论文标题
具有抛物线方程的非侵入性还原基准（NIRB）两网格方法的错误估计值
Error estimate of the Non-Intrusive Reduced Basis (NIRB) two-grid method with parabolic equations
论文作者
论文摘要
经常提出减少基础方法（RBM）以近似参数问题解决方案。它们可用于计算大量参数值（例如参数拟合）的解决方案，并近似于新参数值的解决方案（例如，精度非常高的实时近似）。他们打算降低高保真度（HF）代码的计算成本。我们将重点关注非侵入性降低基础（NIRB）两网格方法。它的主要优点是它专门使用HF代码作为“黑框”，而不是其他需要修改代码的所谓侵入方法。当HF代码是已购买的商业代码时，这是非常方便的，就像行业中经常一样。这种方法的有效性依赖于将其分解为两个阶段，一个离线（如上所述的大多数RBM中经典）和一个在线。离线零件很耗时，但仅执行一次。相反，这种NIRB方法的特异性是，在在线部分期间，它仅在粗网格上解决参数问题，然后提高其精度。结果，它比HF评估便宜得多。该方法最初是针对具有有限元素的椭圆方程开发的，此后已扩展到有限的体积。在本文中，我们将NIRB两网格方法扩展到抛物线方程。我们以$ l^{\ infty}（0，t; h^1（ω））$的方式恢复最佳估计值。然后，我们在热方程和Brusselator问题上提出数值结果。
Reduced Basis Methods (RBMs) are frequently proposed to approximate parametric problem solutions. They can be used to calculate solutions for a large number of parameter values (e.g. for parameter fitting) as well as to approximate a solution for a new parameter value (e.g. real time approximation with a very high accuracy). They intend to reduce the computational costs of High Fidelity (HF) codes. We will focus on the Non-Intrusive Reduced Basis (NIRB) two-grid method. Its main advantage is that it uses the HF code exclusively as a "black-box," as opposed to other so-called intrusive methods that require code modification. This is very convenient when the HF code is a commercial one that has been purchased, as is frequently the case in the industry. The effectiveness of this method relies on its decomposition into two stages, one offline (classical in most RBMs as presented above) and one online. The offline part is time-consuming but it is only performed once. On the contrary, the specificity of this NIRB approach is that, during the online part, it solves the parametric problem on a coarse mesh only and then improves its precision. As a result, it is significantly less expensive than a HF evaluation. This method has been originally developed for elliptic equations with finite elements and has since been extended to finite volume. In this paper, we extend the NIRB two-grid method to parabolic equations. We recover optimal estimates in $L^{\infty}(0,T;H^1(Ω))$ using as a model problem, the heat equation. Then, we present numerical results on the heat equation and on the Brusselator problem.
