Sparse Grids One-dimensional Multilevel Basis

The sparse grid method is a general numerical discretization technique for multivariate problems. This approach, first introduced by the Russian mathematician Smolyak in 1963 [27], constructs a multi-dimensional multilevel basis by a special truncation of the tensor product expansion of a one-dimensional multilevel basis (see Figure 1 for an example of a sparse grid). Discretizations on sparse grids involve only O(N (log N) d−1) degrees of freedom, where d is the problem dimension and N denotes the number of degrees of freedom in one coordinate direction. The accuracy obtained this way is comparable to one using a full tensor product basis involving O(N d) degrees of freedom, if the underlying problem is smooth enough, i.e., if the solution has bounded mixed derivatives. This way, the curse of dimension, i.e., the exponential dependence of conventional approaches on the dimension d, can be overcome to a certain extent. This makes the sparse grid approach particularly attractive for the numerical solution of moderate-and higher-dimensional problems. Still, the classical sparse grid method is not completely independent of the dimension due the above logarithmic term in the complexity. Sparse grid methods are known under various names, such as hyperbolic cross points, discrete blending, Boolean interpolation or splitting extrapo-lation. For a comprehensive introduction to sparse grids, see [5]. In computational finance, sparse grid methods have been employed for the valuation of multi-asset options such as basket [25] or outperformance options [13], various types of path-dependent derivatives and likelihood estimation [20] due to the high dimension of the arising partial differential equations or integration problems. The first ingredient of a sparse grid method is a one-dimensional multilevel basis. In the classical sparse grid approach, a hierarchical basis based on standard hat functions, φ(x) := 1 − |x| if x ∈ [−1, 1] , 0 otherwise , (1) Figure 1: A regular two-dimensional sparse grid of level 7. is used. Then, a set of equidistant grids Ω l of level l on the unit interval ¯ Ω = [0, 1] and mesh width 2 −l is considered. The standard hat function is then taken to generate a family of basis functions φ l,i (x) having support Thereby, the index i indicates the location of a basis function or a grid point. This basis is usually termed nodal basis or Lagrange basis (see Figure 2, bottom). These basis functions are then used to define function spaces V l consisting …