Estimating the trace of the inverse of a large matrix is an important problem in lattice quantum chromodynamics. A multilevel Monte Carlo method is proposed for this problem that uses different degree polynomials for the levels. The polynomials are developed from the GMRES algorithm for solving linear equations. To reduce orthogonalization expense, the highest degree polynomial is a composite or double polynomial found with a polynomial preconditioned GMRES iteration. Added to some of the Monte Carlo pieces is deflation of eigenvalues that reduces the variance. Deflation is also used for finding a reduced degree deflated polynomial. The new Multipolynomial Monte Carlo method can significantly improve the trace computation for matrices that have a difficult spectrum due to small eigenvalues.

翻译：大型矩阵逆矩阵的迹估计是格点量子色动力学中的一个重要问题。本文针对该问题提出了一种多层级蒙特卡洛方法，该方法在不同层级上使用不同次数的多项式。这些多项式基于求解线性方程组的GMRES算法构建。为降低正交化计算开销，最高次多项式采用通过多项式预条件GMRES迭代获得的复合多项式或双重多项式。部分蒙特卡洛计算单元中引入了特征值减缩技术以降低方差，同时利用减缩技术构造降次减缩多项式。这种新型多多项式蒙特卡洛方法能够显著改善因小特征值导致谱分布困难的矩阵的迹计算性能。