We study a weighted \ell^1-regularization technique for solving inverse problems when the forward operator has a significant nullspace. In particular, we prove that a sparse source can be exactly recovered as the regularization parameter \alpha tends to zero. Furthermore, for positive values of \alpha, we show that the regularized inverse solution equals the true source multiplied by a scalar \gamma, where \gamma = 1 - c\alpha. Our analysis is supported by numerical experiments for cases with one and several local sources. This investigation is motivated by a PDE-constrained optimization problem , but the theory is developed in terms of Euclidean spaces. Our results can therefore be applied to many problems.
We study a weighted \ell^1-regularization technique for solving inverse problems when the forward operator has a significant nullspace. In particular, we prove that a sparse source can be exactly recovered as the regularization parameter \alpha tends to zero. Furthermore, for positive values of \alpha, we show that the regularized inverse solution equals the true source multiplied by a scalar \gamma, where \gamma = 1 - c\alpha. Our analysis is supported by numerical experiments for cases with one and several local sources. This investigation is motivated by PDE-constrained optimization problems arising in connection with ECG and EEG recordings , but the theory is developed in terms of Euclidean spaces. Our results can therefore be applied to many problems.
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problems arising in connection with ECG and EEG recordings