1 Introduction

The inverse transport problem,corresponding to the problem that acquires the unknown information of an object by analyzing the detected radiation fields,is always a strong challenge in nonproliferation,arms control verification,and international security[1,2].Thus,great efforts have been dedicated to investigate the inverse transport problem for various application circumstances since the 1960s.It is found that the inverse transport problems can be solved through an optimization approach by searching the unknown system parameters that minimize a cost difference between the quantities of interest measured from a system and the quantities of interest calculated with a direct transport model using a set of postulated parameters[2].

Several optimization methods have been applied to solve the inverse transport problems,including the gradient-based techniques,e.g.,Levenberg–Marquardt(LM)method[3,4]and Schwinger method[5],and the derivative-free techniques,e.g.,differential evolution (DE)method[6,7]and mesh adaptive direct search(MADs)[8].Based on gradient-based techniques,a few inverse radiation transport models have been developed to identify the unknown information of one-dimensional source/shield systems,e.g.,the locations of interfaces between material layers,source composition,shield material identi fication,and material mass density,as well as the combinations of these unknown properties[3–6,9–11].It is found that the gradient-based inverse models can be applied to solve the inverse problems containing relatively few unknowns(approximately smaller than 4)with high success as long as the proper initial guesses for the unknown parameters are used.However,in the problems with several unknown parameters,the gradient-based inverse model is heavily dependent on the accuracy of the initial guess for the unknown parameters and can often fall into the local minima when random initial guesses are used(corresponding to the cases in which there were not enough prior information about the unknown parameter values)[10].The requirement of relatively accurate initial guesses of unknowns can also restrict the application of such gradientbased inverse model to a certain extent.

On the other hand,with the development of the computer and optimization algorithm,several inverse radiation models based on derivative-free techniques have been developed to solve the inverse problem in both spherical and cylindrical geometries[6–8,12].It is found that the derivative-free inverse solvers can be applied for the source systems containing unknowns of more than 4 or having more complicated geometric properties with a much higher success rate than the ones based on gradient-based techniques.Furthermore,such derivative-free inverse calculations usually do not depend on the initial guesses of the unknowns[8,13],indicating that they are more proper for the inverse problems with more unknowns or less prior information.However,in previous studies,it is also pointed out that further investigations of algorithm improvement,exploration of technical feasibility and validation,as well as experimental validation and testing for the derivative-free inverse radiation transport models are always needed.

Recently,based on an enhanced differential evolution algorithm with global and local neighbourhoods(DEGL)[14,15],which is suggested to perform better than the standard differential evolution algorithm adopted in previous inverse solver development[6–8,12],an inverse radiation transport model(IRT-DEGL)has been developed forestimating the unknown shielding layerthicknesses[16].By conducting a series of experiments with a multi-shield point source152Eu and a 60%relative ef ficiency HPGe spectrometer,the numerical tests are carried out for IRT-DEGL model.It is found that in comparison with the traditional gamma-ray absorption method the IRTDEGL model can provide a much more accurate shielding layer thicknesses and may have great potential for complicated inverse problems.

It is known that one of the major advantages for inverse solvers based on the differential evolution algorithm is that their inverse results in principle do not depend on the initial guesses of unknowns,and thus,the present IRT-DEGL model is extended for estimating the unknown layer thicknesses with random initial guesses of unknowns,and the in fluence on the iteration process and the calculation amount is discussed in detail.Furthermore,in real applications supporting nonproliferation, arms control veri fication,and international security,one often has to deal with the inverse problems needed to identify the material information or part of the geometric and material information that the traditional analytical algorithm and the basic iteration method can not provide these accuracy results easily.As an elementary attempt,the IRT-DEGL model is further attended to estimate the unknown material mass densities as well as the combination of unknown layer thickness and mass densities of multi-shielding layers.Using the detected gamma-ray spectra,the numerical tests are implemented and the main in fluence factors for calculation results are analyzed.

