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Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. (English) Zbl 0826.65117
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 rates of convergence for objects known to lie in the bump algebra or in bounded variation than any linear procedure. A brief survey of the subject is also presented; the list of references contains 60 items.

65R20 Numerical methods for integral equations
65R30 Numerical methods for ill-posed problems for integral equations
65R10 Numerical methods for integral transforms
44A12 Radon transform
60H99 Stochastic analysis
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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