Estimates for generalized sparse grid hierarchical basis preconditioners
Provides theoretical guarantees for preconditioners in high-dimensional numerical methods, benefiting computational scientists using sparse grids.
The paper improves estimates for hierarchical basis preconditioners for sparse grid discretizations, extending results to arbitrary dimensions d>1 and generalized sparse grid spaces with monotone index sets, proving sharp bounds up to constants depending only on d.
We reconsider some estimates the paper "M. Griebel, P. Oswald, On additive Schwarz preconditioners for sparse grid discretizations. Numer. Math. 66 (1994), 449-463" concerning the hierarchical basis preconditioner for sparse grid discretizations. The improvement is in three directions: We consider arbitrary space dimensions d>1, give bounds for generalized sparse grid spaces with arbitrary monotone index set, and show that the bounds are sharp up to constants depending only on d, at least for a subclass of generalized sparse grid spaces containing full grid, standard sparse grid spaces, and energy-norm optimized sparse grid spaces.