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The simulation of high-dimensional problems with manageable computationalresource 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 forsolving various types of partial differential equations in high dimensions. Bymaking use of the multiwavelet tensor-product bases on sparse grids inconjunction with the standard DG weak formulation, such a novel method is ableto significantly reduce the computation and storage cost compared with fullgrid DG counterpart, while not compromising accuracy much for sufficientlysmooth solutions. In this paper, we consider the high-dimensional Helmholtzequation with variable coefficients and demonstrate that for such a problem theefficiency of the sparse grid DG method can be further enhanced by exploring asemi-orthogonality property associated with the multiwavelet bases, motivatedby the work [21, 22, 19]. The detailed convergence analysis shows that themodified sparse grid DG method attains the same order accuracy, but theresulting stiffness matrix is much sparser than that by the original method,leading to extra computational savings. Numerical tests up to six dimensionsare provided to verify the analysis.

A sparse grid discontinuous Galerkin method for the high-dimensional Helmholtz equation with variable coefficients