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Tomographic synthetic aperture radar (TomoSAR) inversion of urban areas is an inherently sparse reconstruction problem and, hence, can be solved using compressive sensing (CS) algorithms. This paper proposes solutions for two notorious problems in this field. First, TomoSAR requires a high number of data sets, which makes the technique expensive. However, it can be shown that the number of acquisitions and the signal-to-noise ratio (SNR) can be traded off against each other, because it is asymptotically only the product of the number of acquisitions and SNR that determines the reconstruction quality. We propose to increase SNR by integrating nonlocal (NL) estimation into the inversion and show that a reasonable reconstruction of buildings from only seven interferograms is feasible. Second, CS-based inversion is computationally expensive and therefore, barely suitable for large-scale applications. We introduce a new fast and accurate algorithm for solving the NL L1-L2-minimization problem, central to CS-based reconstruction algorithms. The applicability of the algorithm is demonstrated using simulated data and TerraSAR-X high-resolution spotlight images over an area in Munich, Germany. [ABSTRACT FROM AUTHOR]