2 Theoretical framework of IRT-DEGL model

Inverse procedure of IRT-DEGL model starts with the construction of a direct transport model according to the real detection circumstance and usually containing several unknown parameters of source/shield system to be determined.In order to complicate the direct transport model,a set of vectors,ui（i=1，...，P）,where each vector contains a set of postulated values of the unknown parameters,are generated.The number of the unknown parameters determines the dimension of each vector,ui,and P represents the total number of the vectors,i.e.,the population size.The fitness of each population vector is determined by a cost function,f（u）,between the measured gamma-ray spectrum and the calculated value using the parameters of the population vector,i.e.,

The IRT-DEGL model employs a generational process with mutation,crossover,and selection operations for optimization.The population vectors move from generation h to generation h+1 by first creating trial vectors with mutation operation.One trial vector vi（i=1，...，P）is created for each member of the population according to

After P child vector are created,the selection operation is implemented to choose the vectors in the next generation by a direct competition between the ith parent vectors and the ith child vectors,

where Rgis the measured data,Rg（ui）is the calculated value using postulated parameter set ui,and the sum over g runs over the full gamma-ray spectrum or the strong emission lines.In an inverse problem,one needs to seek to find the population vector with the global minimum,f（ui）.

with the global neighbourhood mutation,g（ih）,and local neighbourhood mutation,

The IRT-DEGL calculation converges after 27 generation evolution,and the final optimal thicknesses of the Cu layer(0.319 cm)and Al layer(0.551 cm)are obtained.The normalized counts calculated with the inverse result for the seven emission lines are shown in comparison with the experimental values in Table 1 as well as the corresponding cost difference,f（ui）.The calculated and experimental normalized counts are the same up to 0.01 with the relativistic difference smaller than 2.126%,which indicates that adopting the final inverse results of the strength of the seven emission lines of the simulated gamma-ray spectrum can reproduce the detected values well.

In Table 2,the thicknesses of the Cu and Al layers calculated by the IRT-DEGL model with and without inputting initial guesses are shown in comparison with the actualvalues.One can see thatboth IRT-DEGLcalculations converge quickly and provide the inverse results within the relative difference from the actual value smaller than 6 and 4%,respectively.The major reason for the deviation between two results comes from the convergence criterion that the change of cost function,f（ui）,is smaller than 0.1%for five generations.When an inverse calculation converges to the area around global minimum and the cost function,f（u）,varies slightly in this area,such criterion can stop the inverse calculation in a few generations and provide a value close to the global minimum.Therefore,using such convergence criterion,both the IRTDEGL calculations with and without the initial guesses could not provide the correct global minimum value,but converge to the global minimum area along the different paths and finally obtain the different results around the global minimum point.Although this criterion is rough and can only provide an approximate global minimum value,for most applications these inverse result with such degree of accuracy is acceptable.Of course,if one needs to obtain the inverse result closer to the global minimum,the more precision converge criterion should be to adopt,e.g.,using criterion where the cost function,f（uopt.）,of the best population vector is less than a small converge value,or allowing the cost function,f（u）,to change a little for large generation steps,or using the combination of them.

根据情景C、D，把种植业结构调整与高效灌溉技术结合起来设定情景E：2020年粮、经、草种植比例为41. 3∶45. 3∶13. 4，微灌面积达70%，2025年粮、经、草比例达41∶40∶19，微灌面积达100%。

After P trial vectors are created,a crossover operation is used to create the child vectors yi（i=1，...，P）with the following rule,

动态规划算法的基本思想：人员在原图像上选择特定的点作为初始点和终止点，对原始图像进行转变从而得到初始价格阵，目标边缘部分对应位置的价格比较低，其他的价格较高，从初始价格阵和规定的初始点计算积聚价格阵，最终由终止点方向反向追踪到初始点，最终获得所需要的范围详情。以往的研究结论中运用DP算法对于不同的医学超声图像进行分割并且得到了相对令人满意的分割结果。文章对着种算法在医学上的超生图像进行实际的分割操作，结果如图一所示。

农村教师由于工作量大，需要完成的任务指标多，对于学生的作业方面重视程度不够。同时，教师对于学生的情况没有很好地把握，布置作业有时急于应付形式，很少考虑到时效性。此外，数学教师参考资料缺乏，不关注学生之间的差异，所有练习采用“一刀切”模式。

我国现阶段正处于社会主义现代化的改革开放进程中，与世界各国之间的联系也变得越来越精密，这就使得跨文化交际背景下的旅游翻译活动变得越来越重要。作为我国和世界联系过程中的重要活动，在高校的教育课程中应当给予一定的重视。

where rjis a random number in[0,1],and CR is a real number in[0,1]denoting the crossover probability and remains the constant throughout the optimization process.i and j denote the ith child or trial or parent vector and jth unknown parameter,respectively.In such a way,the ith child vector could have characteristics of the ith trial vector and the ith parent.

that is,the ith population vector of generation h+1 is chosen as the better fit(lower cost function value)between.Furthermore,according to the self-adaptation scheme,the weight factor,ω,in Eq.(4)for the next generation is updated following the selection operation in Eq.(6),that is,for other cases.

