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Abstract Electroencephalograms (EEG) are invaluable for treating neurological disorders, however, mapping EEG electrode readings to brain activity requires solving a challenging inverse problem. For time series data, the use of ${\ell }_1$regularization quickly becomes intractable for many solvers, and, despite the reconstruction advantages of ${\ell }_1$regularization, ${\ell }_2$-based approaches such as standardized low-resolution brain electromagnetic tomographysLORETAare used in practice. In this work, we formulate EEG source localization as a graphical generalized elastic net inverse problem and present avariable projectedaugmented Lagrangian algorithm (VPAL) suitable for fast EEG source localization. We prove convergence of this solver for a broad class of separable convex, potentially non-smooth functions subject to linear constraints. Leveraging the efficiency of the proposedVPALalgorithm, we introduce a windowed variation,VPAL ${}_W$, that computes time dynamics in sequence suitable for real-time reconstruction. Our proposed methods are compared to state-of-the-art approaches includingsLORETAand other methods for ${\ell }_1$-regularized inverse problems.