The combination technique applied to functionals

TLDR: In this paper , a new sparse grid combination technique was developed to reduce the computational cost of functionals. But it is hard to obtain a concrete error splitting model for complicated approximations.

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. References A. J. Brizard and T. S. Hahm. Foundations of nonlinear gyrokinetic theory. In: Rev. Mod. Phys. 79.2 (2007), pp. 421–468. doi: 10.1103/RevModPhys.79.421 H.-J. Bungartz and M. Griebel. Sparse grids. In: Acta Numer. 13 (2004), pp. 147–269. doi: 10.1017/S0962492904000182 T. Gerstner and M. Griebel. Numerical integration using sparse grids. In: Numer. Algor. 18 (1998), pp. 209–232. doi: 10.1023/A:1019129717644. M. Griebel, M. Schneider, and C. Zenger. A combination technique for the solution of sparse grid problems. In: Iterative methods in linear algebra: Proceedings of the IMACS International Symposium on Iterative Methods in Linear Algebra, 1991. Ed. by P. de Groen and R. Beauwens. North-Holland, Amsterdam, 1992, pp. 263–281. url: https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.33.3530 B. Harding. Fault tolerant computation of hyperbolic partial differential equations with the sparse grid combination technique. PhD thesis. The Australian National University, 2016. url: https://openresearch-repository.anu.edu.au/bitstream/1885/101226/1/Harding%20Thesis%202016.pdf M. Hegland. Adaptive sparse grids. In: Proceedings of the 10th Computational Techniques and Applications Conference CTAC-2001. Ed. by K. Burrage and R. B. Sidje. Vol. 44. 2003, pp. C335–C353. doi: 10.21914/anziamj.v44i0.685 Gene Development Team; F. Jenko et al. The Gyrokinetic Plasma Turbulence Code Gene: User Manual. 2013. url: http://genecode.org/