OPTIMAL ORDER YIELDING DISCREPANCY PRINCIPLE FOR SIMPLIFIED REGULARIZATION IN HILBERT SCALES: FINITE-DIMENSIONAL REALIZATIONS

Simplified regularization using finite-dimensional approximations in the setting of Hilbert scales has been considered for obtaining stable approximate solutions to ill-posed operator equations. The derived error estimates using an a priori and a posteriori choice of parameters in relation to the noise level are shown to be of optimal order with respect to certain natural assumptions on the ill posedness of the equation. The results are shown to be applicable to a wide class of spline approximations in the setting of Sobolev scales.


Introduction.
Many of the inverse problems that occur in science and engineering are ill posed, in the sense that a unique solution that depends continuously on the data does not exist. A typical example of an ill-posed equation that often occurs in practical problems, such as in geological prospecting, computer tomography, steel industry, and so forth, is the Fredholm integral equation of the first kind (cf. [2,6,8]). Many such problems can be put in the form of an operator equation Ax = y, where A : X → Y is a bounded linear operator between Hilbert spaces X and Y with its range R(A) not closed in Y .
Regularization methods are to be employed for obtaining a stable approximate solution for an ill-posed problem. Tikhonov regularization is a simple and widely used procedure to obtain stable approximate solutions to an ill-posed operator equation (2.1). In order to improve the error estimates available in Tikhonov regularization, Natterer [17] carried out error analysis in the framework of Hilbert scales. Subsequently, many authors extended, modified, and generalized Natterer's work to obtain error bounds under various contexts (cf. Neubauer [18], Hegland [7], Schröter and Tautenhahn [20], Mair [10], Nair et al. [16], and Nair [13,15]). Finite-dimensional realizations of the Hilbert scales approach has been considered by Engl and Neubauer [3].
If Y = X and A itself is a positive selfadjoint operator, then the simplified regularization introduced by Lavrentiev is better suited than Tikhonov regularization in terms of speed of convergence and condition numbers of the resulting equations in the case of finite-dimensional approximations (cf. Schock [19]).
In [4], the authors introduced the Hilbert scales variant of the simplified regularization and obtained error estimates under a priori and a posteriori parameter choice strategies which are optimal in the sense of the "best possible worst error" with respect to certain source set. Recently (cf. [5]), the authors considered a new discrepancy principle yielding optimal rates which does not involve certain restrictive assumptions as in [4]. The purpose of this paper is to obtain a finite-dimensional realization of the results in [5].

