论文标题
多种学习和非线性均质化
Manifold Learning and Nonlinear Homogenization
论文作者
论文摘要
我们描述了一个基于非线性多尺度PDE问题的有效域分解框架。该框架的灵感来自多种学习技术，并利用最近邻居跨越的切线空间来压缩本地解决方案歧管。我们的框架应用于具有振荡介质和非线性辐射传递方程的半线性椭圆方程。在这两种情况下，都可以观察到疗效的显着提高。这种新方法不依赖于对多尺度PDE的详细分析理解，例如它们的渐近限制，因此对于一般的多尺度问题而言，它更具用途。
We describe an efficient domain decomposition-based framework for nonlinear multiscale PDE problems. The framework is inspired by manifold learning techniques and exploits the tangent spaces spanned by the nearest neighbors to compress local solution manifolds. Our framework is applied to a semilinear elliptic equation with oscillatory media and a nonlinear radiative transfer equation; in both cases, significant improvements in efficacy are observed. This new method does not rely on detailed analytical understanding of the multiscale PDEs, such as their asymptotic limits, and thus is more versatile for general multiscale problems.
