-POSED PROBLEMS IN VARIABLE HILBERT SCALES 15 { } δ (28) ᾱ := max α i : x αi,δ x αi 1,δ 4C γ. αi 1 Remark 7. As described above, this is an ideal situation, since in general the involved norm might not be computable. However, in practical situations, the regularization is based in discretization, where often these norms can be computed. In order to study the property of the finally chosen regularization xᾱ,δ we need to introduce (29) α := max {α q, Θ(α) δ}, which is not accessible, since ϕ, thus Θ are unknown. We note, that by this definition Θ(α ) δ, while Θ(qα ) > δ, which will be important below. Proposition 4. Fix any 0 < c 1 and let C γ,q := 2C γ (1 + 2 q q 1 ). If ϕ is covered by ρ with constant c, then the following assertions are true, uniformly for x A ϕ. (1) α ᾱ. (2) x xᾱ,δ C γ,q δ/ α. Proof. First, by construction, for α, condition (27) is satisfied, since α = α l for some l, hence x α,δ x αl 1,δ x x αl,δ + x x αl 1,δ ( C γ ϕ(α l ) + δ 4C γ, αl 1 δ αl + ϕ(α l 1 ) + δ ) αl 1

16 16 PETER MATHÉ AND SERGEI V. PEREVERZEV by monotonicity. Thus α ᾱ. Also, ᾱ = α m, for some m l. Using the triangle inequality successively, we arrive at m l x xᾱ,δ x x α,δ + x αm j,δ x αm j+1,δ j=1 m l 1 x x α,δ + 4δC γ αm i = C γ ( ϕ(α ) + δ 2C γ + δ 4C γ α j=1 δ ) + δ 4C m l γ α α α j=1 1 q m l j q δ = C γ,q. q 1 α The proof is complete. The adaptive strategy described above provides the optimal order of accuracy, uniformly over the following class of index functions. Given D <, let F c,d := {ϕ, ϕ(qt) Dϕ(t), t > 0, ϕ is covered by ρ with constant c}. Theorem 3. On the class F c,d the adaptive strategy provides the following error bound, uniformly for x A ϕ. (30) x xᾱ,δ C γ,q D qϕ(θ 1 (δ)), 0 < δ a. Proof. By Proposition 4 we can conclude δ δ x xᾱ,δ C γ,q = C γ,q q α qα which proves (30). C γ,q qϕ(qα ) C γ,q qdϕ(α ) C γ,q qdϕ(θ 1 (δ)), Remark 8. In the classical Hilbert scales, when smoothness is measured in terms of ϕ(t) := t ν, and regularization is classical with ρ(t) = t µ, then ρ covers ϕ if and only if µ ν. Moreover, the respective constant c equals 1. The 2 -conditions for all ϕ as above are then satisfied with D := q µ, which allows to apply the adaptive strategy, if smoothness is unknown, but an upper bound is given. Again, it is interesting to know, how this adaptation proceeds, if the qualification of the chosen regularization does not cover the actual smoothness ϕ.

17 ILL-POSED PROBLEMS IN VARIABLE HILBERT SCALES 17 Corollary 4. Suppose the qualification ρ of the chosen regularization does not cover ϕ. If X ϕ X ρ, then the adaptive strategy yields automatically (31) x xᾱ,δ 6γCρ(Θ 1 (δ)), 0 < δ a, uniformly for x A ϕ, where C is the embedding constant and Θ(t) := tρ(t). Thus xᾱ,δ provides the optimal order on class A ρ. 7. Example: p-concave source conditions We now describe a class of source conditions, obeying certain concavity properties. This allows to prove, that classical regularization with appropriate qualification provides the optimal order of approximation. The following notion generalizes concavity of a real function. Let p 0 be any real number. Definition 3. A function ϕ : R + R + is called concave of order p on the interval (0, t 0 ), if t ϕ(t)/t p is non-decreasing and concave for 0 < t t 0. For p = 0 this resembles the notion of monotonicity and concavity (near 0). The following class of functions covers all cases studied so far. Example. Given some p 0 and µ 0, we let ϕ p,µ (t) := t p log µ (1/t), t > 0. It is straight forward to check, that each ϕ p,µ is concave of order p on (0, e ( µ 1) ), provided that at least one of the parameters p or µ is positive. The following result shows, that Theorem 1 is applicable for such source conditions. Proposition 5. Suppose ϕ is twice differentiable on (0, a). If for some p the function ϕ is concave of order p on (0, t 0 ), then t ϕ 2 ((Θ 2 ) 1 (t)) is concave for t t 0. Proof. Suppose, ϕ is concave of order p on (0, t 0 ). Since it is the product of a concave index function with a polynomial, log ϕ is concave. Thus Proposition 1 applies. For p-concave functions we state the following basic observation. Lemma 2. Suppose ϕ is concave of order p on (0, t 0 ). covered by any classical regularization of order p + 1. Then it is

