论文标题
不确定性的半线性椭圆方程的随机梯度下降
Stochastic Gradient Descent for Semilinear Elliptic Equations with Uncertainties
论文作者
论文摘要
随机性在现代工程中无处不在。不确定性通常被建模为描述潜在物理学的微分方程中的随机系数。在这项工作中，我们描述了一个两步框架，用于用随机系数求解半连续性椭圆形偏微分方程：1）基于直接的变异计算方法将问题重新加密为功能最小化问题； 2）使用随机梯度下降法解决最小化问题。我们为所得随机梯度下降算法提供了收敛标准，并讨论了一些有用的技术来克服不良条件和较大差异的问题。数值实验证明了算法的准确性和效率。
Randomness is ubiquitous in modern engineering. The uncertainty is often modeled as random coefficients in the differential equations that describe the underlying physics. In this work, we describe a two-step framework for numerically solving semilinear elliptic partial differential equations with random coefficients: 1) reformulate the problem as a functional minimization problem based on the direct method of calculus of variation; 2) solve the minimization problem using the stochastic gradient descent method. We provide the convergence criterion for the resulted stochastic gradient descent algorithm and discuss some useful technique to overcome the issues of ill-conditioning and large variance. The accuracy and efficiency of the algorithm are demonstrated by numerical experiments.
