对称正半定矩阵Schatten p-模的蒙特卡罗估计
Monte Carlo Estimators for the Schatten p-norm of Symmetric Positive Semidefinite Matrices
摘要：
给出了半正定矩阵Schatten $p$ -范数的数值计算方法。我们的动机源于反问题的不确定性量化和最优试验设计，其中Schatten $p$规范定义了一个设计准则，称为P-最优准则。计算高维矩阵的Schatten $p$范数在计算上是昂贵的。我们提出了一种无矩阵的方法来估计Schatten $p$ -范数使用蒙特卡罗估计，并推导出收敛结果和误差估计的估计。为了有效地计算非整数和大值的$p$的Schatten $p$ -范数，我们使用一个估计使用切比雪夫多项式近似和扩展我们的收敛性和误差分析，以及这个设置。我们证明了我们提出的估计的性能，几个测试矩阵，并通过应用程序的最佳试验设计的模型反问题。
英文摘要：
We present numerical methods for computing the Schatten $p$-norm of positive semi-definite matrices. Our motivation stems from uncertainty quantification and optimal experimental design for inverse problems, where the Schatten $p$-norm defines a design criterion known as the P-optimal criterion. Computing the Schatten $p$-norm of high-dimensional matrices is computationally expensive. We propose a matrix-free method to estimate the Schatten $p$-norm using a Monte Carlo estimator and derive convergence results and error estimates for the estimator. To efficiently compute the Schatten $p$-norm for non-integer and large values of $p$, we use an estimator using a Chebyshev polynomial approximation and extend our convergence and error analysis to this setting as well. We demonstrate the performance of our proposed estimators on several test matrices and through an application to optimal experimental design of a model inverse problem.
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