PDE-constrained Large Deformation Diffeomorphic Metric Mapping is a\nparticularly interesting framework of physically meaningful diffeomorphic\nregistration methods. Newton-Krylov optimization has shown an excellent\nnumerical accuracy and an extraordinarily fast convergence rate in this\nframework. However, the most significant limitation of PDE-constrained LDDMM is\nthe huge computational complexity, that hinders the extensive use in\nComputational Anatomy applications. In this work, we propose two\nPDE-constrained LDDMM methods parameterized in the space of band-limited vector\nfields and we evaluate their performance with respect to the most related state\nof the art methods. The parameterization in the space of band-limited vector\nfields dramatically alleviates the computational burden avoiding the\ncomputation of the high-frequency components of the velocity fields that would\nbe suppressed by the action of the low-pass filters involved in the computation\nof the gradient and the Hessian-vector products. Besides, the proposed methods\nhave shown an improved accuracy with respect to the benchmark methods.\n