Sobolev Pre-conditioner for optimizing III-conditioned functionals

Methods and apparatus for providing Sobolev pre-conditioning for optimizing ill-conditioned functionals. A power n is initialized to a maximum power (e.g., 8). For k (e.g., 10) iterations of an optimization pipeline, a matrix M is built by considering all powers of the Laplacian matrix up to the power indicated by n, the Sobolev gradient is computed from the standard gradient, and the computed Sobolev gradient is passed to a numerical optimizer. After the k iterations are complete, if n is at a minimum power (e.g., 1), then the algorithm resets n to the maximum power. Otherwise, n is decremented. For the next k iterations, the matrix M is again built by considering all powers of the Laplacian matrix up to the power indicated by the current value of n. This method is continued until all iterations have completed or until some other terminating condition is reached.