The population of the first generation are created according to the initial guesses set by hand or randomly.Then,the generational process proceeds until the converge criterion is achieved,e.g.,either the cost function,f（u）,of the best population vector is less than a set converge value or until the population has little diversity in the unknown parameter values.And finally,the best population vector is thought to be the inverse results.

3 Results and discussion

It is known that one of the advantages for the derivativefree inverse transport model is that its calculation results do not depend on the initial guesses of unknowns,and therefore,one can,in principle,create the initial values randomly or just according to the few pieces of prior information.Although the proper initial guesses from either enough prior information or numerical results from the traditional analytical method may help the inverse calculation converge,sometimes one can not get enough information in one detection,or the source system is so complicated that the analytical method is not suit or can not be solved easily.Thus,for more general inverse cases,it is necessary to directly generate the initial values for the unknown parameters randomly according to the prior information already obtained instead of setting an accurate initial guess.Therefore,in comparison with the inverse investigations in Ref.[16]where the relative proper initial guesses of the unknown layer thicknesses are input to create the population of the first generation,the IRT-DEGL model in this paper generates the first generation randomly just under the constriction of prior information.

Fig.1(Color online)Gamma-ray spectrum detected for the inverse study

Following the typical experiment conducted in Ref.[16],where two shielding layers of copper and aluminium with thicknesses of 0.308 and 0.57 cm,respectively,locate between a point source of152Eu and a 60%relative efficiency HPGe spectrometer,the gamma-ray spectrum detected for the inverse study is shown in Fig.1.Considering the cases where the shielding layers are encapsulated as a whole,it is not easy to distinguish the layer thicknesses.The thicknesses of copper layer and aluminium layer are both assumed to be unknown.The densities of the copper layer and aluminium layer are 8.96 and 2.7 g/cm3,respectively,and the material in the layer is assumed to be homogeneous.The two layers are close to each other,and the copper layer is on the side close to the HPGe detector.The distance between the radiation source and the HPGe detector and the distance between the HPGe detector and one surface of copper layer are already known.Referring to Ref.[16],the seven strongest emission lines of152Eu(122,344,779,964,1086,1112,and 1408 keV)of detected gamma-ray spectrum are used for inverse calculation,and the simulated gamma-ray spectrum during the inverse calculation is provided by a Monte Carlo code MCNP5[17].For all inverse investigations in the present work,the statistical errors for both experimental and simulated full energy peak areas for these emission lines are smaller than 1%.

represents the best vector in the entire population at generation h and random integral numbers a，b ∈ [1，P]with a/=b/=i.represents the best vector in the neighbourhood ofand random integral numbers c，d ∈[i-k，i+k]with c/=d/=i,where the neighbourhood of radius k is a nonzero integer from 0 to （P-1）/2.α and β are the scaling factors.In comparison with Ref.[16],the weight factor,ω,controlling the balance between the exploration and exploitation capability adopted here is calculated in a self-adaptive way[14],

李云[33]重点阐述碑学与帖学书风的风格差异，从其概念及特点进行分析，认为古人对“南帖”、“北碑”、“帖派”、“碑派”的划分及其艺术特色的评价，尽管有时并不十分科学，但从其总体风貌上来把握，还是很有道理的。相对来说，南派重优美，北派尚壮美，重帖者，偏好阴柔之美；重碑者，侧于阳刚之态。“帖”重“书卷气”，“碑”重“金石气”。书法中的“书卷气”是一种性灵、气质、情趣的流露。“金石气”相对于书卷气来说，所倡导的是苍茫、浑厚、朴拙的审美范畴。

where is the weight factor associated with the best vectorand the value of is restricted in the range[0.05,0.95].

