论文标题
数据自适应RKHS Tikhonov正规化操作员学习内核
Data adaptive RKHS Tikhonov regularization for learning kernels in operators
论文作者
论文摘要
我们提出DARTR：用于运算符中功能参数的非参数学习的线性反面问题的数据自适应RKHS Tikhonov正则化方法。一个关键成分是系统固有的数据自适应（SIDA）RKHS，其规范限制了学习在可识别性功能空间中进行的学习。 Dartr使用此规范，并通过L-Curve方法选择正则化参数。我们在具有离散合成数据的示例中说明了其性能。数值结果表明，由于数据网格在不同级别的噪声下，使用$ l^2 $和$ l^2 $规范，由于数据网格在不同级别的噪声下的精制，DARTR会导致数值鲁棒的准确估计器对数值误差的鲁棒性，并且估算器以一致的速率收敛，并且估算值以一致的速率收敛。
We present DARTR: a Data Adaptive RKHS Tikhonov Regularization method for the linear inverse problem of nonparametric learning of function parameters in operators. A key ingredient is a system intrinsic data-adaptive (SIDA) RKHS, whose norm restricts the learning to take place in the function space of identifiability. DARTR utilizes this norm and selects the regularization parameter by the L-curve method. We illustrate its performance in examples including integral operators, nonlinear operators and nonlocal operators with discrete synthetic data. Numerical results show that DARTR leads to an accurate estimator robust to both numerical error due to discrete data and noise in data, and the estimator converges at a consistent rate as the data mesh refines under different levels of noises, outperforming two baseline regularizers using $l^2$ and $L^2$ norms.
