On the Degree of Ill-posedness for Linear Problems with Noncompact Operators

Hofmann, Bernd; Kindermann, Stefan

doi:10.4310/maa.2010.v17.n4.a8

Cited by 18 publications

(11 citation statements)

References 35 publications

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“…where we have used (23) and (25). In a similar fashion one can employ (22) and (24) to show that λ + k (A α ) ≥ λ + k (A 0 ), and we conclude that (26)…”

Section: Let Us Considermentioning

confidence: 74%

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Analysis of the Minimal Residual Method Applied to Ill Posed Optimality Systems

Nielsen¹,

Mardal²

2013

SIAM J. Sci. Comput.

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We analyze the performance of the Minimal Residual Method applied to linear Karush-Kuhn-Tucker systems arising in connection with inverse problems. Such optimality systems typically have a saddle point structure and have unique solutions for all α > 0, where α is the parameter employed in the Tikhonov regularization. Unfortunately, the associated spectral condition number is very large for small values of α, which strongly indicates that their numerical treatment is difficult. Our main result shows that a broad range of linear ill posed optimality systems can be solved with a number of iterations of order O(ln(α −1)). More precisely, in the severely ill posed case the number of iterations needed by the Minimal Residual Method cannot grow faster than O(ln(α −1)) as α → 0. This result is obtained by carefully analyzing the spectrum of the associated saddle point operator: Except for a few isolated eigenvalues, the spectrum consists of bounded intervals. Krylov subspace methods handle such problems very well. We illuminate our theoretical findings with some numerical results for inverse problems involving partial differential equations. Our investigation is inspired by Prof. H. Egger's discussion of similar results valid for the conjugate gradient algorithm applied to the normal equations.

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“…where we have used (23) and (25). In a similar fashion one can employ (22) and (24) to show that λ + k (A α ) ≥ λ + k (A 0 ), and we conclude that (26)…”

Section: Let Us Considermentioning

confidence: 74%

“…This, however, leads to an even more involved analysis, which would have to address non compact operators. (Moreover, for problems with continuous spectra it is not even straightforward to define the concepts severely and mildly ill posed [24]).…”

Section: A6mentioning

confidence: 99%

Analysis of the Minimal Residual Method Applied to Ill Posed Optimality Systems

Nielsen¹,

Mardal²

2013

SIAM J. Sci. Comput.

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“…Hofmann (1994). Moreover, extensions to non-compact linear operators are considered by Hofmann et al (2010), for example.…”

Section: Notion Of Ill-posednessmentioning

confidence: 99%

Solving inverse problems using data-driven models

et al. 2019

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Recent research in inverse problems seeks to develop a mathematically coherent foundation for combining data-driven models, and in particular those based on deep learning, with domain-specific knowledge contained in physical–analytical models. The focus is on solving ill-posed inverse problems that are at the core of many challenging applications in the natural sciences, medicine and life sciences, as well as in engineering and industrial applications. This survey paper aims to give an account of some of the main contributions in data-driven inverse problems.

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“…σ measures the decay rate of the singular values of T relative to the decay rate of the coefficients of f † in the singular system; therefore it measure the local degree of ill-posedness. In contrast the upper bound only depends on the norm of f † and the decay of the singular values, thus it measures the global degree of ill-posedness, further possible definitions of this term are discussed in [HK10].…”

Section: Convergence Rates Theorymentioning

confidence: 99%

Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs

Weidling¹

2019

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That this general procedure yields indeed a stable reconstruction and that f α converges to the true solution f † when g obs converges to the true data for some choice of α has been established e.g. in [Pös08,Fle11] (see Section 1.2.2 for more details). For the choice of α one roughly has to take the following effect into account: the larger we choose α the more stable is the problem we are solving but the less we try to match the data exactly. Hence one has to try to find an optimal balance of the two effects in the sense that the f α is as close as possible to the true solution f † . It is well known, however, in the regularization theory that any bound on the distance f α to f † requires a priori knowledge on f † , i.e. we cannot find such a bound which will hold globally for all possible choices of parameters f † . Such an a priori knowledge is typically formulated as a source condition. E.g. in [EHN96] for the case that F is a linear, compact operators acting between Hilbert spaces the spectral source condition f † = (TT) ν w, ν ∈ (0, 1] have been studied, where the right hand side is defined by the functional calculus.under mild assumptions on the involved quantities, where the observed data g obs has to satisfy g obs − g ≤ δ again. Recent results of [Fle18] illustrate that such a VSC is always fulfilled, but they do not tell us how the function ψ will look like. The subject of this thesis is to investigate conditions which allow to quantify ψ in the VSC and hence the convergence rate.If F is an operator mapping between function spaces F −1 is often not continuous due to the fact that F is a smoothing operator, that is F( f ) is a smoother function than f . It has been observed for such problems that the spectral source condition can be interpreted as a smoothness assumption on f relative to the smoothing properties of the operator. The smoothing properties of the operator are related to the degree of ill-posedness of the operator; e.g. in case of linear, compact operators acting between Hilbert spaces this is typically measured by the speed of the decay of the singular values of the operator.In this thesis we will use the two factors, smoothness of the solution and illposedness of the operator equation, in order to derive explicit forms of variationalfor all f 1 , f 2 in a (usually smooth) subset K. Such estimates are quite common for many interesting problems while results that show that for a given problem and conditions on the true solution f † a VSC for a specific function ψ is fulfilled are still rare in the literature (see Remark 2.28). As VSCs imply stability estimates this allows us to compare our new findings with existing results in the literature.This thesis is structured as follows:• In Chapter 1 we will give a more detailed introduction into inverse problems and Tikhonov regularization.• Chapter 2 will review the results on convergence rates known in the literature. We will discuss the advantages and limitations of various source conditions as well as their relation to each other. Further we w...

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On the Degree of Ill-posedness for Linear Problems with Noncompact Operators

Cited by 18 publications

References 35 publications

Analysis of the Minimal Residual Method Applied to Ill Posed Optimality Systems

Analysis of the Minimal Residual Method Applied to Ill Posed Optimality Systems

Solving inverse problems using data-driven models

Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs

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