Solving large linear least squares problems with linear equality constraints
Solving large linear least squares problems with linear equality constraints
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DOI：
10.1007/s10543-022-00930-2
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发表时间：
2021-06
影响因子：
1.5
通讯作者：
J. Scott;M. Tuma
中科院分区：
文献类型：
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作者：
J. Scott;M. Tuma
我们考虑的问题，解决大规模的线性最小二乘问题，有一个或多个线性约束，必须完全满足。虽然一些经典的方法在理论上是有根据的，但当约束矩阵包含密集行时，或者如果在解决过程中使用的算法变换导致比原始问题更密集的修改问题时，它们可能会面临困难。我们提出修改，强调要求的约束满足一个小的残差。我们研究结合零空间方法与我们最近开发的算法计算零空间基矩阵的“宽”矩阵。我们进一步表明，一个直接消除的方法，通过仔细的旋转可以有效地将问题转化为一个无约束的稀疏密集的最小二乘问题，可以解决现有的直接或迭代方法。我们还提出了一些解决方案的变体，采用增强系统的制定，这可能是有吸引力的解决一系列相关的问题。对实际应用中的问题进行了数值实验，以证明不同方法的有效性。
We consider the problem of solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. While some classical approaches are theoretically well founded, they can face difficulties when the matrix of constraints contains dense rows or if an algorithmic transformation used in the solution process results in a modified problem that is much denser than the original one. We propose modifications with an emphasis on requiring that the constraints be satisfied with a small residual. We examine combining the null-space method with our recently developed algorithm for computing a null-space basis matrix for a “wide” matrix. We further show that a direct elimination approach enhanced by careful pivoting can be effective in transforming the problem to an unconstrained sparse-dense least squares problem that can be solved with existing direct or iterative methods. We also present a number of solution variants that employ an augmented system formulation, which can be attractive for solving a sequence of related problems. Numerical experiments on problems coming from practical applications are used throughout to demonstrate the effectiveness of the different approaches.