非线性问题中的经验插值与降阶方法同时求解
Simultaneous Empirical Interpolation and Reduced Basis method for non-linear problems
AI总结：
本文提出SER算法，通过同时应用经验插值和降阶方法，显著降低计算成本，仅需N+1次有限元求解，提升非线性问题求解效率。
摘要：
本文聚焦于非线性非仿射参数化偏微分方程的降阶方法，其中传统的仿射分解无法应用。为解决此问题，通常先使用经验插值方法（EIM）构建非线性项近似，但计算成本高。本文提出SER算法，通过同时应用EIM和降阶方法，显著降低计算成本，仅需N+1次有限元求解，其中N为降阶近似维度。本文首先回顾EIM和降阶方法在非线性问题中的应用，识别主要问题，然后介绍SER及其变种，并在[9]提出的基准测试中展示其性能。
英文摘要：
In this paper, we focus on the reduced basis methodology in the context of non-linear non-affinely parametrized partial differential equations in which affine decomposition necessary for the reduced basis methodology are not obtained [4, 3]. To deal with this issue, it is now standard to apply the EIM methodology [8, 9] before deploying the Reduced Basis (RB) methodology. However the computational cost is generally huge as it requires many finite element solves, hence making it inefficient, to build the EIM approximation of the non-linear terms [9, 1]. We propose a simultaneous EIM Reduced basis algorithm, named SER, that provides a huge computational gain and requires as little as N + 1 finite element solves where N is the dimension of the RB approximation. The paper is organized as follows: we first review the EIM and RB methodologies applied to non-linear problems and identify the main issue, then we present SER and some variants and finally illustrates its performances in a benchmark proposed in [9].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
