Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations
Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations
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DOI：
10.1007/978-981-13-7669-6
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
M. Nakao;M. Plum;Yoshitaka Watanabe
中科院分区：
文献类型：
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作者：
M. Nakao;M. Plum;Yoshitaka Watanabe
这本书是作者们多年来进行的研究的汇编。所提出的方法的总体目标是获得信息，特别是存在性声明和明确的后验界，真正的解决方案偏微分方程问题，使用分析参数和数值计算相结合。数值部分包括近似解的计算，借助于近似解的存在性陈述和精确的误差界，通过分析定点参数导出真解。这种方法论基础引起了对这种方法的两种略有不同的观点：一方面，它是“经典”数值方法的推广，因为它为这些方法和通常的渐近误差估计增加了严格和明确的误差界。在这种观点下，“验证的数值计算”或“数值验证”的名称经常出现在相应的文献中。另一方面，该方法也可以被视为分析的一部分，因为它提供了偏微分方程问题的存在性结果，包括纯分析方法没有成功的各种例子。考虑到这一点，“计算机辅助证明”这个名字经常被用来描述这种方法。
This book is a compilation of research studies the authors undertook over many years. The overall goal of the presented methods is to gain information, in particular existence statements and explicit a posteriori bounds, for true solutions to partial differential equation problems, using a combination of analytical arguments and numerical computations. The numerical part includes the computation of an approximate solution, with the aid of which an existence statement and explicit rigorous error bounds for a true solution are derived by analytical fixed-point arguments.This methodological basis gives rise to two slightly different views of such an approach: on one hand, it is an extension of “classical” numerical methods, since it adds rigorous and explicit error bounds to these methods and to the usual asymptotic error estimates. Under this view, the name “verified numerical computation” or “numerical verification” often appears in the corresponding literature. On the other hand, the approach can also be viewed as a part of an analysis, since it provides existence results for partial differential equation problems, including various examples where purely analytical methods have not been successful. With this view in mind, the name “computer-assisted proofs” has often been given to such methods.