基于正则化与粒子群算法的PCS纳米颗粒测量反演算法研究
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摘要
纳米颗粒因其具有特殊的电、磁、力、光、热等特性,在国民经济中发挥越来越重要的作用。纳米颗粒的特性与粒径的大小有关,纳米颗粒的测量成了颗粒研究的重点之一。PCS方法是测量纳米颗粒粒度及其分布的有效方法,其中颗粒粒度反演的研究是PCS方法的重点之一,也是难点之一。本文对PCS纳米颗粒测量技术中的粒度反演算法进行了研究,主要研究工作如下：
一、通过光强相关函数反演纳米颗粒粒径需要求解第一类Freholm积分方程,该方程属于病态问题。本文采用迭代正则化方法在不同的噪声水平下对单分散和双分散颗粒进行反演,反演结果表明,在噪声水平小于0.05时,正则化的反演误差为0-10%,迭代正则化的反演误差为O-7%；噪声水平为0.05时,正则化已无法反演出粒径分布,迭代正则化单分散反演的峰值误差不大于8%。双分散峰值反演误差不大于12%；另外,迭代正则化迭代次数一般要求：在噪声小时,迭代次数大；噪声大时,迭代次数小。
二、纳米颗粒粒径的反演从另一个角度可以看作是一个优化问题,本文采用粒子群算法对纳米颗粒进行反演。单峰和双峰分布的颗粒反演结果表明：在噪声水平小于0.05时,误差小于10%；噪声水平为0.05和0.1时峰值处的比例大大超过理论值,使反演所得颗粒粒度分布与真实分布产生偏离。
三、粒子群算法目标函数的约束条件影响颗粒粒度反演的速度和准确度,本文以正则化方法的展平泛函为目标函数,加上可行的约束条件,采用粒子群算法进行了颗粒粒度反演。反演结果表明,采用有光滑约束的目标函数时,粒子群算法给出的颗粒粒度分布光滑,解决了采用无光滑约束的目标函数反演结果中存在颗粒粒径分布在峰值过于集中的现象。
四、在L曲线准则的基础上采用粒子群算法对单峰分布和双峰分布的颗粒进行反演。反演结果表明,在噪声水平较小时,L曲线导出的目标函数所得反演结果给出的解光滑,解决了粒度分布集中现象,与采用展平泛函作为目标函数的粒子群算法相比,无需求取正则化参数,减小了计算量。
在PCS颗粒测量技术中,颗粒粒度的反演是影响颗粒测量准确性的主要原因。目前,纳米颗粒粒度分布测量的准确测量仍然制约这一技术的广泛应用,本文所做工作有助于PCS技术的发展。
Because of its unique electricity, magnetism, force, light, heat and other properties, nanoparticles played an important role in the national economy. The characteristics of nano-particle were related to its size, so the measurement of nanoparticles was important. PCS method was an effective way to measure the particle size and distribution, and the inversion of particle size was one focus of the PCS method, which was also difficulty. In this paper, the inversion algorithm of PCS has been studied, the main research work were as follows:
First, we need to solve the Fredholm integral equations of the first kind in order to get the nanoparticle sizing from the light autocorrelation function. The integral equation is the ill-posed problem. In this paper, iterative regularization method was used to inversed mo-dispersed and bi-dispersed particles at different noise levels. The inversion results indicated that the regularization of the inversion errors were 0-10% and the iterative regularization inversion error of 0 to 7% when noise level less than 0.05; when the noise level was 0.05, the regularization was no longer inverted size distribution but the iterative regularization could inverted, the single peak distribution error was less than 8% and the bimodal peak distribution error was less than 12%. In addition, the iterative regularization required the noise was larger, the number of iterations was smaller.
Second, the inversion of nanoparticle size can be considered as an optimization problem. In this paper, particle swarm optimization inverted the nanoparticles. Particles inversion results of Single and bimodal peak distribution showed that:when the noise level was less than 0.05, the error is less than 10%; when the noise level were 0.05 and 0.1, the ratio of peak values much higher than the theoretical value. So that the particle size distribution deviated from the true distribution.
