Tikhonov Regularization for Finitely and Infinitely Smoothing Operators

Mair, B.A.

doi:10.1137/s0036141092238060

Cited by 84 publications

(88 citation statements)

References 19 publications

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“…C with a ÿxed constant C¿0 otherwise (16) is well deÿned and satisÿes the limit condition lim →0 ( ) = 0: The parameter choice (16) can be rewritten as =2…”

mentioning

confidence: 99%

Approximate source conditions in Tikhonov–Phillips regularization and consequences for inverse problems with multiplication operators

Hofmann

2005

Math Methods in App Sciences

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SUMMARYThe object of this paper is threefold. First, we investigate in a Hilbert space setting the utility of approximate source conditions in the method of Tikhonov-Phillips regularization for linear ill-posed operator equations. We introduce distance functions measuring the violation of canonical source conditions and derive convergence rates for regularized solutions based on those functions. Moreover, such distance functions are veriÿed for simple multiplication operators in L 2 (0; 1): The second aim of this paper is to emphasize that multiplication operators play some interesting role in inverse problem theory. In this context, we give examples of non-linear inverse problems in natural sciences and stochastic ÿnance that can be written as non-linear operator equations in L 2 (0; 1), for which the forward operator is a composition of a linear integration operator and a non-linear superposition operator. The FrÃ echet derivative of such a forward operator is a composition of a compact integration and a non-compact multiplication operator. If the multiplier function deÿning the multiplication operator has zeros, then for the linearization an additional ill-posedness factor arises. By considering the structure of canonical source conditions for the linearized problem it could be expected that di erent decay rates of multiplier functions near a zero, for example the decay as a power or as an exponential function, would lead to completely di erent ill-posedness situations. As third we apply the results on approximate source conditions to such composite linear problems in L 2 (0; 1) and indicate that only integrals of multiplier functions and not the speciÿc character of the decay of multiplier functions in a neighbourhood of a zero determine the convergence behaviour of regularized solutions.

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“…C with a ÿxed constant C¿0 otherwise (16) is well deÿned and satisÿes the limit condition lim →0 ( ) = 0: The parameter choice (16) can be rewritten as =2…”

mentioning

confidence: 99%

Approximate source conditions in Tikhonov–Phillips regularization and consequences for inverse problems with multiplication operators

Hofmann

2005

Math Methods in App Sciences

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“…Subsequently, many authors extended, modified, and generalized Natterer's work to obtain error bounds under various contexts (see, e.g., Natterer [9], Hegland [3], Schröter and Tautenhahn [12], Mair [6], Nair et al [8], and Nair [7]). …”

Section: For Y ∈ R(t ) + R(t )mentioning

confidence: 99%

An optimal order yielding discrepancy principle for simplified regularization of ill‐posed problems in Hilbert scales

George¹,

Nair

2003

International Journal of Mathematics and Mathematical Sciences

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have considered a monotone rule as a parameter choice strategy for choosing the regularization parameter while considering approximate solution of an ill-posed operator equation T x = y, where T is a bounded linear operator between Hilbert spaces. Motivated by this, we propose a new discrepancy principle for the simplified regularization, in the setting of Hilbert scales, when T is a positive and selfadjoint operator. When the data y is known only approximately, our method provides optimal order under certain natural assumptions on the ill-posedness of the equation and smoothness of the solution. The result, in fact, improves an earlier work of the authors (1997).

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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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Tikhonov Regularization for Finitely and Infinitely Smoothing Operators

Cited by 84 publications

References 19 publications

Approximate source conditions in Tikhonov–Phillips regularization and consequences for inverse problems with multiplication operators

Approximate source conditions in Tikhonov–Phillips regularization and consequences for inverse problems with multiplication operators

An optimal order yielding discrepancy principle for simplified regularization of ill‐posed problems in Hilbert scales

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

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