Preliminaries.
Let H be a Hilbert space and A : H → H a positive, bounded selfadjoint operator on H. The inner product and the corresponding norm are denoted by ·, · and · , respectively. Recall that A is said to be a positive operator if Ax, x ≥ 0 for every x ∈ H. For y ∈ R(A), the range of A, consider the operator equation where k(·, ·) is a nondegenerate kernel which is square integrable, that is, 1], and such that the eigenvalues of the corresponding integral operator A : are all nonnegative (cf. [14]). For example, one of the important ill-posed problems which arise in applications is the backward heat equation problem: the problem is to determine the initial temperature ϕ 0 := u(·, 0) from the measurements of the final temperature ϕ T := u(·,T ), where u(ξ, t) satisfies We recall from elementary theory of partial differential equations that the solution u(ξ, t) of the above heat equation is given by (cf. Weinberger [23]) whereφ 0 (n) for n ∈ N are the Fourier coefficients of the initial temperature ϕ 0 (ξ) := u(ξ, 0). Hence, The above equation can be written as e −n 2 π 2 T ϕ 0 ,u n u n (ξ) with u n (ξ) = 2sin(nπ ξ). (2.8) Thus the problem is to solve the operator equation Note that the above integral operator is compact, positive, and selfadjoint with positive eigenvalues e −n 2 π 2 T and corresponding eigenvectors u n (·) for n ∈ N.
For considering the regularization of (2.1) in the setting of Hilbert scales, we consider a Hilbert scale {H t } t∈R generated by a strictly positive operator L : (2.12) By the operator L being strictly positive, we mean that Lx, x > 0 for all nonzero x ∈ H. Recall (cf. [9]) that the space H t is the completion of D := ∞ k=0 D(L k ) with respect to the norm x t , induced by the inner product (2.13) Moreover, if β ≤ γ, then the embedding H γ H β is continuous, and therefore the norm · β is also defined in H γ and there is a constant c β,γ such that (2.14) An important inequality that we require in the analysis is the interpolation inequality (2.16) and the moment inequality where B is a positive selfadjoint operator (cf. [2]).
We assume that the ill-posed nature of the operator A is related to the Hilbert scale {H t } t∈ according to the relation (2.18) for some positive reals a, c 1 , and c 2 .
For the example of the integral operator considered in (2.4), one may take L to be defined by In this case, it can be seen that and the constants a, c 1 , and c 2 in (2.18) are given by a = 1, c 1 = c 2 = 1/π 2 (see Schröter and Tautenhahn [20,Section 4]). The regularized approximation ofx, considered in [4] is the solution of the wellposed equation A + αL s x α = y, α > 0, (2.22) where s is a fixed nonnegative real number. Note that if D(L) = X and L = I, then the above procedure is the simplified or Lavrentiev regularization. Suppose the data y is known only approximately, sayỹ in place of y with y −ỹ ≤ δ for a known error level δ > 0. Then, in place of (2.22), we have (2.23) It can be seen that the solutionx α of the above equation is the unique minimizer of the function One of the crucial results for proving the results in [4,5] as well as the results in this paper is the following proposition, where the functions f and g are defined by Using the above proposition, the following result has been proved by George and Nair [4].
Theorem 2.2 (cf. [4,Theorem 3.2]). Supposex ∈ H t , 0 < t ≤ s + a, and α > 0, and x α is as in (2.23). Then In particular, if α = c 0 δ (s+a)/(t+a) for some constant c 0 > 0, then For proposing a finite-dimensional realization, we consider a family {S h : h > 0} of finite-dimensional subspaces of H k for some k ≥ s, and consider the minimizerx α,h of the map defined in (2.24) when x varies over S h . Equivalently,x α,h is the unique element in S h satisfying the equation (2.31) As in Engl and Neubauer [3], we assume the following approximation properties for S h . There exists a constant κ > 0 such that for every u ∈ H r with r > k ≥ s, As already exemplified in [3], the above assumption is general enough to include a wide variety of approximations spaces, such as spline spaces and finite element spaces.
We will also make use of the following result from Engl and Neubauer [ (2.33)

General error estimates.
For a fixed s > 0, letx α andx α,h be as in (2.23) and (2.31), respectively. We will obtain estimate for x α −x α,h so that we get an estimate for x −x α,h using Theorem 2.2 and the relation In view of the interpolation inequality (2.15), by taking ρ = −a/2, τ = s/2, and λ = 0 in (2.15), we get Thus, we can deduce an estimate for x α −x α,h once we have estimates for x α − x α,h −a/2 and x α −x α,h s/2 . For this purpose, we first prove the following.

Lemma 3.1. Letx α andx α,h be as in (2.23) and (2.31), respectively. Then
Proof. It can be seen (cf. [16]) that defines a complete inner product on D(L). Let · * be the norm induced by ·, · * , that is, Let X be the space D(L) with the inner product ·, · * and let P h be the orthogonal projection of X onto the space S h . Then from (2.23) and (2.31) we have that is, Hence so that Now the result follows using the definition of · * .
Next we obtain estimate for x α −x α,h using the estimates for x α −x α,h −a/2 and x α −x α,h s/2 . We will use the notation and observe that for α > 0, Theorem 3.2. Supposex ∈ H t and assumption (2.32) holds, and letx α andx α,h be as in (2.23) and (2.31), respectively. Then where f and g are as in (2.25), and Proof. First we prove 14) with Ᏺ(s, a), Ᏻ(s,t,a), and Φ(s,h,α) as in the statement of the theorem. By Lemma 3.1 and Proposition 2.1, it follows that (3.17) Note that where Hence, by Lemma 2.3, we have In particular, From these, we obtain (3.14) and (3.15). Now, to prove (3.16), observe from (2.23) and (3.11) that By Proposition 2.1, taking ν = −s/(s + a), we have where it follows from the above relations that (3.29) Thus, (3.25) and (3.29) give Now, the estimates (3.14) and (3.15) together with the interpolation inequality (3.2) give From this, the result follows by making use of the estimate (3.16) forx α .

4.
A priori error estimates. Now we choose the regularization parameter α and discretization parameter h a priori depending on the noise level δ such that optimal order O(δ t/(t+a) ) yields wheneverx ∈ H t . x −x α,h ≤ η(s, t) + ξ(s, t) δ t/(t+a) , Proof. Using the choice (4.1), it is seen that Therefore, by Theorem 3.2, we have Also, from Theorem 2.2, we have x −x α ≤ η(s, t)δ t/(t+a) . (4.6) Thus the result follows from the inequality Remark 4.2. We observe that the error bound obtained is of the same order as of Theorem 2.2, and this order is optimal with respect to the source set in the sense of the best possible worst error (cf. [4]).

