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CONVERGENCE ESTIMATES OF POD-GALERKIN METHODS FOR PARABOLIC PROBLEMS

机译：抛物面问题Pod-Galerkin方法的收敛估计

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Proper orthogonal decomposition (POD) is a Galerkin method which has been introduced in a fluids mechanics context. It is also known as Karhunen-Loeve decomposition and principal component analysis. The idea of POD consists in using a priori known information on the solution u of PDE, for example snapshots u{sub}i - u(t{sub}i), to determine a set of functions which axe the eigenfunctions of an Hilbert-Schmidt operator. This basis can be used to solve the PDE with a smaller amount of computations. Convergence estimates have been proved recently in the parabolic case starting from a particular discretization scheme [7]. Moreover it has been proved that the method converges independently from the scheme [6], We consider the case of a linear parabolic equation. We give a first convergence estimate in a case where u is regular. However classical POD does not look satisfactory and an improvement consists in considering a POD which takes into account the derivative of u. We will also present some insights into the control of the approximation by introducing what will be called a good order of approximation.
机译：适当的正交分解（POD）是一种在流体力学上下文中引入的Galerkin方法。它也被称为Karhunen-Loeve分解和主成分分析。 POD的概念包括在PDE的解决方案U上使用先验的已知信息，例如快照U {Sub} I - U（T {Sub} I），以确定一组函数，该函数X X XIGENT-施密特运营商。此基础可用于解决具有较小计算量的PDE。最近从特定离散化方案开始的抛物面案例中已经证明了收敛估计[7]。此外，已经证明了该方法从方案中独立地收敛[6]，我们考虑线性抛物线方程的情况。在您是常规的情况下，我们提供第一个收敛估计。然而，古典豆荚看起来不令人满意，并且在考虑考虑到u的衍生物的POD中，改善包括。我们还将通过引入将被称为良好的近似顺序的近似来展示对近似的控制来控制。

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