Adaptive Wavelet Galerkin Methods for Linear Inverse Problems

Cohen, Albert; Hoffmann, Marc; Reiß, Markus

doi:10.1137/s0036142902411793

Cited by 84 publications

(119 citation statements)

References 23 publications

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“…Other papers have explored the application of Galerkin-type methods to inverse problems, using an appropriate but fixed wavelet basis [10,14,32]. The underlying intuition is again that if the operator lends itself to a fairly sparse representation in wavelets, e.g., if it is an operator of the type considered in [5], and if the object is mostly smooth with some singularities, then the inversion of the truncated operator will not be too onerous, and the approximate representation of the object will do a good job of capturing the singularities.…”

Section: Related Workmentioning

confidence: 99%

“…The underlying intuition is again that if the operator lends itself to a fairly sparse representation in wavelets, e.g., if it is an operator of the type considered in [5], and if the object is mostly smooth with some singularities, then the inversion of the truncated operator will not be too onerous, and the approximate representation of the object will do a good job of capturing the singularities. In [10] the method is made adaptive, so that the finer-scale wavelets are used where lower scales indicate the presence of singularities.…”

Section: Related Workmentioning

confidence: 99%

“…If we assume that the operator K has particular smoothing properties, then we can use these to derive bounds on the corresponding modulus of continuity, and thus also on the rate of convergence for our regularization algorithm. In particular, let us assume that the operator K is a smoothing operator of order α, a property that can be formulated as an equivalence between the norm K h and the norm of h in a Sobolev space of negative order H −α , i.e., in a Besov space B −α 2,2 ; see, e.g., [10,13,23,32]. In other words, we assume that for some α > 0, there exist constants A and…”

Section: Then the Following Upper And Lower Bounds Hold For M( ρ)mentioning

confidence: 99%

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An iterative thresholding algorithm for linear inverse problems with a sparsity constraint

Daubechies

Defrise

Mol

2004

Comm Pure Appl Math

4,150

3,559

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We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted p -penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem. Use of such p -penalized problems with p < 2 is often advocated when one expects the underlying ideal noiseless solution to have a sparse expansion with respect to the basis under consideration. To compute the corresponding regularized solutions, we analyze an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Then the Following Upper And Lower Bounds Hold For M( ρ)mentioning

confidence: 99%

See 1 more Smart Citation

An iterative thresholding algorithm for linear inverse problems with a sparsity constraint

Daubechies

Defrise

Mol

2004

Comm Pure Appl Math

4,150

3,559

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“…Later, Cohen et al [8] introduced an algorithm combining a Galerkin inversion to a thresholding algorithm.…”

Section: Projection Methodsmentioning

confidence: 99%

“…The following figure (thanks to Paolo Baldi) is an illustration of this phenomenum: it shows a needlet constructed as explained above using Legendre polynomials of degree 2 8 . The highly oscillationg function is a Legendre polynomials of degree 2 8 , whereas the localised one is a needlet centered approximately in the middel of the interval .…”

Section: Localisation Properties This Construction Has Been Performementioning

confidence: 99%

Estimation in inverse problems and second-generation wavelets

Kerkyacharian¹,

Picard²

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. We consider the problem of recovering a function f , when we receive a blurred (by a linear operator) and noisy version : Yε = Kf + εẆ . We will have as guides 2 famous examples of such inverse problems : the deconvolution and the Wicksell problem. The direct problem (K is the identity) isolates the denoising operation. It can't be solved unless accepting to estimate a smoothed version of f : for instance, if f has an expansion on a basis, this smoothing might correspond to stopping the expansion at some stage m. Then a crucial problem lies in finding an equilibrium for m considering the fact that for m large, the difference between f and its smoothed version is small, whereas the random effect introduces an error which is increasing with m. In the true inverse problem, in addition to denoising we have to 'inverse the operator' K, which operation not only creates the usual difficulties, but also introduces the necessity to control the additional instability due to the inversion of the random noise. Our purpose here is to emphasize the fact that in such a problem, there generally exists a basis which is fully adapted to the problem, where for instance the inversion remains very stable : this is the Singular Value Decomposition basis. On the other hand, the SVD basis might be difficult to determine and manipulate numerically, it also might not be appropriate for the accurate description of the solution with a small number of parameters. Also in many practical situations, the signal provides inhomogeneous regularity, and its local features are especially interesting to recover. In such cases, other bases (in particular localised bases such as wavelet bases) may be much more appropriate to give a good representation of the object at hand. Our approach here will be to produce estimation procedures trying to keep the advantages of localisation without loosing the stability and computability of SVD decompositions. We will especially consider two cases. In the first one (which is the case of the deconvolution example) we show that a fairly simple algorithm (WAVE-VD) using an appropriate thresholding technique performed on a standard wavelet system, enables us to estimate the object with rates which are almost optimal up to logarithm factors for any Lp loss function, and on the whole range of Besov spaces. In the second case (which is the case of the Wicksell example where the SVD bases lies in the range of Jacobi polynomials), we prove that quite a similar algorithm (NEED-VD) can be performed provided replacing the standard wavelet system by a second generation wavelet-type basis : the Needlets. We use here the construction (essentially following the work of Petrushev and co-authors) of a localised frame linked with a prescribed basis (here Jacobi polynomials) using a Littlewood Paley decomposition combined with a cubature formula. Section 5 describes the direct case (K = I). It has its own interest and will act as a guide for understanding the 'true' inverse models for a reader which is unfamiliar with nonpar...

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A Brief Introduction to PDE-Constrained Optimization

Antil

Leykekhman

2018

Frontiers in PDE-Constrained Optimization

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Adaptive Wavelet Galerkin Methods for Linear Inverse Problems

Cited by 84 publications

References 23 publications

An iterative thresholding algorithm for linear inverse problems with a sparsity constraint

An iterative thresholding algorithm for linear inverse problems with a sparsity constraint

Estimation in inverse problems and second-generation wavelets

A Brief Introduction to PDE-Constrained Optimization

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