论文标题
蒙特卡林估计器的对称阳性半金属矩阵的schatten p-norm
Monte Carlo Estimators for the Schatten p-norm of Symmetric Positive Semidefinite Matrices
论文作者
论文摘要
我们提出了计算阳性半明确矩阵的schatten $ p $ norm的数值方法。我们的动机源于对反问题的不确定性定量和最佳实验设计，在该问题中，Schatten $ p $ norm定义了一种称为P-最佳标准的设计标准。计算Schatten $ p $ - 高维矩阵的计算在计算上很昂贵。我们提出了一种无基质方法，以使用蒙特卡洛估计器估算schatten $ p $ norm，并得出估算器的收敛结果和错误估计。为了有效地计算非企业的schatten $ p $ norm和$ p $的大值，我们使用chebyshev多项式近似的估算器，并将我们的收敛性和错误分析扩展到此设置。我们通过应用模型逆问题的最佳实验设计来证明我们提出的估计器在几个测试矩阵上的性能。
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.
