Multilevel techniques for compression and reduction of scientific data-the multivariate case

We develop a technique for multigrid adaptive reduction of data (MGARD). Special attention is given to the case of tensor product grids, where our approach permits the use of nonuniformly spaced grids in each direction, which can prove problematic for many types of data reduction methods. An important feature of our approach is the provision of guaranteed, computable bounds on the loss incurred by the reduction of the data. Many users are leery of lossy algorithms and will only consider using them provided that numerical bounds on the pointwise difference between the original and the reduced datasets are given. Accordingly, we develop techniques for bounding the loss measured in the L^∞(Ω) norm, and we show that these bounds are realistic in the sense that they do not significantly overestimate the actual loss. The resulting loss indicators are used to guide the adaptive reduction of the data so that the reduced dataset meets a user-prescribed tolerance or memory constraint. Illustrative numerical examples, including the reduction of data arising from the simulation of a nonlinear reaction-diffusion problem, a turbulent channel flow, and a climate simulation, are provided.