Error Estimates for Regularization Methods in Hilbert Scales

Tautenhahn, Ulrich

doi:10.1137/s0036142994269411

Cited by 71 publications

(69 citation statements)

References 16 publications

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“…In terms of the original concept of a Hilbert scale, as introduced by Krein and Petunin [12] and generated by a densely-defined, unbounded, self-adjoint and strictly positive operator T , such inequalities take the form where u a = T a/2 u . Through the appropriate choice of T and the values for r and s, the corresponding inequality (1) can be used to derive estimates for the error e of the regularized solution of improperly posed operator equations that simultaneously take account of both the compact and smoothing nature of the operator (Groetsch [7], Natterer [17], Schröter and Tautenhahn [21], Tautenhahn [22]). Typically, such inequalities lead to bounds for the error e of the form e ≤ Cδ θ , where δ is a measure of the error in the data.…”

Section: Introductionmentioning

confidence: 99%

Dilational interpolatory inequalities

Hegland

Anderssen

2011

Math. Comp.

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Abstract. Operationally, index functions of variable Hilbert scales can be viewed as generators for families of spaces and norms and, thereby, associated scales of interpolatory inequalities. Using one-parameter families of index functions based on the dilations of given index functions, new classes of interpolatory inequalities, dilational interpolatory inequalities (DII), are constructed. They have ordinary Hilbert scales (OHS) interpolatory inequalities as special cases. They represent a precise and concise subset of variable Hilbert scales interpolatory inequalities appropriate for deriving error estimates for peak sharpening deconvolution. Only for Gaussian and Lorentzian deconvolution do the DIIs take the standard form of OHS interpolatory inequalities. For other types of deconvolution, such as a Voigt, which is the convolution of a Gaussian with a Lorentzian, the DIIs yield a new class of interpolatory inequality. An analysis of deconvolution peak sharpening is used to illustrate the role of DIIs in deriving appropriate error estimates.

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Section: Introductionmentioning

confidence: 99%

Dilational interpolatory inequalities

Hegland

Anderssen

2011

Math. Comp.

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“…Thresholding in the Fourier domain has been used, for example, in a deconvolution problem in Mair and Ruymgaart [1996] or Neumann [1997] and coincides with an approach called spectral cut-off in the numerical analysis literature (c.f. Tautenhahn [1996]). …”

Section: Introductionmentioning

confidence: 99%

Adaptive estimation in circular functional linear models

Comte

Johannes

2010

Math. Meth. Stat.

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We consider the problem of estimating the slope parameter in circular functional linear regression, where scalar responses Y 1 , . . . , Y n are modeled in dependence of 1-periodic, second order stationary random functions X 1 , . . . , X n . We consider an orthogonal series estimator of the slope function β, by replacing the first m theoretical coefficients of its development in the trigonometric basis by adequate estimators. We propose a model selection procedure for m in a set of admissible values, by defining a contrast function minimized by our estimator and a theoretical penalty function; this first step assumes the degree of ill posedness to be known. Then we generalize the procedure to a random set of admissible m's and a random penalty function. The resulting estimator is completely data driven and reaches automatically what is known to be the optimal minimax rate of convergence, in term of a general weighted L 2 -risk. This means that we provide adaptive estimators of both β and its derivatives.

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“…There is substantial literature devoted to inverse problems with operators acting along Hilbert scales. We mention Natterer [25], Neubauer [26], Mair [20], Hegland [14], Tautenhahn [36], and Dicken and Maass [8]. These papers studied only problems with deterministic noise.…”

Section: This Corresponds To (13) With Designmentioning

confidence: 99%

Optimal Discretization of Inverse Problems in Hilbert Scales. Regularization and Self-Regularization of Projection Methods

Mathé¹,

Pereverzev²

2001

SIAM J. Numer. Anal.

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Abstract. We study the efficiency of the approximate solution of ill-posed problems, based on discretized noisy observations, which we assume to be given beforehand. A basic purpose of the paper is the consideration of stochastic noise, but deterministic noise is also briefly discussed. We restrict ourselves to problems which can be formulated in Hilbert scales. Within this framework we shall quantify the degree of ill-posedness, provide general conditions on projection schemes to achieve the best possible order of accuracy. We pay particular attention on the problem of self-regularization vs. Tikhonov regularization.Moreover, we study the information complexity. Asymptotically, any method which achieves the best possible order of accuracy must use at least such amount of noisy observations. Key words. ill-posed problems, inverse estimation, operator equations, Gaussian white noise, information complexity AMS subject classifications. 62G05, 65J10PII. S003614299936175X1. Introduction and statement of the main problem. We study optimal discretizations of ill-posed problems in Hilbert scales. On the class of problems, which will be introduced in section 2, the best order of accuracy for a given noise level is well known. But this quantity does not take into account any discretization. Therefore we address two issues. First, can this best possible order of accuracy be achieved by discretized regularization methods? More precisely, we aim at presenting general conditions, which allow us to achieve this best possible order. This is made explicit for convergence analysis of corresponding schemes of Tikhonov regularization as well as for regularization by projection methods, self-regularization. In particular we pay attention to the limitations of self-regularization and indicate how these naturally occur when the design is given beforehand.A second issue to be addressed concerns the size, say, N = N (δ) of the design, necessary for a given noise level δ > 0, to enable best order of accuracy. This may be understood as the information complexity of the problem, since, in the asymptotic setting, no numerical method can achieve the best order of accuracy using fewer noisy observations. We establish the asymptotic behavior N (δ), δ → 0.We shall study ill-posed problems, where we wish to recover some element x from some real Hilbert space X from indirectly observed data near y = Ax, where A is some injective compact linear operator acting in a real Hilbert space Y . For the sake of convenience we will assume that X = Y , which can be obtained by isometries. Such linear inverse problems often arise in scientific context, ranging from stereological

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Error Estimates for Regularization Methods in Hilbert Scales

Cited by 71 publications

References 16 publications

Dilational interpolatory inequalities

Dilational interpolatory inequalities

Adaptive estimation in circular functional linear models

Optimal Discretization of Inverse Problems in Hilbert Scales. Regularization and Self-Regularization of Projection Methods

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