题目：Gaussian quadrature rules for composite highly oscillatory integrals

报告人：王海永

报告时间：2021年11月20日 9：30-11：30

报告地点：数学与统计学院 235教室

摘要：Highly oscillatory integrals of composite type arise in the numerical simulation of electronic circuits and their calculations remain to be a challenge since they do not fit into the classical pattern of highly oscillatory integrals of Fourier-type. In this talk, we propose two Gaussian quadrature rules for computing such integrals. The first one is constructed based on the classical theory of orthogonal polynomials and its nodes and weights can be computed efficiently by using tools of numerical linear algebra. We prove that the convergence rate of this rule depends solely on the regularity of the smooth part of the integrands. The second one is constructed with respect to a sign-changing function and the classical theory of Gaussian quadrature can not be used anymore. We explore theoretical properties of this Gaussian quadrature, including the trajectories of the quadrature nodes and the convergence rates of these nodes to both endpoints of the integration interval, and prove its asymptotic error estimate under suitable hypotheses. Numerical experiments are presented to demonstrate the performance of the proposed methods.