An overview of high-dimensional collision detection for the purpose of motion planning is in arXiv: 1806.07457 :
Grid-based approaches are resolution complete and often offer optimal solutions. However, the number of grid points grows exponentially in the configuration space dimension, which makes even the state-of-the-art methods inappropriate for very high-dimensional problems. Sampling-based approaches are efficient in most practical problems but offer weaker guarantees. They are probabilistically complete, however, they often require post-processing and can still be inefficient in very complex configuration spaces. Trajectory optimization approaches can solve high-dimensional motion planning problems quickly, but solutions are only locally optimal.