Randomization of Forcing in Large Systems of PDEs for Improvement of Energy Estimates

We consider a class of stochastic PDEs (SPDEs) driven by purely spatial white noise, for which the numerical computation of the energy is desired. Our paper compares the efficiency of two different bases of expansion of white noise, one of a local scale and the other of a “large scale,” for approximating the energy of the SPDE, and we will show that the latter basis dramatically improves the approximation of the energy. Such problems with a local scale basis arise in applications such as electromagnetic wave propagation with incoherent sources, but current approaches to computing the energy have found a roadblock in the sheer size of the problem. Thus, knowledge of the improved efficiency of a large scale basis becomes useful in vastly reducing computational cost while attaining highly accurate approximations of the energy.