Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition
type: Item
instance of: scholarly article
title: Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition
zbMATH Open document ID: 0826.65117
author: David Donoho
published in: Applied and Computational Harmonic Analysis
publication date: 1995-07-20T00:00:00Z
review text: A stochastic version of linear inverse problems is treated. It is assumed that the data upon which the inversion is to be based, \(y(u)\), is given by \(y(u) = (Kf) (u) + z(u)\), \(u \in U\), where \(z\) is a noise, and it is necessary to recover \(f\) from the data \(y\). The \(L^2\) norm is used to measure the quality of recovery. In the cases which are of most interest, \(K\) is not invertible and the problem is ill-posed. The well-known standard approach to the problem is the so-called singular value decomposition (SVD) of the inverse problem; this approach is based on a regularization of A. N. Tikhonov's type. The author describes the wavelet-vaguelette decomposition (WVD) of a linear inverse problem and uses it instead of the SVD. He proposes to solve the problem by nonlinear ``shrinking'' the WVD coefficients of the noisy, indirect data. The author shows that the WVD exists for a class of special inverse problems of homogeneous type (numerical differentiation, inversion of Abel-type transforms, certain convolution transforms, and the Radon transform). Orthogonal wavelet bases which serve as unconditional bases for any of the spaces in the Besov and Triebel-Lisorkin scales are used (by the way, wavelet bases are useful for data compressions). The author's approach offers significant advantages over traditional SVD inversion in recovering spatially inhomogeneous objects. The author supposes that observations are contaminated by white noise and the object, \(f\) is an unknown element of a Besov space. He proves that nonlinear WVD shrinkage can be tuned to attain the minimax rate of convergence, for \(L^2\) loss, over the entire scale of Besov spaces. The case of Besov spaces \(B^\sigma_{p,q}\), \(p < 2\), which model spatial inhomogeneity, is included. In comparison, linear procedures (SVD included) cannot attain optimal rates of convergence over such classes in the case \(p < 2\). This is the main result of the paper. In particular, these methods achieve faster…
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65R30
Mathematics Subject Classification ID: 65R10
Mathematics Subject Classification ID: 44A12
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 45E10
zbMATH DE Number: 772195
zbMATH Keywords: ill-posed problem
zbMATH Keywords: numerical differentiation
zbMATH Keywords: orthogonal wavelet bases
zbMATH Keywords: Besov scales
zbMATH Keywords: stochastic version
zbMATH Keywords: linear inverse problems
zbMATH Keywords: singular value decomposition
zbMATH Keywords: regularization
zbMATH Keywords: wavelet-vaguelette decomposition
zbMATH Keywords: nonlinear ``shrinking''
zbMATH Keywords: inversion of Abel-type transforms
zbMATH Keywords: convolution transforms
zbMATH Keywords: Radon transform
zbMATH Keywords: Triebel-Lisorkin scales
zbMATH Keywords: data compressions
zbMATH Keywords: white noise
zbMATH Keywords: Besov space
zbMATH Keywords: convergence
reviewed by: Grigori L. Litvinov
MaRDI profile type: Publication
full work available at URL: acha.1995.1008
OpenAlex ID: W2092543127
DOI: 10.1006/ACHA.1995.1008
Recommended article: Solution of inverse problems for wave equation with a nonlinear coefficient
Recommended article: Q4310325
Recommended article: Wavelet projection methods for solving pseudodifferential inverse problems
Recommended article: A wavelet multiscale method for the inverse problem of a nonlinear convection-diffusion equation
Recommended article: WAVELETS AND INVERSE PROBLEMS
Recommended article: Nonlinear approximations using wavelet decompositions
Recommended article: Q4947182
Recommended article: Approximation-theoretic methods for nonlinear deconvolution and inversion
Recommended article: Resolution of an inverse problem by Haar basis and Legendre wavelet methods
Recommended article: Q6075195
description: scientific article; zbMATH DE number 772195