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

*(English)*Zbl 1077.65055The linear inverse problem corresponding to an equation \(Kf=h\) with a bounded linear operator \(K:H\to L^2(\Omega)\) on a Hilbert space \(H\) consists in finding an estimate of \(f\) from not exactly given \(h\). One of the best known methods of solving such (possibly ill-posed) problems is the Tikhonov regularization method where a regularization functional has to be minimized which is a sum of discrepancy and additional quadratic constraints multiplied by the regularization parameter.

The authors present a new method which is a generalization of the Tikhonov method. Namely, in place of classical quadratic constraints, a weighted \(l^p\)-norm (\(1\leq p\leq2\)) of expansion coefficients of \(f\) with respect to a particular orthogonal basis in \(H\) is taken as a penalization term added to the discrepancy in the regularization functional. The proposed minimization procedure promotes sparsity of the expansion of \(f\) with respect to the basis under consideration. Wavelet bases as well as other orthogonal bases are taken into account.

To compute the corresponding regularized solution, the authors propose an iterative algorithm for solving a system of nonlinear equations related to the minimizing problem for the regularization functional. The detailed and full mathematical analysis of the proposed method is given. The strong convergence of successive iterates to the minimizer of the regularization functional is proved and stability of regularized solution is shown. The presented general regularization theorem establishes the convergence to the exact solution as the noise level tends to zero. Under some additional a priori assumptions error estimates are derived. Moreover, possible generalizations as well as the numerical complexity of the algorithm are discussed. In the Appendix a brief review of basic definitions of wavelets and their connection with Besov spaces is given.

The authors present a new method which is a generalization of the Tikhonov method. Namely, in place of classical quadratic constraints, a weighted \(l^p\)-norm (\(1\leq p\leq2\)) of expansion coefficients of \(f\) with respect to a particular orthogonal basis in \(H\) is taken as a penalization term added to the discrepancy in the regularization functional. The proposed minimization procedure promotes sparsity of the expansion of \(f\) with respect to the basis under consideration. Wavelet bases as well as other orthogonal bases are taken into account.

To compute the corresponding regularized solution, the authors propose an iterative algorithm for solving a system of nonlinear equations related to the minimizing problem for the regularization functional. The detailed and full mathematical analysis of the proposed method is given. The strong convergence of successive iterates to the minimizer of the regularization functional is proved and stability of regularized solution is shown. The presented general regularization theorem establishes the convergence to the exact solution as the noise level tends to zero. Under some additional a priori assumptions error estimates are derived. Moreover, possible generalizations as well as the numerical complexity of the algorithm are discussed. In the Appendix a brief review of basic definitions of wavelets and their connection with Besov spaces is given.

Reviewer: Teresa Regińska (Warszawa)

##### MSC:

65J10 | Numerical solutions to equations with linear operators (do not use 65Fxx) |

65J22 | Numerical solution to inverse problems in abstract spaces |

65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |

47A52 | Linear operators and ill-posed problems, regularization |

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

65T60 | Numerical methods for wavelets |

##### Keywords:

linear inverse problems; regularization; generalization of the Tikhonov method; nonquadratic constraints; sparse extension; wavelets; iterative algorithm; thresholding; ill-posed problem; convergence; stability; Hilbert space
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\textit{I. Daubechies} et al., Commun. Pure Appl. Math. 57, No. 11, 1413--1457 (2004; Zbl 1077.65055)

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