18 18 PETER MATHÉ AND SERGEI V. PEREVERZEV Proof. By assumption, since ϕ p : t ϕ(t)/t p is concave, we have ϕ p (s)/s ϕ p (t)/t, whenever 0 < s t. Thus the function t t p+1 /ϕ(t) is monotonically increasing, which ensures property (20) with constant c = 1. We conclude our study with the following discussion. First, for the index functions from the above example, we can provide the asymptotically optimal error and indicate the respective parameter choice for appropriate regularization. Let us abbreviate A p,µ := A ϕp,µ. Theorem 4. Let 0 p < and µ > 0. (1) As δ 0 we have e(a p,µ, δ) ( (δ 2 ) p log µ (1/δ 2 ) 1/(2p+1)) 1/(2p+1) (1 + O(δ)). (2) Any regularization of qualification at least p+1 with choice of α := ( δ 2 log 2µ (1/δ 2 ) 1/(2p+1)) 1/(2p+1), provides the optimal order of approximation. The proof is based on the following assertions, which can easily be verified. Lemma 3. Let q, µ > 0 and introduce ψ q,µ (s) := s 1/q log µ/q 1/s 1/q, s > 0. For the inverse function ϕ 1 q,µ we have ϕ 1 q,µ(s) lim s 0 ψ q,µ (s) = 1. Moreover, if Θ p,µ (t) := tϕ p,µ (t), t > 0, then lim δ 0 ϕp,µ 2p+1 (Θ 1 p,µ(δ)) (δ 2 ) p log µ (1/δ 2 ) 1/(2p+1) = 1. Remark 9. The assertions from Theorem 4 extend naturally to µ = 0, yielding the classical estimates. For µ > 0 and p = 0, this provides the asymptotic error e(a 0,µ, δ) log µ (1/δ 2 )(1 + O(δ)). Also, Tikhonov regularization with α := δ 2 log 2µ (1/δ 2 ), provides the optimal order. However, the error is not very sensitive with respect to over-smoothing, i.e., choosing α too large, for instance α δ r, for 0 < r < 2. Such choice still provides the optimal order. This is consistent with [8] and [22].

19 ILL-POSED PROBLEMS IN VARIABLE HILBERT SCALES 19 Finally, we finish with an immediate consequence for concave source conditions. We state the following interesting Corollary 5. Tikhonov regularization ḡ α (λ) := 1/(α + λ), used with Θ(α) = δ/r is optimal, uniformly for all source conditions, given by concave index functions. With this regularization parameter its error is bounded by (32) e(a ϕ (R), ḡ α, δ) 3/2Rϕ(Θ 1 (δ/r)). Moreover, since concave functions belong to F(1, q), the adaptive strategy applies and adapts to the optimal rate of convergence. References [1] A. B. Bakushinskiĭ. On the rate of convergence of iterative processes for nonlinear operator equations. Zh. Vychisl. Mat. Mat. Fiz., 38(4): , [2] Peter Deuflhard, Heinz W. Engl, and Otmar Scherzer. A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions. Inverse Problems, 14(5): , [3] Helmut Gfrerer. An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates. Math. Comp., 49(180): , S5 S12, [4] Alexander Goldenshluger and Sergei V. Pereverzev. Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations. Probab. Theory Related Fields, 118(2): , [5] C. W. Groetsch. The theory of Tikhonov regularization for Fredholm equations of the first kind, volume 105 of Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston, MA, [6] Markus Hegland. An optimal order regularization method which does not use additional smoothness assumptions. SIAM J. Numer. Anal., 29(5): , [7] Markus Hegland. Variable Hilbert scales and their interpolation inequalities with applications to Tikhonov regularization. Appl. Anal., 59(1-4): , [8] Thorsten Hohage. Regularization of exponentially ill-posed problems. Numer. Funct. Anal. Optim., 21(3-4): , [9] V. Ivanov and T. Korolyuk. Error estimates for solutions of incorrectly posed linear problems. Zh. Vychisl. Mat. i Mat. Fiz., 9:30 41, [10] O. V. Lepskiĭ. A problem of adaptive estimation in Gaussian white noise. Teor. Veroyatnost. i Primenen., 35(3): , [11] B. A. Mair. Tikhonov regularization for finitely and infinitely smoothing operators. SIAM J. Math. Anal., 25(1): , [12] Bernard A. Mair and Frits H. Ruymgaart. Statistical inverse estimation in Hilbert scales. SIAM J. Appl. Math., 56(5): , [13] J. T. Marti. An algorithm for computing minimum norm solutions of Fredholm integral equations of the first kind. SIAM J. Numer. Anal., 15(6): , 1978.

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