Approximation on non-tensor domains including squircles, Part III: Polynomial hyperinterpolation and radial basis function interpolation on Chebyshev-like grids and truncated uniform grids
Single-domain spectral methods have been largely restricted to tensor product bases on a tensor product grid. To break the tensor barrier, we study approximation in a domain bounded by a squircle, the zero isoline of B(x,y)=x2?+y2??1 B ( x , y ) = x 2 ? + y 2 ? ? 1 . The boundary varies smoothly from a circle [?=1 ? = 1 ] to the square [?=? ? = ? ]. Polynomial least-squares hyperinterpolation converges geometrically as long as the number of points P is (at least) double the number of basis functions N. The polynomial grid was made denser near the boundaries (Chebyshev-like) by depositing grid points along wisely chosen contours of B . Gaussian radial basis functions (RBFs) were more robust in the sense that they, too, converged geometrically, but hyperinterpolation (P>N P > N ) and a Chebyshevized grid were unnecessary. A uniform grid, truncated to include only those points within the squircle, was satisfactory even without interpolation points on the boundary (although boundary points are a cost-effective improvement). For a given number of points P , however, RBF interpolation was only slightly more accurate than polynomial hyperinterpolation, and needed twice as many basis functions. Interpolation costs can be greatly reduced by exploiting the invariance of the squircle-bounded domain to the eight elements D4 D 4 dihedral group.
hyperinterpolation radial basis functions dihedral group pseudospectral chebyshev polynomials zernike polynomials group theory d4 d 4 d4 d 4 d4 d 4 d 4