This paper gives a refinement of a general method for the construction of multivariate numerical integration formulas that use merely boundary points as evaluation points. Boundary quadrature formulas are constructed by using the optimal dimensionality-reducing expansion and quadrature formulas for the integrals of periodic functions with wavelet weights. Boundary quadrature formulas are also used to solve boundary value problems of partial differential equations.
Tian-Xiao He, Tian-Xiao He,
"Construction of boundary quadrature formulas using wavelets", Proc. SPIE 2569, Wavelet Applications in Signal and Image Processing III, (1 September 1995); doi: 10.1117/12.217636; https://doi.org/10.1117/12.217636