Numerical methods for multivariate highly oscillatory integrals

In this paper, new algorithms are proposed for numerical evaluation of multivariate highly oscillatory integrals having critical point(s) over piecewise smooth rectangular and non-rectangular domains. Multi-resolution analysis is used to take care of the critical point(s). Theoretical error bounds of the component methods of the new algorithms are found. Accuracy of the proposed algorithms is validated numerically on benchmark tests. Numerical results are compared with the methods available in the literature to ascertain stable and accurate performance of the new algorithms.