大波数Helmholtz方程新型、高效积分方程解法的研究

Numerical simulation of the solutions of Helmholtz equations with large wavenumbers arises in many applications in mathematics, physics and engineering, which has attracted a growing interest from scientific computation community, and is a hot and difficult topic in this area. There are a number of challenging problems that require mathematical insights to solve. The main focus of this proposal is on developing new and efficient integral equation methods for solving Helmholtz equations with large wavenumbers, where the solutions of the Helmholtz equations are formulated by boundary integral equations. One feature of these integral equations is of particular relevance: the solutions are more highly oscillatory as the wavenumber increases. Numerical approaches for these equations have been constructed by Galerkin methods, collocation methods, etc. These discretisations of oscillatory integral equations invariably involve the evaluation of a large number of (singularly) oscillatory integrals. Therefore, the problems of evaluating oscillatory integrals and of solving oscillatory integral equations are highly correlated. Consequently, efficient solution methods for integral equations require efficient methods for the calculation of such integrals. However,the computation of these highly oscillatory integrals by standard quadrature methods is exceedingly difficult and the cost steeply increases with large values of the wavenumber. This means that the numerical methods based on standard numerical quadrature formulas, such as Gauss, Clenshaw-Curtis and Nystr?m, are not feasible for solving these equations. These are particularly challenging examples of the general problem of computing oscillatory integrals, which has a long history and has seen a great deal of research interest in recent years. Here, we will devote to develop new Clenshaw-Curtis-Filon-type methods for approximation of such integrals with singularities, which are in simplicity and can be implemented using O(NlogN) operations with the fast Fourier transform (FFT), and share that the higher the wavenumber, the higher the accuracy. Furthermore, we will analyze the asymptotics of the solution of the Helmholtz equation and study the hybrid methods combined with conventional piecewise polynomial approximations, orthogonal polynomials, trigonometric functions or Bessel functions with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions. They have the potential to solve scattering problems accurately and efficiently. Finally, together with the above analysis and efficient methods, we will design efficient (spectral) Galerkin, collocation and Filon methods for solving the boundary value problems for these Helmholtz equations formulated by boundary integral equations, and consider their convergence and error bounds.

大波数Helmholtz方程的数值计算在数学、物理、工程领域有着广泛而重要的应用，是科学计算领域非常重要、被公认为难的国际热点研究课题，存在许多挑战性问题亟待解决。本项目集中研究大波数Helmholtz方程的高效积分方程解法，其与传统方法的区别在于：第一，针对Galerkin、配置法等离散格式中的（Hilbert变换、对数奇异、高阶奇异）高振荡积分,发展新型、高效的积分算法，而非传统的Gauss、Clenshaw-Curtis或Nystr?m等算法（当波数很大时，这些传统算法的计算效率极低、计算精度较差）；第二，强调大波数问题解的渐进分析，针对其渐进性分析,构造混合方法中基于正交多项式等的高振荡混合基函数的逼近。基于这些理论与算法分析，结合（谱）Galerkin、配置、Filon方法，构建出大波数问题的新型、高精度的高效积分方程解法。

项目针对具有广泛应用背景的大波数Helmholtz方程，集中研究了其高效数值积分方程解法。针对传统Galerkin、配置法等离散格式产生的含对数奇异、Hilbert、高阶奇异的高振荡积分发展快速高效数值积分格式，分析其关于波数的渐进性，并根据渐进性分析构造混合方法的高振荡基函数逼近。克服了传统的Gauss、Clenshaw-Curtis、Nyström等算法对高振荡问题的计算复杂性高、计算精度差等缺陷,一些算法可应用于医疗图像、电磁散射反问题中Volterra积分方程的研究,解决了一般Jacobi点多项式插值的最优阶估计以及Hermite-Fejer重心插值格式的快速计算，推动了古典数值分析理论的发展。研究结果发表于SIAM J. Numer. Math.、SIAM J. Sci. Comput.、Math. Prog.等顶级期刊，已发表标注基金资助SCI论文23篇。

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