Table 1 Calculated normalized counts obtained with the thicknesses of copper and aluminium layers from IRT-DEGL model in comparison with the experimental values.The cost function value for each emission line,f（ui）,is also shown

Line energy Normalized counts Cost function(keV) Cal. Exp. f（ui）(%)122 0.264 0.263 0.576 344 0.283 0.289 1.846 779 0.095 0.093 1.459 964 0.098 0.096 1.769 1086 0.063 0.063 0.090 1112 0.084 0.082 2.126 1408 0.112 0.113 0.924

Table 2 Estimated thicknesses of copper and aluminium layers obtained by the IRT-DEGL model with and without inputting initial guesses in comparison with the actual values.The inverse results with initial guesses are obtained from Ref.[16]

Unknown Thickness_Cu(cm) Thickness_Al(cm)Initialization 0.508– 0.700–Generation 19 27 19 27 IRT-DEGL 0.316 0.319 0.534 0.551 Actual value 0.308 0.308 0.57 0.57 Relative difference(%) 2.60 3.57 6.32 3.33

Furthermore,in Table 2 one can also see that with proper initial guesses,i.e.,0.508 cm for copper layer(corresponding to the relative difference of 64.9%)and 0.700 cm for aluminium layer(corresponding to the relative difference of 22.8%),the IRT-DEGL calculation converges after 19 generation evolutions,while for the case without an initial guess,where the initial population is generated randomly with the prior information introduced above,the inverse calculation converges after 27 generation evolutions and provides relatively accurate results as the inverse values obtained with proper initial guesses.It indicates that the IRT-DEGL model can be applied for the inverse problems with and withoutenough prior information,and the computing time of the inverse calculation can be saved greatly when the proper initial guesses are used.

我国《国家安全法》第52条规定：“国家安全机关、公安机关、有关军事机关根据职责分工，依法搜集涉及国家安全的情报信息。”特别在第四章第二节“情报信息”第51条～第54条中，分别规定了“情报工作制度”“各部门搜集上报情报信息职责”“情报信息工作运用现代科技手段和加强研判分析”“情报信息的报送要求”等内容。“该法的立法模式呈现‘原始型’（即分散式立法）向‘混合型Ⅰ’（即分散式＋专门式立法）再向‘混合型Ⅱ’（即分散式＋专门式＋综合式立法）发展的趋势”。《国家安全法》是我国国家安全领域的基础性法律，也是统筹、引领国家安全领域立法工作的综合性法律之一，该法涵盖了国家安全多个领域。

In addition to identifying the unknown shielding layer thicknesses,the IRT-DEGL model in the present work is further applied for the inverse problems of estimating the unknown matter mass densities of shielding layers.The detected gamma-ray spectrum for identifying unknown shielding layer thicknesses is used for the present inverse study of the densities.Assuming that the geometric parameters of the source system are already known,three test cases for mass densities of shielding layers,including only copper density(8.96 g/cm3)or aluminium density(2.7 g/cm3)unknowns,as well as both copper and aluminium densities unknown,are considered.

2.孩子们能感受到对联的无穷魅力，柯添恺、廖俊茗、陈妍等多个孩子着迷于对联创作，每天一两副，乐此不疲。柯添恺妈妈说，孩子玩耍时、吃饭时、睡觉时都想着对对子，欲罢不能。

The IRT-DEGL calculations are performed for the three inverse cases with the detected full energy peak areas of the seven strongest emission lines of152Eu(122,344,779,964,1086,1112,and 1408 keV).The inverse results are shown in Table 3 with the actual values for comparison.It can be seen that for three test cases the IRT-DEGL calculations can provide the relatively accurate results with the differences from the actual values smaller than 9%.The deviation between the copperand aluminium densities calculated in different cases mainly comes from the de finition of convergence criterion as discussed above,while the deviation between the inverse results and actual values is mainly induced by statistical error,extracting error for the experimental full energy peak area,theoretical simulation of the HPGe detector and detection process,and the lack of enough spectrum characteristics to be analyzed in inverse calculation.Of course,it should be pointed out that the discussion about the mass density of the shielding material only having one component is insigni ficant,however,the density is related to the material composition and therefore,the inverse investigation for the density of the shielding material containing several components can help to determine the weight fractions of each material component.