Third, the objective function of particle swarm optimization affected the speed and accuracy of the inversion. This paper, the flattening functional as the objective function and combined with practical constraints, the particle swarm algorithm was used to invert. The inversion results showed that the particle swarm algorithm could get smooth distribution. The particle saize distribution which over concentrated on the peak was solved.
Fourth, based on the L curve criterion, particle swarm optimization was used to invert the single-peak distribution and bimodal distribution of particles. The results showed that the solution were smooth, which solved the size distribution of concentration. This algorithm didn't need a regularization parameter, reducing the amount of calculation.
In the PCS technique, the inversion of particle size was the main reason which affected the accuracy of particle measurement.Currently, nano-particle size distribution measurement was still constrainted the broad application of the technology. This will help the development of PCS technology.
一、通过光强相关函数反演纳米颗粒粒径需要求解第一类Freholm积分方程,该方程属于病态问题。本文采用迭代正则化方法在不同的噪声水平下对单分散和双分散颗粒进行反演,反演结果表明,在噪声水平小于0.05时,正则化的反演误差为0-10%,迭代正则化的反演误差为O-7%；噪声水平为0.05时,正则化已无法反演出粒径分布,迭代正则化单分散反演的峰值误差不大于8%。双分散峰值反演误差不大于12%；另外,迭代正则化迭代次数一般要求：在噪声小时,迭代次数大；噪声大时,迭代次数小。
二、纳米颗粒粒径的反演从另一个角度可以看作是一个优化问题,本文采用粒子群算法对纳米颗粒进行反演。单峰和双峰分布的颗粒反演结果表明：在噪声水平小于0.05时,误差小于10%；噪声水平为0.05和0.1时峰值处的比例大大超过理论值,使反演所得颗粒粒度分布与真实分布产生偏离。
三、粒子群算法目标函数的约束条件影响颗粒粒度反演的速度和准确度,本文以正则化方法的展平泛函为目标函数,加上可行的约束条件,采用粒子群算法进行了颗粒粒度反演。反演结果表明,采用有光滑约束的目标函数时,粒子群算法给出的颗粒粒度分布光滑,解决了采用无光滑约束的目标函数反演结果中存在颗粒粒径分布在峰值过于集中的现象。
四、在L曲线准则的基础上采用粒子群算法对单峰分布和双峰分布的颗粒进行反演。反演结果表明,在噪声水平较小时,L曲线导出的目标函数所得反演结果给出的解光滑,解决了粒度分布集中现象,与采用展平泛函作为目标函数的粒子群算法相比,无需求取正则化参数,减小了计算量。
在PCS颗粒测量技术中,颗粒粒度的反演是影响颗粒测量准确性的主要原因。目前,纳米颗粒粒度分布测量的准确测量仍然制约这一技术的广泛应用,本文所做工作有助于PCS技术的发展。
Because of its unique electricity, magnetism, force, light, heat and other properties, nanoparticles played an important role in the national economy. The characteristics of nano-particle were related to its size, so the measurement of nanoparticles was important. PCS method was an effective way to measure the particle size and distribution, and the inversion of particle size was one focus of the PCS method, which was also difficulty. In this paper, the inversion algorithm of PCS has been studied, the main research work were as follows:
First, we need to solve the Fredholm integral equations of the first kind in order to get the nanoparticle sizing from the light autocorrelation function. The integral equation is the ill-posed problem. In this paper, iterative regularization method was used to inversed mo-dispersed and bi-dispersed particles at different noise levels. The inversion results indicated that the regularization of the inversion errors were 0-10% and the iterative regularization inversion error of 0 to 7% when noise level less than 0.05; when the noise level was 0.05, the regularization was no longer inverted size distribution but the iterative regularization could inverted, the single peak distribution error was less than 8% and the bimodal peak distribution error was less than 12%. In addition, the iterative regularization required the noise was larger, the number of iterations was smaller.
