Sparse grids, adaptivity, and symmetry
Sparse grid methods represent a powerful and efficient technique for the representation and approximation of functions and particularly the solutions of partial differential equations in moderately high space dimensions. To extend the approach to truly high-dimensional problems as they arise in quantum chemistry, an additional property has to be brought into play, the symmetry or antisymmetry of the functions sought there. In the present article, an adaptive sparse grid refinement scheme is developed that takes full advantage of such symmetry properties and for which the amount of work and storage remains strictly proportional to the number of degrees of freedom. To overcome the problems with the approximation of the inherently complex antisymmetric functions, augmented sparse grid spaces are proposed.
sparse grids adaptivity symmetry antisymmetry