Besides the cases where only geometric information or material information is unknown,in the applications that support nonproliferation,arms control veri fication,andinternational security,sometimes one has to deal with the inverse problem that part of the geometric and material information are needed to be identi fied simultaneously.Again,adopting the detected gamma-ray spectrum shown in Fig.1 and assuming the cases where the thicknesses of both aluminium and copper layers as well as the copper density are unknown,the inverse calculations are performed with IRT-DEGL model.

新型职业农民作为促进农业农村经济发展的重要力量，对“三农”问题的解决将会发挥不可替代的作用［4］。特别是新型职业农民“双创”能力培养对激活农村地区发展活力、促进现代农业转型升级与提高农民收入的作用更为明显。

Table 3 Material mass densities of Cu and Al shielding layers estimated by IRT-DEGL model for three test cases in comparison with the actual values in unit of g/cm3

Unknown ρCu ρAl ρCu ρAl IRT-DEGL 9.113 2.922 9.266 2.710 Actual 8.96 2.7 8.96 2.7 Relative difference(%) 1.71 8.22 3.41 0.37

The optimal thicknesses for the aluminium layer and copper layer as well as the optimal copper density estimated by the IRT-DEGL model are shown as a function of generation in Fig.2.The actual values are represented as dotted lines in each panel.Following the de finition in Ref.[16],the 0 generation is used to denote the initial guesses for the unknown parameters,and since the IRTDEGL calculations in the present work generate the first generation randomly instead of inputting the initial values,the inverse results in each panel of Fig.2 start from 1 generation.

In the figure,one can see that the estimated aluminium and copper layer thicknesses and copper density approach the actual values gradually.After 10 generations,all calculated values are located in the area around the actual values with relative difference from actual values smaller than 10%,e.g.,8.42%for aluminium layer thickness,4.33%for copper layer thickness and 6.46%for copper density,indicating the IRT-DEGL model can search the correct value area rapidly.For the applications that the rough values are required,the inverse calculation can be stopped manually and provide the estimating values.The whole inverse calculation stops after 67 generations of evolution,because the value of the cost function,f（uopt.）,is less than the converged value.As shown in Table 4,the final optimal thicknesses of the aluminium layer and copper layer are 0.314 cm and 0.551 cm corresponding to the relative difference from the actual value 1.95 and 3.33%,and the final optimal copper density is 9.090 g/cm3corresponding to the relative difference of 1.45%.

4 Summary and conclusion

In summary,the derivative-free inverse radiation transport model based on the differential evolution algorithm with globaland localneighbourhoods(IRT-DEGL)developed recently has been further extended for the inverse problems of identifying the unknown thicknesses with random initial guesses and material mass densities of multi-shielding layers as well as their combination.Using the detected gamma-ray spectrum,the illustration calculations are implemented.It is found that when using either random or proper initial guesses the IRT-DEGL model can provide the same shielding layer thicknesses,but the computing time of the inverse calculation can be saved greatly with proper initial values.For identifying the unknown mass densities and the combination of unknown layer thickness and mass densities of multi-shielding layers,the IRT-DEGL model can provide the relatively accurate values within 9 and 5%,respectively,of their actual values.

Fig.2(Color online)The optimal aluminium layer thickness(a),optimal copper layer thickness(b),optimal copper density(c),estimated by the IRT-DEGL model as a function of generation.The dotted line denotes actual thicknesses of copper and aluminium layers in panels(a)and(b),and the actual copper density in panel(c)

Table 4 Estimated thicknesses of aluminium and copper layers and the estimated copper density by the IRT-DEGL model in comparison with the actual values

ce(%)ρCu(g/cm) 8.96 9.090 1.45

Furthermore,as discussed above,the IRT-DEGL model solves the unknown parameters of the source system by iteratively adjusting the hypothetical direct transport model to match the measured spectrum,and thus,the variance for solving different inverse problems is only correlated with the construction of the direct transport model while the framework and calculation process of the solver are identical,and therefore,the framework of IRT-DEGL model is general in its applications and thus it in principle can be easily extended for the complicated source systems.However,it should be also pointed out that the application of the IRT-DEGL model so far is limited to the simple source system,e.g.,point source with one or two shielding layers.When it is applied to the complicated source systems,e.g.,multi-shielded volume source or irregular source systems,more numerical problems,including the construction of the direct transport model and the method to reduce the numerical deviation,will have to be dealt with,and therefore,a great deal of work will need to be done in the future.

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