The Minimax Estimation Method for a Class of Inverse Helmholtz Transmission Problems

Oleksandr Nakonechnyi; Yuri Podlipenko; Yury Shestopalov

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The Minimax Estimation Method for a Class of Inverse Helmholtz Transmission Problems

机译：一类的极小极大估计方法逆亥姆霍兹传输问题

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摘要

We present complete mathematical statements and perform detailed investigations of the minimax estimation problems of unknown data for the Helmholtz transmission problems from indirect noisy observations of their solutions. We construct optimal, in certain sense, estimates, which are called minimax mean-square estimates, of the values of linear functionals from unknown data. It is established that when unknown data and correlation functions of errors in observations belong to special sets, the minimax mean square estimates are expressed via solutions to certain transmission problems for systems of Helmholtz equations. We prove that these systems are uniquely solvable. Several possible generalizations of the techniques and results are proposed including applications to the problems with incomplete data and pointwise observations.

机译：我们提供完整的数学陈述和执行极大极小的详细调查估计未知数据的问题亥姆霍兹从间接传输问题嘈杂的观察他们的解决方案。构造最优,在一定意义上,估计,这被称为极小极大均方估计,从未知较之线性泛函更具普遍意义的价值数据。和相关函数的错误观察属于特殊集,极大极小通过解决方案均方估计表达为系统的某些传输问题亥姆霍兹方程。唯一可以解决的。概括的技术和成果提出了包括应用程序的问题不完整的数据和逐点的观测。

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《Minimax theory and its applications》 |2019年第2期|305-328|共24页
作者
Oleksandr Nakonechnyi; Yuri Podlipenko; Yury Shestopalov;
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作者单位

National Taras Shevchenko University, Kiev, Ukraine;

University of G?vle,G?vle,Sweden;

T. Shevchenko University, Kyiv, Ukraine;

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原文格式 PDF
正文语种英语
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关键词
minimax estimation; Helmholtz equations;

机译：极大极小估计,亥姆霍兹方程;

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The Minimax Estimation Method for a Class of Inverse Helmholtz Transmission Problems

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