改进有理逼近的贪心算法
Improving Greedy Algorithms for Rational Approximation
AI总结：
本文提出两种改进的贪心算法，用于在一致范数下对分数阶拉普拉斯算子进行有理逼近，以构建对称正定的移位拉普拉斯算子序列，并有效预条件Darcy-Stokes耦合问题。
摘要：
在开发多物理场问题的鲁棒预条件子时，经常会出现拉普拉斯算子的分数阶函数，并且需要对其求逆。一致范数下的有理逼近可用于将这些分数阶算子的求逆转化为对一系列移位拉普拉斯算子的求逆。在逼近过程中必须小心处理，以确保移位拉普拉斯算子保持对称正定，从而使其具有更好的条件数。在本工作中，我们研究了两种贪心算法，用于寻找此类分数阶算子的有理逼近。第一种算法通过在该过程中增加一个一致范数下的最小化步骤，改进了[Li等人，SISC，2024]中讨论的正交贪心算法。第二种方法采用一致范数下的弱切比雪夫贪心算法。两种方法都能产生非递增的误差。数值结果证实了我们所提出算法的有效性，这些算法也具有灵活性，可适用于其他逼近问题。此外，借助对分数阶算子的有效有理逼近，所得算法在预条件处理Darcy-Stokes耦合问题时表现出良好的性能。
英文摘要：
When developing robust preconditioners for multiphysics problems, fractional functions of the Laplace operator often arise and need to be inverted. Rational approximation in the uniform norm can be used to convert inverting those fractional operators into inverting a series of shifted Laplace operators. Care must be taken in the approximation so that the shifted Laplace operators remain symmetric positive definite, making them better conditioned. In this work, we study two greedy algorithms for finding rational approximations to such fractional operators. The first algorithm improves the orthogonal greedy algorithm discussed in [Li et al., SISC, 2024] by adding one minimization step in the uniform norm to the procedure. The second approach employs the weak Chebyshev greedy algorithm in the uniform norm. Both methods yield non-increasing error. Numerical results confirm the effectiveness of our proposed algorithms, which are also flexible and applicable to other approximation problems. Moreover, with effective rational approximations to the fractional operator, the resulting algorithms show good performance in preconditioning a Darcy-Stokes coupled problem.
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