本项目研究Helmholtz边界积分方程的多尺度快速算法。Helmholtz边界积分方程在实际问题中具有广泛的应用，然而，当波数增大时，方程的数值求解极其困难。本项目将从三个重要方面：系数矩阵的截断、高振荡奇异积分的数值计算以及大规模代数方程组的求解，来解决Helmholtz边界积分方程在数值求解中的困难。首先，对于系数矩阵的截断问题，由于多尺度基底离散化的代数方程其系数矩阵是满矩阵，具有数值稀疏性，根据积分核的奇异位置以及振荡性，本项目将设计一种新的矩阵截断策略，用截断后的矩阵代替原来的满矩阵，从而提高系数矩阵的计算效率。其次，针对高振荡奇异积分导致传统的数值求积公式失效这一问题，本项目基于多尺度基底将构造满足整体拟线性计算复杂度的有效求积公式。在大规模代数方程组的求解方面，本项目利用截断后的系数矩阵具有“分层”的特点，建立多层扩充格式，使得大规模方程组求解的计算复杂度为拟线性的。

This subject focuses on the fast multiscale algorithms for the boundary integral equations of the Helmholtz equation. The numerical computations for these equations have wide applications in the engineering. However, many difficulties are encountered when studying the numerical solution of these equations as the wave number increases. First, since the matrix is numerical sparse based on the discretion of multiscale bases, we introduce a new truncation strategy of matrix according to the oscillating property and the positions of singularity to increasing the efficient of algorithms. Secondly,not only the integral kernels have high oscillating property but also the integral kernels have the singularity, which yield the classical integration rules don’t work.We construct an efficient integration scheme satisfying the quasi linear computational complexity order. Finally, we construct a fast multiscale method for the boundary integral equation of the Helmholtz equation since the matrix has the “level”character. We will apply the new algorithms in the electromagnetic scattering problems.