态密度估计的准确性：一项比较研究
Accurate Density of States Estimation: A Comparative Study
- Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究通过比较多种方法，提出利用累积态密度的单调分段三次插值（中点样条）来高效准确估计大型稀疏矩阵的态密度，该方法无需固定带宽高斯平滑，且能以较少矩阵-向量乘积获得高精度结果。
AI中文摘要：
我们研究大型稀疏实对称矩阵的态密度（DOS）估计，通过拟合累积态密度（CDOS）并对拟合结果进行微分。我们使用随机Lanczos方法提供Ritz值和权重，这些值在多次Lanczos运行中取平均，以构建CDOS中点数据。对这些数据进行单调分段三次插值，产生一个平滑的CDOS近似，可以轻松微分，得到非负、归一化的分段二次DOS，而无需在构造中使用固定带宽的高斯平滑。我们还考虑使用高斯过程回归（GPR）近似CDOS，采用单高斯和双高斯协方差核，产生带有不确定性信息的近似。然后通过基于GPR的CDOS的解析导数获得DOS近似。我们对来自不同科学应用的矩阵进行实验，将这些估计器与高斯展宽随机Lanczos和Jackson阻尼核多项式近似进行比较。在共同的高斯验证分辨率下，我们使用误差度量来评估局部差异、积分误差以及整体谱形状的一致性。结果表明，中点样条可以用相对较少的矩阵-向量乘积产生准确的DOS近似，支持使用累积插值作为实用的DOS估计方法。
英文摘要：
We study density of states (DOS) estimation for large sparse real symmetric matrices by fitting the cumulative density of states (CDOS) and differentiating the fit. We use a stochastic Lanczos method to supply Ritz values and weights, which we average across Lanczos runs to construct CDOS midpoint data. Monotone piecewise-cubic interpolation of these data yields a smooth CDOS approximation that can be easily differentiated to yield a nonnegative, normalized, piecewise-quadratic DOS without requiring a Gaussian smoothing with a fixed bandwidth in its construction. We also consider approximating CDOS by using Gaussian-process regression (GPR) with single- and double-Gaussian covariance kernels, yielding an approximation with uncertainty information. DOS approximation is then obtained by analytic derivative of the GPR-based CDOS. Experiments on matrices from diverse scientific applications are performed to compare these estimators with Gaussian-broadened stochastic Lanczos and Jackson-damped kernel polynomial approximations. At a common Gaussian validation resolution, we use error measures that assess local discrepancies, integrated errors, and agreement in the overall spectral shape. The results demonstrate that the midpoint spline can produce accurate approximations to the DOS with relatively few matrix--vector products, supporting the use of cumulative interpolation as a practical DOS estimation.
