We present a nonlinear optimization method for large-scale 3D elastic full-waveform seismic inversion. The method combines outer Gauss–Newton nonlinear iterations with inner conjugate gradient linear iterations, globalized by an Armijo backtracking line search, solved on a sequence of finer grids and higher frequencies to remain in the vicinity of the global optimum, inexactly terminated to prevent oversolving, preconditioned by L-BFGS/Frankel, regularized by a total variation operator to capture sharp interfaces, finely discretized by finite elements in the Lamé parameter space to provide flexibility and avoid bias, implemented in matrix-free fashion with adjoint-based computation of reduced gradient and reduced Hessian-vector products, checkpointed to avoid full spacetime waveform storage, and partitioned spatially across processors to parallelize the solutions of the forward and adjoint wave equations and the evaluation of gradient-like information. Several numerical examples demonstrate the grid independence of linear and nonlinear iterations, the effectiveness of the preconditioner, the ability to solve inverse problems with up to 17 million inversion parameters on up to 2048 processors, the effectiveness of multiscale continuation in keeping iterates in the basin of attraction of the global minimum, and the ability to fit the observational data while reconstructing the model with reasonable resolution and capturing sharp interfaces.