标题： 用于大型对称矩阵密集谱的Krylov投影算法 显示英文标题

标题： A Krylov projection algorithm for large symmetric matrices with dense spectra

摘要： 我们考虑对大型s.p.d.的$B^T (A+sI)^{-1} B$进行近似。$A\in\mathbb{R}^{n\times n}$具有密集谱和$B\in\mathbb{R}^{n\times p}$、$p\ll n$。我们针对连续谱测度问题的大规模离散化中的多输入多输出(MIMO)传递函数的计算，例如无界域上的线性时不变(LTI)PDE。传统的Krylov方法，如Lanczos或CG算法，已知对于具有实正$s$的$(A+sI)^{-1}B$的计算是最佳的，导致适应于明显离散和非均匀谱。然而，对于具有密集谱的矩阵，这种适应会减弱。 在[Zimmerling, Druskin, Simoncini, 《科学计算杂志》103(1), 5 (2025)]中已经证明，使用块Lanczos方法计算的Gau{\ss }和Gau\ss -Radau求积法的平均值可以显著减少此类问题的近似误差。 在这里，我们引入了一种自适应的Kreĭn-Nudelman扩展到（块）Lanczos递推公式，使得在几乎不增加$o(n)$成本的情况下进一步加速。 类似于Gau\ss -Radau求积法，对（块）Lanczos矩阵应用了一个低秩修正。 然而，与Gau\ss -Radau求积法不同，这种修正依赖于$\sqrt{s}$并可以在Hermite-Padé逼近的框架下进行考虑，这些逼近已知对于具有分支切割的问题是高效的，可以作为密集谱区间的良好近似。 在无界域中热扩散和准静磁 Maxwell 算子的大规模离散化的数值结果证实了所提出方法的效率。 显示英文摘要

摘要： We consider the approximation of $B^T (A+sI)^{-1} B$ for large s.p.d. $A\in\mathbb{R}^{n\times n}$ with dense spectrum and $B\in\mathbb{R}^{n\times p}$, $p\ll n$. We target the computations of Multiple-Input Multiple-Output (MIMO) transfer functions for large-scale discretizations of problems with continuous spectral measures, such as linear time-invariant (LTI) PDEs on unbounded domains. Traditional Krylov methods, such as the Lanczos or CG algorithm, are known to be optimal for the computation of $(A+sI)^{-1}B$ with real positive $s$, resulting in an adaptation to the distinctively discrete and nonuniform spectra. However, the adaptation is damped for matrices with dense spectra. It was demonstrated in [Zimmerling, Druskin, Simoncini, Journal of Scientific Computing 103(1), 5 (2025)] that averaging Gau{\ss} and Gau\ss -Radau quadratures computed using the block-Lanczos method significantly reduces approximation errors for such problems. Here, we introduce an adaptive Kre\u{i}n-Nudelman extension to the (block) Lanczos recursions, allowing further acceleration at negligible $o(n)$ cost. Similar to the Gau\ss -Radau quadrature, a low-rank modification is applied to the (block) Lanczos matrix. However, unlike the Gau\ss -Radau quadrature, this modification depends on $\sqrt{s}$ and can be considered in the framework of the Hermite-Pad\'e approximants, which are known to be efficient for problems with branch-cuts, that can be good approximations to dense spectral intervals. Numerical results for large-scale discretizations of heat-diffusion and quasi-magnetostatic Maxwell's operators in unbounded domains confirm the efficiency of the proposed approach.