论文标题
Tikhonov正则化的自适应交叉近似一般形式
Adaptive cross approximation for Tikhonov regularization in general form
论文作者
论文摘要
科学和工程方面的许多问题都会引起第一类的线性积分方程，并具有光滑的内核。积分运算符的离散化产生了一个矩阵，其奇异值群集在原点处。我们通过自适应交叉近似来描述此类矩阵的近似值，从而避免形成整个矩阵。讨论了自适应交叉近似步骤数的选择。离散的右侧表示通常受测量误差污染的数据。因此获得的方程式线性系统的解并不有意义，因为由自适应交叉近似确定的矩阵是缺陷的。我们通过使用Tikhonov正则化来解决这一困难，并讨论如何使用相当普遍的正则矩阵。计算的示例表明，使用与身份不同的正则矩阵可以显着提高计算的近似解决方案的质量。
Many problems in Science and Engineering give rise to linear integral equations of the first kind with a smooth kernel. Discretization of the integral operator yields a matrix, whose singular values cluster at the origin. We describe the approximation of such matrices by adaptive cross approximation, which avoids forming the entire matrix. The choice of the number of steps of adaptive cross approximation is discussed. The discretized right-hand side represents data that commonly are contaminated by measurement error. Solution of the linear system of equations so obtained is not meaningful because the matrix determined by adaptive cross approximation is rank-deficient. We remedy this difficulty by using Tikhonov regularization and discuss how a fairly general regularization matrix can be used. Computed examples illustrate that the use of a regularization matrix different from the identity can improve the quality of the computed approximate solutions significantly.