Second, the inversion of nanoparticle size can be considered as an optimization problem. In this paper, particle swarm optimization inverted the nanoparticles. Particles inversion results of Single and bimodal peak distribution showed that:when the noise level was less than 0.05, the error is less than 10%; when the noise level were 0.05 and 0.1, the ratio of peak values much higher than the theoretical value. So that the particle size distribution deviated from the true distribution.
Third, the objective function of particle swarm optimization affected the speed and accuracy of the inversion. This paper, the flattening functional as the objective function and combined with practical constraints, the particle swarm algorithm was used to invert. The inversion results showed that the particle swarm algorithm could get smooth distribution. The particle saize distribution which over concentrated on the peak was solved.
Fourth, based on the L curve criterion, particle swarm optimization was used to invert the single-peak distribution and bimodal distribution of particles. The results showed that the solution were smooth, which solved the size distribution of concentration. This algorithm didn't need a regularization parameter, reducing the amount of calculation.
In the PCS technique, the inversion of particle size was the main reason which affected the accuracy of particle measurement.Currently, nano-particle size distribution measurement was still constrainted the broad application of the technology. This will help the development of PCS technology.
引文
[1]王乃宁等.颗粒粒径的光学测量技术及应用[M].北京：原子能出版社,2000(1).
[2]S. P. Lee, W. Tscharnuter, B. Chu. Calibration of an optical self-beating spectrometer by polystyrene latex spheres and confirmation of the Stokes-Einstein formula[J]. J. Polym. Sci. Phys,1972,10:2453-2459.
[3]Tadakazu Maeda, Satoru Fujime. Dynamic light-scattering study of suspensions of fdvirus. Application of a theory of light-scattering spectrum of weakly bending filaments[J].Macromolecules, 1985,18(12):2430-2437.
[4]Pusey, P. N., Van Megen, W. Detection of small polydispersities by photon correlation spectroscopy[J]. The Journal of Chemical Physics,1984,80(8):3513-3520.
[5]M. Corti, V. Degiorgio. Measurement of the thermal diffusivity of pure fluids by Rayleigh scattering of laser light [J]. J. Phys. C:Solid State Phys.1975,8:953-960.
[6]Kenneth S. Schmitz, Donald J. Ramsay. A QELS-SEF study on high molecular weight poly(lysine) field strength dependent apparent diffusion coefficient and the ordinary-extraordinary phase transition[J]. Macromolecules,1985,18(5):933-938.
[7]Lal J,Abernathy D, Auvray L. Dynamics and correlations in magnetic colloidal systems studied by X-ray photon correlation spectroscopy[J]. Eur. Phys. J. E 2001,4(3):263-271.
[8]G. Gonzalez-Gaitano, P. Rodriguez, J.R. Isasiet al. The aggregation of cyclodextrins as studied by photon correlation spectroscopy[J]. Journal of Inclusion Phenomena and Macrocyclic Chemistry.2002, 44:101-105.
[9]胡松青,李琳,郭祀远等.现代颗粒粒度测量技术[J].现代化工,2002,22(1)：58.
[10]周俊虎,王乃宁.用累积法确定光子相关光谱测量中的超细颗粒粒度[J].上海机械学院学报.1992,14(1)：l-6.
[11]Barbara J.Frisken. Revisiting the method of cumulantsfor the analysis of dynamic light scattering data[J]. APPLIED OPTICS,20 August 2001,Vol.40, No.24:4087-4089.
[12]Stephen W, PROVENCHER.A constrained regularization method for inverting data represented by linear algebraic or integral equations[J].Computer Physics Communications 1982,27:213-242.
[13]Provencher,S W, BioPhys [J].1976(16):27.
[14]ProvenCher, S W, Chem.Phys[J].1976 (64):2772.
[15]Provencher, S W, HendriX J, DeMaeyer L and Paulussen N,Chem.Phys[J].1978, (69):4273.
