The AZ algorithm for least squares systems with a known incomplete generalized inverse
The AZ algorithm for least squares systems with a known incomplete generalized inverse
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具有已知不完全广义逆的最小二乘系统的 AZ 算法
DOI：
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发表时间：
2019
影响因子：
1.5
通讯作者：
M. Webb
中科院分区：
文献类型：
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作者：
Vincent Coppé;D. Huybrechs;Roel Matthysen;M. Webb
给出了一个矩形线性系统$Ax=b$的最小二乘解的算法，其中$A$可能是任意病态的。我们假设一个互补矩阵$Z$是已知的，使得$A-AZ^*A$在数值上是低阶的。不严格地说，$Z^*$的作用类似于$A$的广义逆，直到数值上的低阶误差。我们给出了几个$(A，Z)$组合在函数逼近中的例子，其中我们可以在许多非标准设置下实现高阶逼近：具有不规则形状的区域上的函数的逼近，具有高度偏斜权的加权最小二乘问题，以及具有局部奇异性的函数的谱逼近。当$A$和$Z^*$具有快速的矩阵-向量乘法并且当$A-AZ^*A$的数值秩小时，该算法是最有效的。
We introduce an algorithm for the least squares solution of a rectangular linear system $Ax=b$, in which $A$ may be arbitrarily ill-conditioned. We assume that a complementary matrix $Z$ is known such that $A - AZ^*A$ is numerically low rank. Loosely speaking, $Z^*$ acts like a generalized inverse of $A$ up to a numerically low rank error. We give several examples of $(A,Z)$ combinations in function approximation, where we can achieve high-order approximations in a number of non-standard settings: the approximation of functions on domains with irregular shapes, weighted least squares problems with highly skewed weights, and the spectral approximation of functions with localized singularities. The algorithm is most efficient when $A$ and $Z^*$ have fast matrix-vector multiplication and when the numerical rank of $A - AZ^*A$ is small.