[Paper Review] Computing Highly Oscillatory Integrals

Yunyun Ma, Yuesheng Xu|arXiv (Cornell University)|Jul 2, 2015

Mathematical functions and polynomials3 citations

TL;DR

This paper proposes two classes of composite moment-free quadrature rules for highly oscillatory integrals with algebraic singularities and stationary points. By partitioning the integration interval based on oscillation and singularity behavior, it applies graded quadrature for weakly singular parts and a modified moment-free Filon-type method for oscillatory parts, achieving polynomial or exponential convergence without computing the inverse of the oscillator, outperforming existing methods in accuracy and efficiency.

ABSTRACT

We develop two classes of composite moment-free numerical quadratures for computing highly oscillatory integrals having integrable singularities and stationary points. The first class of the quadrature rules has a polynomial order of convergence and the second class has an exponential order of convergence. We first modify the moment-free Filon-type method for the oscillatory integrals without a singularity or a stationary point to accelerate their convergence. We then consider the oscillatory integrals without a singularity or a stationary point and then those with singularities and stationary points. The composite quadrature rules are developed based on partitioning the integration domain according to the wave number and the singularity of the integrand. The integral defined on a subinterval has either a weak singularity without rapid oscillation or oscillation without a singularity. The classical quadrature rules for weakly singular integrals using graded points are employed for the singular integral without rapid oscillation and the modified moment-free Filon-type method is used for the oscillatory integrals without a singularity. Unlike the existing methods, the proposed methods do not have to compute the inverse of the oscillator. Numerical experiments are presented to demonstrate the approximation accuracy and the computational efficiency of the proposed methods. Numerical results show that the proposed methods outperform methods published most recently.

Motivation & Objective

• To address the challenge of efficiently computing highly oscillatory integrals with singularities and stationary points when traditional methods become computationally expensive at high wave numbers.
• To develop quadrature rules that avoid computing moments or the inverse of the oscillator, which are bottlenecks in existing Filon-type and Levin-type methods.
• To achieve higher convergence orders—polynomial and exponential—by adapting the number of quadrature nodes based on local behavior of the integrand.
• To improve computational efficiency and approximation accuracy compared to state-of-the-art methods like CFCC and FCC formulas.
• To provide a robust numerical framework applicable to real-world problems in quantum chemistry, tomography, and fluid dynamics involving oscillatory integrals.

Proposed method

• The method partitions the integration interval into subintervals based on the wave number and the location of singularities and stationary points, ensuring each subinterval contains either weak singularity without rapid oscillation or oscillation without singularity.
• For subintervals with weak singularities, classical quadrature rules using graded points are applied to handle the integrand's non-smooth behavior.
• For subintervals with rapid oscillations but no singularity, a modified moment-free Filon-type method is used, avoiding the need to compute moments of the oscillator.
• The composite quadrature rules are constructed using piecewise polynomial interpolants with either fixed or variable numbers of nodes to achieve polynomial or exponential convergence rates.
• The approach avoids computing the inverse of the nonlinear oscillator $ g $, which is required in methods like FCC and CFCC, thus reducing computational cost.
• Theoretical convergence analysis is supported by numerical experiments comparing the proposed method with CMFE, FCC, and CFCC formulas across varying wave numbers and integrand types.

Experimental results

Research questions

• RQ1 Can composite quadrature rules be designed to achieve high convergence rates for highly oscillatory integrals with both singularities and stationary points?
• RQ2 How can the inverse of the oscillator be avoided in moment-free Filon-type methods while maintaining high accuracy?
• RQ3 What is the optimal partitioning strategy for the integration interval that balances accuracy and computational cost based on local oscillation and singularity behavior?
• RQ4 How do the proposed composite moment-free quadrature rules compare in accuracy and efficiency to existing methods like CFCC and FCC?
• RQ5 Can exponential convergence be achieved without computing moments or inverting the oscillator in nonlinear cases?

Key findings

• The proposed composite moment-free quadrature rules achieve polynomial or exponential convergence rates depending on whether a fixed or variable number of interpolation nodes is used.
• Numerical results show that the CMFE formula achieves higher approximation accuracy than the CFCC formula, especially when the integrand has a singularity with index $ \mu < 0 $ and a stationary point of order $ r > 0 $.
• The CMFE method consumes significantly less CPU time than the FCC and CFCC formulas—e.g., 3.35e-2 seconds vs. 2.84e-1 seconds for $ \kappa = 10^2 $ with $ f_1(x) = 1 $, despite using more functional evaluations.
• For $ f_2(x) = \ln x $, the CMFE formula uses 88 functional evaluations and 4.10e-2 seconds, while the CFCC formula uses 73 evaluations and 3.90e-1 seconds, demonstrating superior efficiency.
• The CMFE method does not require computing $ g^{-1} $, which saves substantial computational time compared to FCC and CFCC methods, particularly for nonlinear oscillators.
• In all tested cases, the CMFE method maintains high accuracy across a wide range of wave numbers $ \kappa $, from $ 10^2 $ to $ 10^7 $, with consistent convergence behavior.