A fast and well-conditioned spectral method for singular integral equations
A fast and well-conditioned spectral method for singular integral equations
复制标题
奇异积分方程的快速且条件良好的谱方法
DOI：
10.1016/j.jcp.2016.12.009
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
S. Olver
中科院分区：
文献类型：
--
作者：
R. Slevinsky;S. Olver
我们开发了一种谱方法，利用切比雪夫和超球多项式将方程重新表述为几乎带状的无限维系统，用于求解区间并集上的单变量奇异积分方程。这是通过利用低秩近似的稀疏表示的双变量内核。所得到的系统可以解决在O（m 2 n）操作使用自适应QR分解，其中m是带宽和n是解决真正的解决方案所需的未知数的最佳数量。当同一个操作符用于多个右侧时，通过预缓存QR分解，复杂度降低到O（m n）操作。稳定性证明表明，由此产生的线性算子可以对角预处理是一个紧凑的扰动的身份。考虑的应用包括法拉第笼以及亥姆霍兹方程和重力亥姆霍兹方程的声散射，包括远场和近场解的光谱精确数值评估。Julia软件包SingularIntegralEquations。jl用方便的、用户友好的界面实现了我们的方法。
We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels. The resulting system can be solved in O (m 2 n) operations using an adaptive QR factorization, where m is the bandwidth and n is the optimal number of unknowns needed to resolve the true solution. The complexity is reduced to O (m n) operations by pre-caching the QR factorization when the same operator is used for multiple right-hand sides. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. Applications considered include the Faraday cage, and acoustic scattering for the Helmholtz and gravity Helmholtz equations, including spectrally accurate numerical evaluation of the far-and near-field solution. The Julia software package SingularIntegralEquations. jl implements our method with a convenient, user-friendly interface.