混合与混合有限元方法结合区域分解和谱方法在KPZ模型重整化研究中的应用
Combined use of mixed and hybrid finite elements method with domain decomposition and spectral methods for a study of renormalization for the KPZ model
AI总结：
本文研究KPZ模型的非线性随机演化问题，提出使用混合与混合有限元结合区域分解和噪声平滑进行离散化，并通过数值实验验证了KPZ方程的病态性及Hairer重整化程序的必要性。
摘要：
本工作的重点是初边值问题相关的含时偏微分方程的数值逼近。本硕士论文主要关注Kardar-Parisi-Zhang (KPZ) 模型支配的非线性随机演化问题解的实际计算。此外，论文旨在通过大量数值实验来佐证：具有周期边界条件的KPZ方程初边值问题是病态的，且该方程需要重整化。KPZ方程的离散化方法是使用混合与混合有限元结合区域分解过程，以及对噪声进行适当的平滑处理。所得解与具乘性噪声的随机热方程通过Cole-Hopf变换给出的著名解进行了比较。我们验证了两个解表现出良好的一致性，但随着平滑子的支撑集减小，会出现一个增长的偏移。对于随机热方程的数值逼近，我们使用了一种最先进的半线性随机偏微分方程数值方法，该方法结合了谱技术、Taylor展开和对底层噪声的特殊数值处理。此外，使用了Martin Hairer引入的最先进的重整化程序对KPZ方程进行重整化，并通过非平凡的数值实验进行了验证。
英文摘要：
The focus of this work is the numerical approximation of time-dependent partial differential equations associated to initial-boundary value problems. This master dissertation is mostly concerned with the actual computation of the solution to nonlinear stochastic evolution problems governed by Kardar-Parisi-Zhang (KPZ) models. In addition, the dissertation aims to contribute to corroborate, by means of a large set of numerical experiments, that the initial-boundary value problem with periodic boundary conditions for the equation KPZ is ill-posed and that such equation needs to be renormalized. The approach to discretization of KPZ equation perfomed by means of the use of hybrid and mixed finite elements with a domain decomposition procedure along with a pertinent mollification of the noise. The obtained solution is compared with the well known solution given by the Cole-Hopf transformation of the stochastic heat equation with multiplicative noise. We were able to verify that both solutions exhibit a good agreement, but there is a shift that grows as the support of the mollifier decreases. For the numerical aproximation of the stochastic heat equation we use a state-of-the-art numerical method for evaluating semilinear stochastic PDE , which in turn combine spectral techniques, Taylor's expantions and particular numerical treatment to the underlying noise. Furthermore, a state-of-the-art renormalization procedure introduced by Martin Hairer is used to renormalize KPZ equation that is validated with nontrivial numerical experiments.
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