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浙江大学学报(工学版)
Journal of ZheJiang University (Engineering Science)  2022, Vol. 56 Issue (3): 613-621    DOI: 10.3785/j.issn.1008-973X.2022.03.021
    
Improved sparse grid collocation method for uncertainty quantification of EIT conductivity distribution
Ying LI1,2(),Guan-xiong WANG1,Wei YAN1,Ying-ge ZHAO2,Chong-lei MA1
1. State Key Laboratory of Reliability and Intelligence of Electrical Equipment, Hebei University of Technology, Tianjin 300130, China
2. Tianjin Key Laboratory of Bioelectromagnetic Technology and Intelligent Health, Hebei University of Technology, Tianjin 300130, China
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Abstract  

An improved sparse grid collocation method based on sensitivity analysis was proposed to quantify the uncertainty, aiming at the uncertainty problem of conductivity distribution in electrical impedance tomography (EIT) research. The four-layer concentric circular head model was taken for simulation, and the variance-based global sensitivity analysis method was used to analyze the model. The results of sensitivity analysis show that the conductivity changes of each layer have different effects on the output potential. Furthermore, the influence of the input variables of each dimension in the model on the output results were considered, the traditional sparse grid collocation method was improved. The input variables of each dimension were assigned with different accuracy levels in the improved method, the implicit expression of the EIT model was transformed into an explicit expression and the high-precision substitute model was constructed. Compared with the Monte Carlo (MC) method, polynomial chaos expansion (PCE) method and traditional sparse grid collocation method, the results show that the improved method can obtain the more accurate quantified results with less calculation cost. The simulation results were given to verify the efficiency of the proposed improved method.

Key words： electrical impedance tomography (EIT)      uncertainty quantification      Monte Carlo (MC) method      polynomial chaos expansion (PCE) method      sensitivity analysis method      sparse grid collocation method     
Received: 19 April 2021      Published: 29 March 2022
CLC:  TM 152  
Fund:  河北省自然科学基金资助项目（E2015202050）
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Ying LI
Guan-xiong WANG
Wei YAN
Ying-ge ZHAO
Chong-lei MA
Cite this article:

Ying LI,Guan-xiong WANG,Wei YAN,Ying-ge ZHAO,Chong-lei MA. Improved sparse grid collocation method for uncertainty quantification of EIT conductivity distribution. Journal of ZheJiang University (Engineering Science), 2022, 56(3): 613-621.

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https://www.zjujournals.com/eng/10.3785/j.issn.1008-973X.2022.03.021     OR     https://www.zjujournals.com/eng/Y2022/V56/I3/613


改进的稀疏网格配点法对EIT电导率分布的不确定性量化

针对电阻抗成像（EIT）研究中电导率分布的不确定性问题，提出基于灵敏度分析的改进稀疏网格配点法以量化不确定性. 以4层同心圆头模型为算例，采用基于方差的全局灵敏度分析法对其进行分析，发现各层电导率的变化对输出电位的影响程度各不相同. 考虑模型中各维输入变量对输出结果不同程度的影响，改进传统稀疏网格配点法. 改进方法对各维输入变量配置不同的精度水平，将EIT模型的隐式表达式转化为显式表达式，构造出高精度的替代模型. 与蒙特卡洛（MC）法、混沌多项式展开（PCE）法和传统稀疏网格配点法相比，改进方法能够以更少的计算成本获得较高精度的量化结果. 仿真结果验证了所提改进方法的高效性.


关键词： 电阻抗成像(EIT),  不确定性量化,  蒙特卡洛(MC)法,  混沌多项式展开(PCE)法,  灵敏度分析法,  稀疏网格配点法 
Fig.1 Four-layer concentric circular head model
部位 r/cm σ/(S·m?1)
大脑层 8.00 0.33
脑脊液层 8.50 1.00
颅骨层 9.20 0.42×10?2
头皮层 10.00 0.33
Tab.1 Radius values and conductivity values of each layer in four-layer concentric circular head model
Fig.2 Potential distribution diagram of each electrode on boundary
Fig.3 Convergence diagram of potential mean of electrode point 1 and sample number of MC method
部位 S1 S2 S3 S4
头皮层 0.723 989 0.577 689 0.561 464 0.625 605
颅骨层 0.271 172 0.413 325 0.430 142 0.369 696
脑脊液层 0.000 169 0.000 221 0.000 287 0.000 327
大脑层 0.002 858 0.004 513 0.004 987 0.003 958
Tab.2 Sensitivity index of conductivity of each layer of tissue
Fig.4 Probability density function distribution diagram of potential of electrode point 1
Fig.5 Probability density function distribution diagram of potential of electrode point 2
Fig.6 Probability density function distribution diagram of potential of electrode point 3
Fig.7 Probability density function distribution diagram of potential of electrode point 4
UQ方法 1号电极 2号电极 3号电极 4号电极 N
$\mu/{\rm{mV}}$ s $\mu/{\rm{mV}}$ s $\mu/{\rm{mV}}$ s $\mu/{\rm{mV}}$ s
MC 28.892 9 2.407 8 22.401 0 1.818 2 18.842 3 1.535 2 16.400 6 1.359 2 100 000
PCE 28.867 3 2.332 2 22.374 0 1.745 4 18.809 5 1.445 7 16.372 0 1.273 1 512
稀疏网格配点法，K = [4,4,4,4] 28.880 7 2.348 0 22.394 6 1.764 6 18.826 5 1.474 1 16.389 0 1.296 3 401
稀疏网格配点法，K = [5,5,5,5] 28.884 6 2.367 2 22.402 2 1.776 9 18.833 7 1.485 4 16.394 3 1.305 8 1 105
改进稀疏网格配点法，K = [3,3,4,4] 28.878 3 2.347 9 22.390 6 1.762 7 18.824 2 1.470 2 16.386 1 1.295 6 385
改进稀疏网格配点法，K = [3,2,4,5] 28.882 2 2.361 2 22.395 6 1.765 5 18.831 7 1.482 2 16.393 2 1.301 0 845
Tab.3 Statistical information of each electrode potential under different uncertainty quantification methods
UQ方法 ${\varepsilon _1}$ ${\varepsilon _2}$ ${\varepsilon _3}$
稀疏网格配点法，K = [4,4,4,4] 0.012 2 0.059 8 0.062 3
稀疏网格配点法，K = [5,5,5,5] 0.008 3 0.040 6 0.016 0
改进稀疏网格配点法，K = [3,3,4,4] 0.014 6 0.059 9 0.066 3
改进稀疏网格配点法，K = [3,2,4,5] 0.010 7 0.046 6 0.020 6
PCE 0.025 6 0.075 6 0.079 1
Tab.4 Error values of uncertainty quantification methods
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