Hutchinson's estimator is a randomized algorithm that computes an $\epsilon$-approximation to the trace of any positive semidefinite matrix using $\mathcal{O}(1/\epsilon^2)$ matrix-vector products. An improvement of Hutchinson's estimator, known as Hutch++, only requires $\mathcal{O}(1/\epsilon)$ matrix-vector products. In this paper, we propose a generalization of Hutch++, which we call ContHutch++, that uses operator-function products to efficiently estimate the trace of any trace-class integral operator. Our ContHutch++ estimates avoid spectral artifacts introduced by discretization and are accompanied by rigorous high-probability error bounds. We use ContHutch++ to derive a new high-order accurate algorithm for quantum density-of-states and also show how it can estimate electromagnetic fields induced by incoherent sources.

翻译：Hutchinson 估计器是一种随机算法，它使用 $\mathcal{O}(1/\epsilon^2)$ 次矩阵-向量乘积来计算任意半正定矩阵迹的 $\epsilon$ 近似。Hutchinson 估计器的一种改进版本，称为 Hutch++，仅需 $\mathcal{O}(1/\epsilon)$ 次矩阵-向量乘积。本文提出了一种 Hutch++ 的推广方法，我们称之为 ContHutch++，它利用算子-函数乘积来高效估计任意迹类积分算子的迹。我们的 ContHutch++ 估计避免了离散化引入的频谱伪影，并附有严格的高概率误差界。我们使用 ContHutch++ 推导了一种新的高精度量子态密度算法，并展示了如何用它来估计非相干源感生的电磁场。