Celebratio Mathematica — Daubechies

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Works connected to Sergey Voronin

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F. J. Simons, I. Loris, G. Nolet, I. C. Daubechies, S. Voronin, J. S. Judd, P. A. Vetter, J. Charléty, and C. Vonesch: “Solving or resolving global tomographic models with spherical wavelets, and the scale and sparsity of seismic heterogeneity,” Geophys. J. Int. 187 : 2 (November 2011), pp. 969–988. ArXiv 1104.3151 article

We propose a class of spherical wavelet bases for the analysis of geophysical models and for the tomographic inversion of global seismic data. Its multiresolution character allows for modelling with an effective spatial resolution that varies with position within the Earth. Our procedure is numerically efficient and can be implemented with parallel computing. We discuss two possible types of discrete wavelet transforms in the angular dimension of the cubed sphere. We describe benefits and drawbacks of these constructions and apply them to analyse the information in two published seismic wave speed models of the mantle, using the statistics of wavelet coefficients across scales. The localization and sparsity properties of wavelet bases allow finding a sparse solution to inverse problems by iterative minimization of a combination of the \( \ell_2 \) norm of the data residuals and the \( \ell_1 \) norm of the model wavelet coefficients. By validation with realistic synthetic experiments we illustrate the likely gains from our new approach in future inversions of finite-frequency seismic data.

Ignace Loris	Related
Jean Charléty	Related
J. Stephen Judd	Related
Frederik J. Simons	Related
Philip Adam Vetter	Related
Cédric Vonesch	Related
Sergey Voronin	Related
Auguste (Guust) Nolet	Related

S. Voronin and I. Daubechies: “An iteratively reweighted least squares algorithm for sparse regularization,” pp. 391–411 in Functional analysis, harmonic analysis, and image processing: A collection of papers in honor of Björn Jawerth. Edited by M. Cwikel and M. Milman. Contemporary Mathematics 693. American Mathematical Society (Providence, RI), 2017. MR 3682621 Zbl 1392.65074 ArXiv 1511.08970 incollection

We present a new algorithm and the corresponding convergence analysis for the regularization of linear inverse problems with sparsity constraints, applied to a new generalized sparsity promoting functional. The algorithm is based on the idea of iteratively reweighted least squares, reducing the minimization at every iteration step to that of a functional including only \( \ell_2 \)-norms. This amounts to smoothing of the absolute value function that appears in the generalized sparsity promoting penalty we consider, with the smoothing becoming iteratively less pronounced. We demonstrate that the sequence of iterates of our algorithm converges to a limit that minimizes the original functional.

Björn David Jawerth	Related
Michael Ćwikel	Related
Mario Marcos Milman	Related
Sergey Voronin	Related