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.
翻译:在本文中,我们讨论了一种基于Laplace变异应用的预测线性进化PDE(MOR)的预测模型减少线性进化PDE(MOR)方法。这种方法的主要优点是,与时间阶方法不同,如龙格-库塔混凝土,拉普变异允许在特定瞬间直接计算解决办法,这可以通过通过适当的梯度公式对与反拉普特变异相关的等离子元集成部分进行对等。从某些传统的摩尔方法来看,这决定了削减阶段的重大改进,如基于古典正态正对流分解(POD)的方法,因为用于分解的矢量数量急剧减少,因为它并不包含通过时间阶法在集成电网上产生的所有中间解决办法。我们用融资产生的一些示例parpolic PDE方法展示了该方法的有效性,并且提供了一些证据,表明在对线性对方对等方公式应用时,我们建议的方法并不因单数值的缓慢衰败问题,而影响正在形成时的磁度。