Academic Journal
Efficient direct sampling type methods for some inverse problems
| Title: | Efficient direct sampling type methods for some inverse problems |
|---|---|
| Contributors: | Han, Fuqun (author.), Zou, Jun , 1962- (thesis advisor.), Chinese University of Hong Kong Graduate School. Division of Mathematics. (degree granting institution.) |
| Publication Year: | 2020 |
| Collection: | The Chinese University of Hong Kong: CUHK Digital Repository / 香港中文大學數碼典藏 |
| Subject Terms: | Inverse problems (Differential equations)--Numerical solutions, Sampling (Statistics), QA378.5 .H36 2020eb |
| Description: | M.Phil. ; In this thesis, we propose two novel direct sampling type methods for highly ill-posed inverse problems. Direct sampling method (DSM) is one type of numerical method leverages on an important almost orthogonality property between the fundamental solution of the forward problems and a proper family of probing functions under certain Sobolev duality products. The main feature of the DSM is that it can provide a satisfactory reconstruction result under noisy and extremely limited boundary measurements with an index function that can be computed in a fast, stable, and highly parallelable manner. ; The first part of the thesis introduces a new DSM for an elliptic inverse problem that aims to recover inhomogeneities of different nature which is a highly unstable and ill-posed inverse problem. A main highlight of the new method is that it designs two distinct sets of probing functions, i.e., the monopole and dipole probing functions, which help decouple the respective signals coming from the monopole-type and dipole-type sources located in the sampling domain. ; In the second part of the thesis, we present a new DSM for inverting the Radon transform under noisy and inadequate measurement data. The novel DSM considers the reconstruction with fractional order Sobolev duality products which enhances the robustness of reconstruction against noise. The proposed method can be regarded as a weighted back projection with a special weighting function in the frequency domain and hence can be computed in a very fast manner. ; Numerical experiments are conducted for each proposed method to demonstrate their accuracy, efficiency, and stability. ; 在本論文中，我們將會提出兩種新的直接取樣類的數值方法以解決兩個高度不適定的反問題。直接取樣法是一類解決反問題的數值方法，這一類方法是建立在正問題的基本解和一組特殊的探針函數在特定的索博列夫空間內的幾乎垂直性質之上的。直接取樣法的主要特點是即使只有非常少的測量值，它也可以提供一個大致滿意的解，並且這個解可以非常快速，穩定，並行的被得到。 ; 在本論文的第一部分，我們考慮在一個橢圓型偏微分方程下，去同時重建物體內不均勻介質的兩個不同的物理性質。這一個反問題在數學上常是不穩定並且不適定的。我們創新的提出並且驗證了一類名為獨立幾乎垂直的性質。通過這一性質，我們新建了兩組截然不同的探針函數分辨不均勻介質的兩個不同的物理性質。 ... |
| Document Type: | text |
| File Description: | electronic resource; remote; 1 online resource (ix, 81 leaves) : illustrations (chiefly color); computer; online resource |
| Language: | English Chinese |
| Availability: | https://julac.hosted.exlibrisgroup.com/primo-explore/search?query=addsrcrid,exact,991040013774503407,AND&tab=default_tab&search_scope=All&vid=CUHK&mode=advanced&lang=en_US https://repository.lib.cuhk.edu.hk/en/item/cuhk-2628012 |
| Rights: | Use of this resource is governed by the terms and conditions of the Creative Commons "Attribution-NonCommercial-NoDerivatives 4.0 International" License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
| Accession Number: | edsbas.41C09FAA |
| Database: | BASE |
| Description not available. |