核心概念
This paper develops tree-based nonlinear reduced modeling approaches to efficiently approximate the solution manifold of parametric partial differential equations (PDEs). The proposed strategies rely on building a hierarchical library of linear or nonlinear approximation spaces, which can handle a wider range of PDEs compared to classical linear subspace-based methods.
摘要
The paper focuses on developing efficient model order reduction techniques for parametric PDEs. The key insights are:
Classical linear subspace-based model reduction methods are limited in their applicability, especially for transport-dominated or weakly coercive PDEs.
The authors propose a tree-based library approach that can use both linear and nonlinear approximation spaces to represent the solution manifold. This allows handling a broader class of PDEs.
Two tree-based library construction algorithms are presented:
The first algorithm assumes the parameter domain has a tensor product structure and performs dyadic splits along the parameter dimensions.
The second algorithm is more general and does not rely on the shape of the parameter domain, building the tree in a greedy fashion.
The tree-based representation of the library allows for a compressed storage and efficient evaluation of the reduced model, compared to a flat library approach.
Numerical experiments demonstrate the effectiveness of the proposed tree-based strategies in approximating the solution manifold for various types of PDEs, including diffusion, convection-diffusion, and transport-dominated problems.
統計資料
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引述
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