In this work we present and compare deterministic and statistical algorithms for efficiently solving large-scale contaminant source inversion problems. The underlying equations of contaminant transport are assumed linear but unsteady and defined over complex geometries. The algorithms presented are accelerated through discrete adjoint solutions that are pre-computed efficiently in an offline stage, yielding savings in the time-critical online stage of several orders of magnitude in computational time. In the deterministic case, adjoints accelerate the application of the Hessian matrix, while in the statistical case, adjoints are used to directly evaluate samples. To address deterioration of statistical sampling efficiency for anisotropic posteriors, we present an application of a recently developed ensemble Markov chain Monte Carlo method. Results for two- and three-dimensional problems demonstrate the feasibility of statistical inversion for large-scale problems and show the advantage of statistical results over single-point deterministic results.