Inverse Obstacle Scattering with Non-Over-Determined Scattering Data
暫譯: 非過度確定散射數據的逆障礙散射

Ramm, Alexander G., Krantz, Steven G.

Inverse Obstacle Scattering with Non-Over-Determined Scattering Data
  • 出版商: Morgan & Claypool
  • 出版日期: 2019-06-12
  • 售價: $2,090
  • 貴賓價: 9.5 折 $1,986
  • 語言: 英文
  • 頁數: 69
  • 裝訂: Hardcover - also called cloth, retail trade, or trade
  • ISBN: 1681735903
  • ISBN-13: 9781681735900

    • 海外代購書籍(需單獨結帳)

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商品描述

The inverse obstacle scattering problem consists of finding the unknown surface of a body (obstacle) from the scattering ��(��;��;��), where ��(��;��;��) is the scattering amplitude, ��;�� �� �� is the direction of the scattered, incident wave, respectively, �� is the unit sphere in the ℝ3 and k > 0 is the modulus of the wave vector.

The scattering data is called non-over-determined if its dimensionality is the same as the one of the unknown object. By the dimensionality one understands the minimal number of variables of a function describing the data or an object. In an inverse obstacle scattering problem this number is 2, and an example of non-over-determined data is ��(��): = ��(��;��₀;��₀). By sub-index 0 a fixed value of a variable is denoted.

It is proved in this book that the data ��(��), known for all �� in an open subset of �� , determines uniquely the surface �� and the boundary condition on ��. This condition can be the Dirichlet, or the Neumann, or the impedance type.

The above uniqueness theorem is of principal importance because the non-over-determined data are the minimal data determining uniquely the unknown ��. There were no such results in the literature, therefore the need for this book arose. This book contains a self-contained proof of the existence and uniqueness of the scattering solution for rough surfaces.

商品描述(中文翻譯)

反向障礙散射問題的目標是從散射振幅 \( S(\theta; k) \) 中找出物體（障礙物）的未知表面，其中 \( S(\theta; k) \) 是散射振幅，\( \theta \) 是散射波和入射波的方向，分別為 \( \theta \) 和 \( k \) 是波向量的模。\( S \) 是 \( \mathbb{R}^3 \) 中的單位球，且 \( k > 0 \)。

如果散射數據的維度與未知物體的維度相同，則稱該數據為非過度確定的。這裡的維度是指描述數據或物體的函數的最小變數數量。在反向障礙散射問題中，這個數字是 2，而非過度確定數據的一個例子是 \( S(\theta) := S(\theta; k_0) \)。下標 0 表示變數的固定值。

本書證明了在 \( \mathbb{S} \) 的開子集內，對於所有 \( \theta \) 已知的數據 \( S(\theta) \) 唯一確定了表面 \( \Gamma \) 及其邊界條件。這個條件可以是 Dirichlet、Neumann 或阻抗類型。

上述唯一性定理具有重要意義，因為非過度確定數據是唯一確定未知 \( \Gamma \) 的最小數據。文獻中並沒有這樣的結果，因此本書的需求應運而生。本書包含了對粗糙表面散射解的存在性和唯一性的自足證明。

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