A fast solver for Fredholm equations of the second kind with weakly singular kernels
A fast solver for Fredholm equations of the second kind with weakly singular kernels
复制标题
具有弱奇异核的第二类 Fredholm 方程的快速求解器
DOI：
10.1515/jnma.2002.13
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发表时间：
2002
期刊：
影响因子：
--
通讯作者：
Chi
中科院分区：
文献类型：
--
作者：
R. Chan;F. Lin;Chi
摘要本文研究了第二类Fredholm积分方程的解，其中核函数是渐近光滑的，或者这类函数与高振荡系数函数的乘积。我们提出了一种基于多项式插值的方法，利用这些积分算子的离散化来逼近矩阵a。我们的近似矩阵B是通过将定义核函数的域划分为不同大小的子域，并在每个子域上用切比雪夫点处的插值多项式逼近核函数得到的。虽然B是密集的，但它仍然可以在O（nk）次操作中构造，需要O（nk）个存储空间，并且可以在O（nk logn）次操作中获得乘积B y，其中n是矩阵的大小，k是所使用的插值多项式的程度。证明了对于光滑核（包括log|x - t|），如果k是O(logε - 1)，对于弱奇异核（如|x - t| - 1/2），如果k是O(loglog + logε - 1)，则F≥ε。与Alpert等人的类小波方法比较[j]。[com[14]: 159-184, 1993]表明我们的方法需要更少的内存和更准确。
Abstract In this paper, we consider solutions of Fredholm integral equations of the second kind where the kernel functions are asymptotically smooth or products of such functions with highly oscillatory coefficient functions. We present a scheme based on polynomial interpolation to approximate matrices A from the discretization of these integral operators. Our approximation matrix B is obtained by partitioning the domain on which the kernel function is defined into subdomains of different sizes and approximating the kernel function at each subdomain by interpolation polynomial at the Chebyshev points. Although B is dense, it can still be constructed in O(nk) operations, requires O(nk) storage and the product B y can be obtained in O(nk logn)operations, where n is the size of the matrix and k is the degree of the interpolation polynomial used. We prove that the Frobenius norm ‖A – B‖ F ⩽ ε if k is of O(logε –1) for smooth kernels (including log|x – t|) and of O(loglogn + logε –1) for weakly singular kernels such as |x – t|–1/2. Comparison with the wavelet-like method by Alpert et al. [SIAM J. Sci. Comp 14: 159–184, 1993] shows that our method requires less memory and is more accurate.
影响因子：
2
作者：
ALPERT, BK
通讯作者：
ALPERT, BK