A fast and well-conditioned computation of multivariate highly oscillatory integrals

Highly oscillatory integrals (HOIs) address several applied phenomena in engineering and applied sciences. The stable and efficient numerical computation of HOIs is an ongoing research problem, especially for large-scale data, large frequency, and high dimensions. The Levin method with global radial basis functions (RBFs) is an effective numerical method for dealing with HOIs in both one and high dimensions. It is well-known that the global RBFs produce dense and ill-conditioned interpolation matrices. In the literature, a new basis function known as compactly supported RBFs has been developed to eliminate the stability issues faced by global RBFs, particularly for large data points and high-dimensional problems. Therefore, this work connects compactly supported RBFs (CS-RBFs) with the Levin method to address the stability and computational issues that arise when the Levin method is used with global RBFs in high-dimensional problems involving HOIs. Several experiments numerically verify the fast and stable behaviour, as well as the convergence order, of the Levin method with CS-RBFs.