[16]Provencher, S W, Makronlol.Chem[J].1979, (180),201.
[17]Provencher, S W, "CONTIN Users Manual" EBL tchnical report DA05, EuroPean Molecular Biology Laboratory, Heidelberg.1982.
[18]J. G. McWhirter. A stabilized model-fitting approach to the proceeding of laser anemometry and other photon-correlation data[C]. Optica Acta,1980,27:83-105.
[19]S. H. Chen, B. Chu, R. Nossal. Scattering Techniques Applied to Supramolecular and Nonequilibrium Systems[M]. Plenum Press, N.Y.,1981.
[20]N. Ostrowsky, D. Sornette. Exponential sampling method for light scattering polydispersity analysis[J]. Optica Acta,1981,28:1050-1070.
[21]岳成凤,杨冠玲,何振江.动态光散射光强自相光函数与颗粒分布关系及算法比较[J].光电子技术与信息.2004,17(1)：10-14.
[22]Dahneke B E,et al.Measurement of Suspended Particles by Quasi-Elastic Light Scattering[M].New York:Wiley-Interscience,1983.
[23]Morrison I.D etc. Improved Technique for Particle Size Determination by QELS[J]. Langoniu.1985, No.1
[24]Particle size analysis-Photon correlation spectroscopy[J]. ISO 13321:1996(E).
[25]Instruction Manual for BI-9000AT Digital Autocorrelator[R].Brookhaven Instruments Corporation,IX-8.
[26]韩秋燕,申晋,孙贤明等.基于Tikhonov正则参数后验选择策略的PCS颗粒粒度反演方法[J].光子学报,2009,38(11)：2917-2926.
[27]喻雷寿,杨冠玲,何振江等.用于动态光散射颗粒测量的迭代CONTIN算法[J].光电工程,2006,33(8),64-69.
[28]喻雷寿,杨冠玲,何振江等.颗粒粒径测量中约束正则CONTIN算法分析[J].激光生物学报,2007,16(1),74-78.
[29]王少清,陶冶薇,董学仁等.由光子相关谱反演微粒体系粒径分布方法的分析与比较[J].中国粉体技术,2005,11(1)：28-32.
[31]Engl.H.W, Hanke.M, Ncubauer.Regularization of Inverse problems [J]. Ddrdrecht: Kluwer.1996.
[32]Groesch C W. The theory of Tikhonov Regularization for Fredholm equations of the first kind [J]. Pitman Advanced Publising Program.1984.
[33]Groesch H.. An A-Posteriori Parameter Choice for Ordinary and Iterated Tikhonov Regularization of Ⅲ-Posed Problems Leading to Optimal Convergence Rates[J]. Mathematics of Computation.1987, 49(180),507-522.
[34]King J.T., Chillingworth, D. Approximation generalized inverse by iterated regularization. Numer Funct Anal Optimiz[J].2 (1979),499-513.
[35]肖庭延,于慎根,王彦飞.反问题的数值解法[M].北京：科学出版社,2003.
[36]Morozov V A.On Regularization of Ⅲ-Problems and selection of Regularization Parameter.J. Comp[J]. Math.Phys.1966 (1),170-175.
[37]Tikhonov A N,Arsenin V Y. Solutions of Ill-Posed Problems[M]. John Wiley and Sons, New York.1977.
[38]Hansen P C,Analysis of discrete ill-posed problems by means of the 1-curve[J],SIAM Review,1992,1(34):561-580.
[39]何坚勇.最优化方法[M].北京：清华大学出版社,2007.
[40]黄平,孟永钢.最优化理论与方法[M].北京：清华大学出版社,2009.
[41]莫愿斌.粒子群优化算法的扩展与应用[D].(博士论文).浙江大学.2006.
[42]纪震,廖惠连,吴青华.粒子群算法及应用[M].北京：科学出版社.2009.
