Approximate Solution of Ill-Posed Equations: Arbitrarily Slow Convergence vs. Superconvergence

Schock, Eberhard

doi:10.1007/978-3-0348-9317-6_20

Cited by 55 publications

(45 citation statements)

References 5 publications

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“…Note that lim →0 f( ) = 0 for all x 0 ∈ X as proven by using spectral theory for general linear regularization schemes in Reference [4, p. 72, Theorem 4.1] (see also Reference [10, p. 45, Theorem 5.2]), but the decay rate of f( ) → 0 as → 0 depends on x 0 and can be arbitrarily slow (see References [4,11,Proposition 3.11]). However, the analysis of proÿle functions (7) expressing the relative smoothness of x 0 with respect to the operator A yields convergence rates of regularized solutions.…”

Section: General and Approximate Source Conditions For Convergence Ramentioning

confidence: 94%

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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Section: General and Approximate Source Conditions For Convergence Ramentioning

confidence: 94%

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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“…The choice of the basis {ϕ i } is suggested by the special structure of the operator A, namely its relation to (27). It is easy to see that for u(z) = cos(π(i − 1)z), b(z, t) ≡ 0 the solution of (27) is…”

Section: Initial Temperature Reconstruction For a Nonlinear Heat Equamentioning

confidence: 99%

“…This gives the discretized source term. Then we run a numerical procedure for (27) with that discretized source term and zero as initial condition.…”

Section: A Appendixmentioning

confidence: 99%

Regularized fixed-point iterations for nonlinear inverse problems

Pereverzyev¹,

Pinnau²,

Siedow³

2005

Inverse Problems

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In this paper we introduce a derivative-free, iterative method for solving nonlinear illposed problems F x = y, where instead of y noisy data y δ with y − y δ ≤ δ are given and F : D(F ) ⊆ X → Y is a nonlinear operator between Hilbert spaces X and Y . This method is defined by splitting the operator F into a linear part A and a nonlinear part G, such that F = A + G. Then iterations are organized as Au k+1 = y δ − Gu k . In the context of ill-posed problems we consider the situation when A does not have a bounded inverse, thus each iteration needs to be regularized. Under some conditions on the operators A and G we study the behavior of the iteration error. We obtain its stability with respect to the iteration number k as well as the optimal convergence rate with respect to the noise level δ, provided that the solution satisfies a generalized source condition. As an example, we consider an inverse problem of initial temperature reconstruction for a nonlinear heat equation, where the nonlinearity appears due to radiation effects. The obtained iteration error in the numerical results has the theoretically expected behavior. The theoretical assumptions are illustrated by a computational experiment.

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“…If rf denotes the Moore-Penrose generalized inverse (see, e.g., [17]), the best approximate solution is given by T^y. For nonclosed range R(T) of T, the problem of determining T^y is ill posed.…”

mentioning

confidence: 99%

An A Posteriori Parameter Choice for Ordinary and Iterated Tikhonov Regularization of Ill-Posed Problems Leading to Optimal Convergence Rates

Gfrerer¹

1987

Mathematics of Computation

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Approximate Solution of Ill-Posed Equations: Arbitrarily Slow Convergence vs. Superconvergence

Cited by 55 publications

References 5 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

Regularized fixed-point iterations for nonlinear inverse problems

An A Posteriori Parameter Choice for Ordinary and Iterated Tikhonov Regularization of Ill-Posed Problems Leading to Optimal Convergence Rates

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