使用无限Arnoldi方法计算非线性特征值问题的部分Schur分解
Computing a partial Schur factorization of nonlinear eigenvalue problems using the infinite Arnoldi method
AI总结：
提出一种基于无限Arnoldi方法的技术，通过将非线性特征值问题转化为无限维线性算子并利用结构保持的隐式重启，计算其部分Schur分解，显著降低计算时间和内存消耗。
摘要：
部分Schur分解可用于以数值鲁棒的方式表示矩阵的多个特征对。Arnoldi方法的不同变体常被用于计算部分Schur分解。本文提出一种计算非线性特征值问题（NEP）的部分Schur分解的技术。该技术受文献[8]中算法的启发，现称为无限Arnoldi方法。无限Arnoldi方法是为NEP设计的方法，可解释为应用于线性无限维算子的Arnoldi方法，该算子的倒数特征值是NEP的解。首先，我们证明算子的不变对与NEP的不变对等价。我们刻画了算子不变对的结构，并展示了如何通过尊重该结构来修改无限Arnoldi方法。这也使我们能够自然地添加称为锁定（locking）的特性。我们将该算法嵌套在外迭代中，其中针对特定类型结构化函数的无限Arnoldi方法被适当地重启。重启利用了结构，并受标准特征值问题中著名的隐式重启Arnoldi方法的启发。最终算法应用于基准集合中的例子，表明重启技术可以显著减少处理时间和内存消耗。
英文摘要：
The partial Schur factorization can be used to represent several eigenpairs of a matrix in a numerically robust way. Different adaptions of the Arnoldi method are often used to compute partial Schur factorizations. We propose here a technique to compute a partial Schur factorization of a nonlinear eigenvalue problem (NEP). The technique is inspired by the algorithm in [8], now called the infinite Arnoldi method. The infinite Arnoldi method is a method designed for NEPs, and can be interpreted as Arnoldi's method applied to a linear infinite-dimensional operator, whose reciprocal eigenvalues are the solutions to the NEP. As a first result we show that the invariant pairs of the operator are equivalent to invariant pairs of the NEP. We characterize the structure of the invariant pairs of the operator and show how one can carry out a modification of the infinite Arnoldi method by respecting the structure. This also allows us to naturally add the feature known as locking. We nest this algorithm with an outer iteration, where the infinite Arnoldi method for a particular type of structured functions is appropriately restarted. The restarting exploits the structure and is inspired by the well-known implicitly restarted Arnoldi method for standard eigenvalue problems. The final algorithm is applied to examples from a benchmark collection, showing that both processing time and memory consumption can be considerably reduced with the restarting technique.
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