[43]Kennedy J and EberhartRC.partieles warm optimization.[C] In Porceeding of the IEEE International Conference on Neural Nctworks.1995,1942 - 1948.
[44]Shi,Y.H.and Eberhart,R.C. A modified partiele swarm optimizer[C]. In Porceeding of the IEEE International Conference on Evolutionary Computation.1998,69-73.
[45]Roig, A.R & Alessandrini, J.L. Praticle size distribution from static light scattering with regularized non-negative least squares constraints[J]. Particle and particle systems characterization,2007,23(6),431-437.
[46]Engl, H.W. Discrepancy Principles for Tikhonov Regularization of Ill-Posed problems leading to optimal convergence rates[J]. Journal of Optimization and Theory Application,1987,52(2),209-215.
[2]S. P. Lee, W. Tscharnuter, B. Chu. Calibration of an optical self-beating spectrometer by polystyrene latex spheres and confirmation of the Stokes-Einstein formula[J]. J. Polym. Sci. Phys,1972,10:2453-2459.
[3]Tadakazu Maeda, Satoru Fujime. Dynamic light-scattering study of suspensions of fdvirus. Application of a theory of light-scattering spectrum of weakly bending filaments[J].Macromolecules, 1985,18(12):2430-2437.
[4]Pusey, P. N., Van Megen, W. Detection of small polydispersities by photon correlation spectroscopy[J]. The Journal of Chemical Physics,1984,80(8):3513-3520.
[5]M. Corti, V. Degiorgio. Measurement of the thermal diffusivity of pure fluids by Rayleigh scattering of laser light [J]. J. Phys. C:Solid State Phys.1975,8:953-960.
[6]Kenneth S. Schmitz, Donald J. Ramsay. A QELS-SEF study on high molecular weight poly(lysine) field strength dependent apparent diffusion coefficient and the ordinary-extraordinary phase transition[J]. Macromolecules,1985,18(5):933-938.
[7]Lal J,Abernathy D, Auvray L. Dynamics and correlations in magnetic colloidal systems studied by X-ray photon correlation spectroscopy[J]. Eur. Phys. J. E 2001,4(3):263-271.
[8]G. Gonzalez-Gaitano, P. Rodriguez, J.R. Isasiet al. The aggregation of cyclodextrins as studied by photon correlation spectroscopy[J]. Journal of Inclusion Phenomena and Macrocyclic Chemistry.2002, 44:101-105.
[9]胡松青,李琳,郭祀远等.现代颗粒粒度测量技术[J].现代化工,2002,22(1)：58.
[10]周俊虎,王乃宁.用累积法确定光子相关光谱测量中的超细颗粒粒度[J].上海机械学院学报.1992,14(1)：l-6.
[11]Barbara J.Frisken. Revisiting the method of cumulantsfor the analysis of dynamic light scattering data[J]. APPLIED OPTICS,20 August 2001,Vol.40, No.24:4087-4089.
[12]Stephen W, PROVENCHER.A constrained regularization method for inverting data represented by linear algebraic or integral equations[J].Computer Physics Communications 1982,27:213-242.
[13]Provencher,S W, BioPhys [J].1976(16):27.
[14]ProvenCher, S W, Chem.Phys[J].1976 (64):2772.
[15]Provencher, S W, HendriX J, DeMaeyer L and Paulussen N,Chem.Phys[J].1978, (69):4273.
[16]Provencher, S W, Makronlol.Chem[J].1979, (180),201.
[17]Provencher, S W, "CONTIN Users Manual" EBL tchnical report DA05, EuroPean Molecular Biology Laboratory, Heidelberg.1982.
[18]J. G. McWhirter. A stabilized model-fitting approach to the proceeding of laser anemometry and other photon-correlation data[C]. Optica Acta,1980,27:83-105.
[19]S. H. Chen, B. Chu, R. Nossal. Scattering Techniques Applied to Supramolecular and Nonequilibrium Systems[M]. Plenum Press, N.Y.,1981.
