The combination technique applied to functionals

Abstract

Functionals related to a solution of a problem, usually modelled by partial differential equations, can be important quantities used to capture features of the problem. For high dimensional problems the computational cost of the functionals can be large since the numerical solution of a high dimensional partial differential equation is usually expensive to compute. We develop a new sparse grid combination technique to reduce the computational cost of such functionals. Our method is based on error splitting models of the functionals. However, it is hard to obtain a concrete error splitting model for complicated approximations. We show the connection between the decay of the surpluses and the error splitting models. By using the connection, we can also apply our combination technique to functionals when we only know their computed surpluses. Numerical experiments are provided to illustrate our idea and test the performance of our method.