基于力的准连续算子在均匀周期链上的谱
The Spectrum of the Force-Based Quasicontinuum Operator for a Homogeneous Periodic Chain
AI总结：
研究线性化基于力的准连续算子的谱性质，证明其谱为正且与次近邻相互作用下的准非局域准连续算子谱相同，并建立了特征向量矩阵条件数的一致界，从而验证并改进了基于数值实验的猜想。
摘要：
我们在一般条件下证明，线性化的基于力的准连续（QCF）算子具有正谱，且在次近邻相互作用情况下，该谱与准非局域准连续（QNL）算子的谱相同。此外，我们建立了特征向量矩阵条件数的一个界，该界在原子数和原子区域大小上一致。这些结果验证并改进了近期基于数值实验的猜想（[arXiv:0907.3861, 猜想2] 和 [arXiv:0910.2013, 猜想8]）。作为我们结果的直接推论，我们获得了（预处理）GMRES算法收敛速度的严格估计，以及QCF方法的一个新的稳定性估计。
英文摘要：
We show under general conditions that the linearized force-based quasicontinuum (QCF) operator has a positive spectrum, which is identical to the spectrum of the quasinonlocal quasicontinuum (QNL) operator in the case of second-neighbour interactions. Moreover, we establish a bound on the condition number of a matrix of eigenvectors that is uniform in the number of atoms and the size of the atomistic region. These results establish the validity of and improve upon recent conjectures ([arXiv:0907.3861, Conjecture 2] and [arXiv:0910.2013, Conjecture 8]) which were based on numerical experiments. As immediate consequences of our results we obtain rigorous estimates for convergence rates of (preconditioned) GMRES algorithms, as well as a new stability estimate for the QCF method.
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