用色噪声恒温器求正定矩阵的函数
Evaluating functions of positive-definite matrices using colored noise thermostats
摘要：
计算科学中的许多应用需要计算大型矩阵的函数的元素。一种常用的方法是基于特征值分解的评估，这是一项任务，一般来说，涉及的计算时间与矩阵大小的立方成比例。我们在这里提出了一种方法，可以用来评估一个函数的正定矩阵的元素与缩放是线性的稀疏矩阵和二次在一般情况下。这种方法是基于一个多维谐波势耦合色噪声广义朗之万方程（GLE）恒温器的动力学特性。这种“$f-$恒温器”（FTH）方法允许我们通过仔细剪裁随机动力学的性质来直接计算正定矩阵的函数的元素。我们证明了这种方法的缩放和精度密集和稀疏的问题，并与其他既定的方法的结果进行比较。
英文摘要：
Many applications in computational science require computing the elements of a function of a large matrix. A commonly used approach is based on the the evaluation of the eigenvalue decomposition, a task that, in general, involves a computing time that scales with the cube of the size of the matrix. We present here a method that can be used to evaluate the elements of a function of a positive-definite matrix with a scaling that is linear for sparse matrices and quadratic in the general case. This methodology is based on the properties of the dynamics of a multidimensional harmonic potential coupled with colored noise generalized Langevin equation (GLE) thermostats. This "$f-$thermostat" (FTH) approach allows us to calculate directly elements of functions of a positive-definite matrix by carefully tailoring the properties of the stochastic dynamics. We demonstrate the scaling and the accuracy of this approach for both dense and sparse problems and compare the results with other established methodologies.
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