  The simulation of high-dimensional problems with manageable computational
resource represents a long-standing challenge. In a series of our recent work
[25, 17, 18, 24], a class of sparse grid DG methods has been formulated for
solving various types of partial differential equations in high dimensions. By
making use of the multiwavelet tensor-product bases on sparse grids in
conjunction with the standard DG weak formulation, such a novel method is able
to significantly reduce the computation and storage cost compared with full
grid DG counterpart, while not compromising accuracy much for sufficiently
smooth solutions. In this paper, we consider the high-dimensional Helmholtz
equation with variable coefficients and demonstrate that for such a problem the
efficiency of the sparse grid DG method can be further enhanced by exploring a
semi-orthogonality property associated with the multiwavelet bases, motivated
by the work [21, 22, 19]. The detailed convergence analysis shows that the
modified sparse grid DG method attains the same order accuracy, but the
resulting stiffness matrix is much sparser than that by the original method,
leading to extra computational savings. Numerical tests up to six dimensions
are provided to verify the analysis.
