Development of SVD Algorithm for Turbulence Tomography

摘要

To realize a nuclear fusion reactor，magnetic confinement of plasma has been investigated for a long time to produce high performance (or high density and temperature) plasma. It has been found that plasma turbulence degrades the property of plasma confinement. Therefore，plasma turbulence has been one of the important problems to be studied in the field of magnetically confinement fusion. In the present research of plasma turbulence，it is necessary to observe the turbulence not only in local region of plasma but also over the entire plasma because the plasma turbulence consists of the fluctuations of various scales from ion Lamor radius to the full size of plasma cross-section. Tomography is one of possible diagnostics to observe the entire plasma and to realizelocal and fine measurements of turbulence. We have constructed a prototype of such tomography system and installed on a linear cylindrical device，named Plasma Assenbly for Nonlinear Turbulence Analysis (PANTA) in Kyushu University. We are now developing a tomography algorithm using Singular Value Decomposition (SVD) in addition to the already developed ones，Maximum Likelihood-Expectation Method (ML-EM)，Algebraic Reconstruction Technique (ART)，Fourier-Bessel series and Cormack expansion. The paper reports simulation and experimental results of the tomography using SVD algorithm.
　　Chapter 2 describes the mathematical base of SVD method together with ML-EM method that is used for our prototype system as a standard. In tomography，the relation between the line-integrated values giand local emissionεi on i-th grid is expressed asg=hε,where h is a matrix. The tomography reconstruction can be performed if the inverse relation is obtained. The SVD is used to obtain the relation by factorizing the matrix h. More precisely saying，the emission can be obtained with a least square method to minimize the modifiedχ2-function Φ，with assuming Φ=∥g-hε∥2/M+?∥Cε∥2,whereγand∥∥represent the regularization parameter and the norm，respectively，and C is selected to be the Laplacian matrix here. The solution is expressed as ε=(hTh+MγCTC)-1hTg，where M is the number of data. In this process the SVD is used and the regularization parameterγneeds to be determined using Generalized Cross Validation (GCV).
　　Chapter 3 presents the comparison results between newly applied tomography algorithm，SVD and already developed one，ML-EM.In order to evaluate new algorithm，SVD method，the SVD method is applied on sinograms (or a set of line-integrate data) calculated from given emission images. The reconstruction is performed under the;nbsp;condition that the detector arrays are located at 16 different azimuthal angle positions with 101 channels for each array，respectively，thus，the number of observation channels is assumed to totally 1616. In comparison between SVD and MLEM，both algorithms have sufficient ability to reconstruct overall structure and sufficient spatial resolution to resolve a fine structure represented by a sharp peak. More detailed comparison is made in evaluating the residual inference error.
　　Chapter 4 presents the results of the application of SVD method on the real experimental data. A prototype system of turbulence tomography is explained with the linear cylindrical device on which the prototype is installed. The cylindrical plasma has the diameter of approximately 100 mm and the length of 4 m. In the tomography system，the plasma emission is measured with four light-guide arrays installed at azimuthal angle，0°，45°，90°，and 135°. Each light-guide is equipped with four sets of detector arrays，each of which has 33ch viewpoints lined up in every 5 mm，and covers the entire plasma. In the experiment the target is argon plasma，and the emissions of ArI and ArII are measured simultaneously. The tomography reconstruction of SVD method is successfully performed with optimizing the regularization parameterthe GCV.
　　Finally，summary is given in chapter 5. Tomography algorithm based on SVD method is being developed，and its application to an assumed emission data well reproduces the original image. Moreover，the tomographic reconstruction on the experimental data is performed in PANTA and the results are successfully obtained with optimizing the regularization parameterto minimize GCV. In comparison with MLEM method iteration process，the SVD method should be a faster algorithm in tomography reconstruction of temporal series of sonogram data. This is because the SVD just repeats the calculation using the same matrix obtained in advance，while the MLEM needs to apply iterative calculations to each slice of temporal series of experimental sinograms.

目录

Chapter 1:Introduction and Review

1.1.Plasma Science and Energy Problem

1.2.Progress toward the Realization of Nuclear Fusion Reactor

1.3.Review of Drift Wave and Zonal Flow

1.4.Motivation

Chapter 2:Review of Tomography Algorithms

2.1.Overview

2.2.Direct Least Square Fitting

2.3.Least square method and Regularization

2.4.Regularization by means of Penalty Function Method

2.5. Singular Value Decomposition (SVD)

2.6. Phillips-Tikhonov Regularization and Generalized Cross Validation (GCV) Criterion

2.7. Maximization Likelihood-Expectation Maximum (ML-EM)

Chapter 3:Test of Algorithms using Assumed Images

3.4. Consideration of Detector Alignment

3.5. Brief Summary of Tomography Algorithms and Detector Configuration

Chapter 4:Application of SVD Algorithm on Experimantal Results

4.1. Experimental Devices

4.2. Application of Developed Algorithms on Experiments

Chapter 5:Summary

参考文献

致谢

Academic Achievements

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