[20]N. Ostrowsky, D. Sornette. Exponential sampling method for light scattering polydispersity analysis[J]. Optica Acta,1981,28:1050-1070.
[21]岳成凤,杨冠玲,何振江.动态光散射光强自相光函数与颗粒分布关系及算法比较[J].光电子技术与信息.2004,17(1)：10-14.
[22]Dahneke B E,et al.Measurement of Suspended Particles by Quasi-Elastic Light Scattering[M].New York:Wiley-Interscience,1983.
[23]Morrison I.D etc. Improved Technique for Particle Size Determination by QELS[J]. Langoniu.1985, No.1
[24]Particle size analysis-Photon correlation spectroscopy[J]. ISO 13321:1996(E).
[25]Instruction Manual for BI-9000AT Digital Autocorrelator[R].Brookhaven Instruments Corporation,IX-8.
[26]韩秋燕,申晋,孙贤明等.基于Tikhonov正则参数后验选择策略的PCS颗粒粒度反演方法[J].光子学报,2009,38(11)：2917-2926.
[27]喻雷寿,杨冠玲,何振江等.用于动态光散射颗粒测量的迭代CONTIN算法[J].光电工程,2006,33(8),64-69.
[28]喻雷寿,杨冠玲,何振江等.颗粒粒径测量中约束正则CONTIN算法分析[J].激光生物学报,2007,16(1),74-78.
[29]王少清,陶冶薇,董学仁等.由光子相关谱反演微粒体系粒径分布方法的分析与比较[J].中国粉体技术,2005,11(1)：28-32.
[31]Engl.H.W, Hanke.M, Ncubauer.Regularization of Inverse problems [J]. Ddrdrecht: Kluwer.1996.
[32]Groesch C W. The theory of Tikhonov Regularization for Fredholm equations of the first kind [J]. Pitman Advanced Publising Program.1984.
[33]Groesch H.. An A-Posteriori Parameter Choice for Ordinary and Iterated Tikhonov Regularization of Ⅲ-Posed Problems Leading to Optimal Convergence Rates[J]. Mathematics of Computation.1987, 49(180),507-522.
[34]King J.T., Chillingworth, D. Approximation generalized inverse by iterated regularization. Numer Funct Anal Optimiz[J].2 (1979),499-513.
[35]肖庭延,于慎根,王彦飞.反问题的数值解法[M].北京：科学出版社,2003.
[36]Morozov V A.On Regularization of Ⅲ-Problems and selection of Regularization Parameter.J. Comp[J]. Math.Phys.1966 (1),170-175.
[37]Tikhonov A N,Arsenin V Y. Solutions of Ill-Posed Problems[M]. John Wiley and Sons, New York.1977.
[38]Hansen P C,Analysis of discrete ill-posed problems by means of the 1-curve[J],SIAM Review,1992,1(34):561-580.
[39]何坚勇.最优化方法[M].北京：清华大学出版社,2007.
[40]黄平,孟永钢.最优化理论与方法[M].北京：清华大学出版社,2009.
[41]莫愿斌.粒子群优化算法的扩展与应用[D].(博士论文).浙江大学.2006.
[42]纪震,廖惠连,吴青华.粒子群算法及应用[M].北京：科学出版社.2009.
[43]Kennedy J and EberhartRC.partieles warm optimization.[C] In Porceeding of the IEEE International Conference on Neural Nctworks.1995,1942 - 1948.
[44]Shi,Y.H.and Eberhart,R.C. A modified partiele swarm optimizer[C]. In Porceeding of the IEEE International Conference on Evolutionary Computation.1998,69-73.
[45]Roig, A.R & Alessandrini, J.L. Praticle size distribution from static light scattering with regularized non-negative least squares constraints[J]. Particle and particle systems characterization,2007,23(6),431-437.
[46]Engl, H.W. Discrepancy Principles for Tikhonov Regularization of Ill-Posed problems leading to optimal convergence rates[J]. Journal of Optimization and Theory Application,1987,52(2),209-215.