Reduced Basis Methods for Partial Differential Equations: An Introduction
Reduced Basis Methods for Partial Differential Equations: An Introduction
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DOI：
10.1007/978-3-319-15431-2
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发表时间：
2015-09
期刊：
影响因子：
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通讯作者：
A. Quarteroni;A. Manzoni;Federico Negri
中科院分区：
文献类型：
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作者：
A. Quarteroni;A. Manzoni;Federico Negri
本书提供了降基（RB）方法的基本介绍，用于解决工程和应用科学中涉及偏微分方程（PDE）重复求解的问题，例如取决于多个参数的 PDE 和 PDE 约束优化。本书介绍了 RB 方法的一般数学公式，分析了它们的基本理论特性，讨论了相关的算法和实现方面，并强调了它们内置的代数和几何结构。更具体地说，作者讨论了使用贪婪算法和适当的正交分解技术构建精确 RB 空间的替代策略，研究了它们的近似特性并分析了旨在降低计算复杂性的离线-在线分解策略。此外，他们还进行先验和后验误差分析。通过在线性和非线性偏微分方程的背景下使用具有应用兴趣的代表性示例，整个数学演示变得更加刺激。此外，包含许多伪代码使读者能够轻松实现全文中说明的算法。这本书非常适合高年级本科生，以及更广泛的对科学计算感兴趣的人。所有这些伪代码实际上都是在 MATLAB 包中实现的，该包可在 https://github 上免费获取。 com/redbkit
This book provides a basic introduction to reduced basis (RB) methods for problems involving the repeated solution of partial differential equations (PDEs) arising from engineering and applied sciences, such as PDEs depending on several parameters and PDE-constrained optimization. The book presents a general mathematical formulation of RB methods, analyzes their fundamental theoretical properties, discusses the related algorithmic and implementation aspects, and highlights their built-in algebraic and geometric structures. More specifically, the authors discuss alternative strategies for constructing accurate RB spaces using greedy algorithms and proper orthogonal decomposition techniques, investigate their approximation properties and analyze offline-online decomposition strategies aimed at the reduction of computational complexity. Furthermore, they carry out both a priori and a posteriori error analysis. The whole mathematical presentation is made more stimulating by the use of representative examples of applicative interest in the context of both linear and nonlinear PDEs. Moreover, the inclusion of many pseudocodes allows the reader to easily implement the algorithms illustrated throughout the text. The book will be ideal for upper undergraduate students and, more generally, people interested in scientific computing. All these pseudocodes are in fact implemented in a MATLAB package that is freely available at https://github. com/redbkit