Multilevel sparse grid kernels collocation with radial basis functions for elliptic and parabolic problems

Radial basis functions (RBFs) are well-known for the ease implementation as they are the mesh-free method [31, 37, 71, 72]. In this thesis, we modify the multilevel sparse grid kernel interpolation (MuSIK) algorithm proposed in [48] for use in Kansa’s collocation method (referred to as MuSIK-C) to solve elliptic and parabolic problems. The curse of dimensionality is a significant challenge in high dimension approximation. A full grid collocation method requires O(Nd) nodal points to construct an approximation; here N is the number of nodes in one direction and d means the dimension. However, the sparse grid collocation method in this thesis only demand O(N logd􀀀1(N)) nodes. We save much more memory cost using sparse grids and obtain a good performance as using full grids. Moreover, the combination technique [20, 54] allows the sparse grid collocation method to be parallelised. When solving parabolic problems, we follow Myers et al.’s suggestion in [90] to use the space-time method, considering time as one spatial dimension. If we apply sparse grids in the spatial dimensions and use time-stepping, we still need O(N2 logd􀀀1(N)) nodes. However, if we use the space-time method, the total number of nodes is O(N logd(N)). In this thesis, we always compare the performance of multiquadric (MQ) basis function and the Gaussian basis function. In all experiments, we observe that the collocation method using the Gaussian with scaling shape parameters does not converge. Meanwhile, in Chapter 3, there is an experiment to show that the space-time method with MQ has a similar convergence rate as a time-stepping method using MQ in option pricing. From the numerical experiments in Chapter 4, MuSIK-C using MQ and the Gaussian always give more rapid convergence and high accuracy especially in four dimensions (T R3) for PDEs with smooth conditions. Compared to some recently proposed mesh-based methods, MuSIK-C shows similar performance in low dimension situation and better approximation in high dimension. In Chapter 5, we combine the Method of Lines (MOL) and our MuSIK-C to obtain good convergence in pricing one asset European option and the Margrabe option, that have non-smooth initial conditions.