Sparsity regularization for inverse problems with nullspaces
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We study a weighted ℓ^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 α tends to zero. Furthermore, for positive values of α, we show that the regularized inverse solution equals the true source multiplied by a scalar γ, where γ = 1 - cα. 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.
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