This paper studies the shallow Ritz method for solving the one-dimensional diffusion problem. It is shown that the shallow Ritz method improves the order of approximation dramatically for non-smooth problems. To realize this optimal or nearly optimal order of the shallow Ritz approximation, we develop a damped block Newton (dBN) method that alternates between updates of the linear and non-linear parameters. Per each iteration, the linear and the non-linear parameters are updated by exact inversion and one step of a modified, damped Newton method applied to a reduced non-linear system, respectively. The computational cost of each dBN iteration is $O(n)$. Starting with the non-linear parameters as a uniform partition of the interval, numerical experiments show that the dBN is capable of efficiently moving mesh points to nearly optimal locations. To improve efficiency of the dBN further, we propose an adaptive damped block Newton (AdBN) method by combining the dBN with the adaptive neuron enhancement (ANE) method [26].
翻译：本文研究用于求解一维扩散问题的浅层Ritz方法。研究表明，对于非光滑问题，浅层Ritz方法能显著提升逼近阶数。为实现浅层Ritz逼近的最优或接近最优阶，我们提出一种阻尼块牛顿（dBN）方法，该方法在线性参数与非线性参数更新之间交替进行。每次迭代中，线性参数通过精确求逆更新，非线性参数则通过对简化非线性系统应用一步改进的阻尼牛顿法进行更新。每次dBN迭代的计算成本为$O(n)$。以区间均匀划分作为非线性参数初始值，数值实验表明dBN方法能有效将网格点移动至接近最优位置。为进一步提升dBN效率，我们通过将dBN与自适应神经元增强（ANE）方法[26]相结合，提出了自适应阻尼块牛顿（AdBN）方法。
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