Discrepancy principle.
In this section, we consider a discrepancy principle to choose the regularization parameter α depending on the noise level δ and the discretization parameter h. This is a finite-dimensional variant of the discrepancy principle considered in [5].
We assume throughout that y = 0. Suppose thatỹ ∈ H is such that for a known error level δ > 0 and P hỹ = 0, where P h is the orthogonal projection of H onto S h . We assume, throughout this section, that for some c 3 > 0, independent of h. Let We will make use of the relation L −s/2 P h x is nonzero for every x ∈ H with P h x = 0, so that the function F(α,x) is well defined for all such x. We observe that the assumption P h x = 0 is satisfied for x = 0 and h small enough, if P h x → x as h → 0 for every x ∈ H.
In the following, we assume that h is such that P hỹ = 0. In order to choose the regularization parameter α, we consider the discrepancy principle for some b, d > 0. In the due course, we will make use of the relation which can easily be derived from Proposition 2.1. First we prove the monotonicity of the function F(α,x) defined in (5.5).

Theorem 5.1. Let x ∈ H be such that the function α F(α,x) for α > 0 in (5.5) is well defined. Then, F(·,x) is increasing and it is continuously differentiable with
Proof. Using the definition (5.5) of F(α,·), we have Let {E λ : 0 ≤ λ ≤ a} be the spectral family of A s , where a ≥ A s . Then (5.10) Similarly, we obtain Therefore, from (5.9), by using (5.10) and (5.11), we get (5.12) The above equation can be rewritten as we see that (5.15) Also, we have Hence, To prove the last part of the theorem, we observe that We note that (5.20) it follows that From this we can conclude that This completes the proof.
For the next theorem, in addition to (5.1), we assume that the inexact dataỹ satisfies the relation This assumption is satisfied for small enough h and δ, if, for example, In order to obtain an estimate for the error x −x α,h with the parameter choice strategy (5.30), we will make use of (3.31). The next lemma gives an error estimate for x α s in terms of α = α(δ, h), δ, and h.  /(2s + 2a)).
Proof. By (3.30), we have To obtain an estimate for αR α A −s/(2s+2a) s L −s/2ỹ , we will make use of the moment inequality (2.17). Precisely, we use (2.17) with Then, since we have .
Therefore, if α := α(δ, h) is the unique solution of (5.30), then we have This completes the proof.
Theorem 5.5. Under the assumptions in Lemma 5.4, for any fixed τ > 0, This completes the proof.
Corollary 5.8. If t, s, a satisfy max{0, 1 − a} < t ≤ s and τ is large enough such that with c 6 as in Theorem 5.7.
Proof. Let ζ, µ be as in Theorems 5.5 and 5.7, respectively. Then we observe that Hence the result follows from Theorem 5.7.
6. Order optimality of the error estimates. In order to measure the quality of an algorithm to solve an equation of the form (2.1), Micchelli and Rivlin [12]  where is the best possible worst error. Here the stabilizing set M is assumed to be convex such that M = −M with 0 ∈ M (see also, Vaȋnikko and Veretennikov [22]), and infimum is taken over all algorithms R : Y → X. Since H is a Hilbert space and A is assumed to be selfadjoint and positive, we, in fact, have (cf. Melkman and Micchelli [11]) Now using the assumption (2.18), and taking r = −a, λ = 0 in the interpolation inequality (2.15), we obtain Therefore, for the set It is known that the above estimate for e(M t,ρ ,δ) is sharp (cf. Vainikko [21]). In view of the above observations, an algorithm is called an optimal order yielding algorithm with respect to M t,ρ and the assumption (2.18), if it yields an approximationx corresponding to the dataỹ with y −ỹ ≤ δ satisfying x −x = O δ t/(t+a) , x ∈ H t . (6.8) Clearly, Corollary 5.8 shows that if h = O(δ) and if max{0, 1 − a} < t ≤ s and τ is large enough such that then we obtain the optimal order.

Applications.
For r ≥ 2, denote by S h the space of r th-order splines on the uniform mesh of width h = 1/n, that is, S h consists of functions in C r −1 [0,1]  so that assumption (2.32) is satisfied. We take L as in (2.19), that is, In this case, (H t ) t∈R is given as in (2.21). It can be seen that Thus we get (7.8) withĉ = k 0,s .