Mathematics – Numerical Analysis

Scientific paper

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2012-02-15

Mathematics

Numerical Analysis

21 pages, 3 figures

Scientific paper

We propose an iterative algorithm for the minimization of a $\ell_1$-norm penalized least squares functional, under additional linear constraints. The algorithm is fully explicit: it uses only matrix multiplications with the three matrices present in the problem (in the linear constraint, in the data misfit part and in penalty term of the functional). None of the three matrices must be invertible. Convergence is proven in a finite-dimensional setting. We apply the algorithm to a synthetic problem in magneto-encephalography where it is used for the reconstruction of divergence-free current densities subject to a sparsity promoting penalty on the wavelet coefficients of the current densities. We discuss the effects of imposing zero divergence and of imposing joint sparsity (of the vector components of the current density) on the current density reconstruction.

**Loris Ignace**

Physics – High Energy Physics – High Energy Physics - Theory

Scientist

**Verhoeven Caroline**

Nonlinear Sciences – Exactly Solvable and Integrable Systems

Scientist

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