论文标题
参数PDE的轮廓积分方法的模型订单降低
Model order reduction in contour integral methods for parametric PDEs
论文作者
论文摘要
在本文中，我们讨论了一类参数线性进化PDE的投影模型订购方法（MOR）方法，该方法基于拉普拉斯变换的应用。这种方法的主要优点在于以下事实：与时间步进方法不同，像runge-kutta集成剂一样，拉普拉斯变换允许在给定的瞬间直接计算解决方案，这可以通过合适的quadrature公式近似与逆laplace变换相关的轮廓积分来完成。就某种经典的方法学而言，这决定了还原阶段的显着改善（例如基于经典正交分解（POD）的阶段），因为分解适用的向量的数量大大减少，因为它不包含所有沿着时间步进方法沿着集成网格产生的中间求解方法。我们通过财务引起的一些说明性抛物线PDE显示了该方法的有效性，还提供了一些证据表明，当我们提出的方法应用于线性对流方程时，不会遇到奇异值的缓慢衰减的问题，而奇异值的慢衰减会影响时间阶梯式方法，从而导致cauchy问题出于空间离散而产生的Cauchy问题。
In this paper we discuss a projection model order reduction (MOR) method for a class of parametric linear evolution PDEs, which is based on the application of the Laplace transform. The main advantage of this approach consists in the fact that, differently from time stepping methods, like Runge-Kutta integrators, the Laplace transform allows to compute the solution directly at a given instant, which can be done by approximating the contour integral associated to the inverse Laplace transform by a suitable quadrature formula. In terms of some classical MOR methodology, this determines a significant improvement in the reduction phase - like the one based on the classical proper orthogonal decomposition (POD) - since the number of vectors to which the decomposition applies is drastically reduced as it does not contain all intermediate solutions generated along an integration grid by a time stepping method. We show the effectiveness of the method by some illustrative parabolic PDEs arising from finance and also provide some evidence that the method we propose, when applied to a linear advection equation, does not suffer the problem of slow decay of singular values which instead affects time stepping methods for the numerical approximation of the Cauchy problem arising from space discretization.
