带噪声椭圆方程组的算子移位
Operator Shifting for Noisy Elliptic Systems
摘要：
在计算科学中，人们必须经常从受噪声和不确定性影响的数据中估计模型参数，从而导致不准确的结果。为了提高含噪声参数模型的精度，研究了椭圆线性系中算子被噪声污染时的误差减小问题。我们假设噪声保持正定性，但除此之外，我们不对噪声的结构做任何额外的假设。在这些假设下，我们提出了运营商转移框架，一个易于实现的算法，增加了一个嘈杂的逆算子减去一个额外的辅助项的集合。以与James-Stein估计器类似的方式，这具有使噪声逆算子更接近地面真实值的效果，并且因此通过减少偏差和方差两者来减少误差。我们开发的引导蒙特卡罗算法估计所需的增强幅度在嘈杂的系统中的最佳误差减少。为了提高这些算法的易处理性，我们提出了几个近似的多项式展开的算子逆，并证明了理想的收敛性和单调性的性质，这些扩展。我们还证明了定理，量化的误差减少操作员增强。除了理论结果，我们提供了一组不同的图形和网格拉普拉斯系统的数值实验，都证明了我们的方法的有效性。
英文摘要：
In the computational sciences, one must often estimate model parameters from data subject to noise and uncertainty, leading to inaccurate results. In order to improve the accuracy of models with noisy parameters, we consider the problem of reducing error in an elliptic linear system with the operator corrupted by noise. We assume the noise preserves positive definiteness, but otherwise, we make no additional assumptions the structure of the noise. Under these assumptions, we propose the operator shifting framework, a collection of easy-to-implement algorithms that augment a noisy inverse operator by subtracting an additional auxiliary term. In a similar fashion to the James-Stein estimator, this has the effect of drawing the noisy inverse operator closer to the ground truth, and hence reduces error by reducing both bias and variance. We develop bootstrap Monte Carlo algorithms to estimate the required augmentation magnitude for optimal error reduction in the noisy system. To improve the tractability of these algorithms, we propose several approximate polynomial expansions for the operator inverse, and prove desirable convergence and monotonicity properties for these expansions. We also prove theorems that quantify the error reduction obtained by operator augmentation. In addition to theoretical results, we provide a set of numerical experiments on four different graph and grid Laplacian systems that all demonstrate effectiveness of our method.
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