Abstract

We present results of simulation research on the constrained regularized least-squares (RLS) solution of the ill-conditioned inverse problem in turbidimetric measurements. The problem is formulated in terms of the discretized Fredholm integral equation of the first kind. The inverse problem in turbidimetric measurements consists in determining particle size distribution (PSD) function of particulate system on the basis of turbidimetric measurements. The desired PSD should satisfy two constraints: non negativity of PSD values and normalization of PSD to unity when integrated over the whole range of particle size. Incorporating the constraints into the RLS method leads to the constrained regularized least-squares (CRLS) method, which is realized by means of an active set algorithm of quadratic programming. Results of simulation research prove that the CRLS method performs considerably better with reconstruction of PSD than the RLS method in terms of better fidelity and smaller uncertainty.

© 2010 Optical Society of America

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References

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  1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).
  2. A. R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. 25, 1–53 (1999).
    [CrossRef]
  3. R. Xu, Particle Characterization: Light Scattering Methods (Kluwer Academic, 2000).
  4. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).
  5. O. C. Lingjearde and N. Christophersen, Regularization Principles: Solving Ill-Posed Inverse Problems, Lecture Notes to IN INVPAR, Department of Infomatics, University of Oslo (1998).
  6. P. C. Hansen, “Regularization tools: a MATLAB package for analysis and solution of discrete ill-posed problems,” Numer. Algorithms 6, 1–35 (1994).
    [CrossRef]
  7. J. G. Crump and J. H. Seinfeld, “A new algorithm for inversion of aerosol size distribution data,” Aerosol Sci. Technol. 1, 15–34 (1981).
    [CrossRef]
  8. M. Kandlikar and G. Ramachandran, “Inverse methods for analysing aerosol spectrometer measurements: a critical review,” J. Aerosol Sci. 30, 413–437 (1999).
    [CrossRef]
  9. F. Stout and J. H. Kalivas, “Tikhonov regularization in standardized and general form for multivariate calibration with application towards removing unwanted spectral artifacts,” J. Chemometrics 20, 22–33 (2006).
    [CrossRef]
  10. S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101(1963).
    [CrossRef]
  11. D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
    [CrossRef]
  12. P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
    [CrossRef]
  13. P. J. Mc Carthy, “Direct analytic model of the L-curve for Tikhonov regularization parameter selection,” Inverse Probl. 19, 643–663 (2003).
    [CrossRef]
  14. L. Eldén, “A weighted pseudoinverse, generalized singular values, and constrained least squares problems,” BIT Numer. Math 22, 487–502 (1982).
    [CrossRef]
  15. G. H. Golub and U. von Matt, “Quadratically constrained least squares and quadratic problems,” Numer. Math. 59, 561–580 (1991).
    [CrossRef]
  16. B. Martos, Nonlinear Programming: Theory and Methods (Akadémiai Kiadó, 1975).
  17. W. I. Zangwill, Nonlinear Programming: a Unified Approach (Prentice-Hall, 1969).
  18. P. E. Gill, W. Murray, M. A. Saunders, and M. H. Wright, “Procedures for optimization problems with a mixture of bounds and general linear constraints,” ACM Trans. Math. Software 10, 282–298 (1984).
    [CrossRef]
  19. P. E. Gill, W. Murray, and M. H. Wright, Numerical Linear Algebra and Optimization (Addison-Wesley, 1991), Vol. 1.
  20. C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974).
  21. A. R. Roig and J. L. Alessandrini, “Particle size distributions from static light scattering with regularized non-negative least squares constraints,” Part. Part. Syst. Charact. 23, 431–437 (2006).
    [CrossRef]
  22. K. F. Ren, F. Xu, X. Cai, and J. M. Dorey, “Development of a precise and in situ turbidity measurement system,” Chem. Eng. Commun. 197, 250–259 (2010).
    [CrossRef]
  23. G. H. Golub, M. Heath, and H. Wahba, “Generalized cross validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–224 (1979).
    [CrossRef]
  24. “Index of refraction,” Technical note 007 (Duke Scientific Corporation, 1 December 1996).
  25. “Release on the refractive index of ordinary water substance as a function of wavelength, temperature and pressure” (The International Association for the Properties of Water and Steam, September 1997).

2010 (1)

K. F. Ren, F. Xu, X. Cai, and J. M. Dorey, “Development of a precise and in situ turbidity measurement system,” Chem. Eng. Commun. 197, 250–259 (2010).
[CrossRef]

2006 (2)

A. R. Roig and J. L. Alessandrini, “Particle size distributions from static light scattering with regularized non-negative least squares constraints,” Part. Part. Syst. Charact. 23, 431–437 (2006).
[CrossRef]

F. Stout and J. H. Kalivas, “Tikhonov regularization in standardized and general form for multivariate calibration with application towards removing unwanted spectral artifacts,” J. Chemometrics 20, 22–33 (2006).
[CrossRef]

2003 (1)

P. J. Mc Carthy, “Direct analytic model of the L-curve for Tikhonov regularization parameter selection,” Inverse Probl. 19, 643–663 (2003).
[CrossRef]

1999 (2)

M. Kandlikar and G. Ramachandran, “Inverse methods for analysing aerosol spectrometer measurements: a critical review,” J. Aerosol Sci. 30, 413–437 (1999).
[CrossRef]

A. R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. 25, 1–53 (1999).
[CrossRef]

1994 (1)

P. C. Hansen, “Regularization tools: a MATLAB package for analysis and solution of discrete ill-posed problems,” Numer. Algorithms 6, 1–35 (1994).
[CrossRef]

1993 (1)

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

1991 (1)

G. H. Golub and U. von Matt, “Quadratically constrained least squares and quadratic problems,” Numer. Math. 59, 561–580 (1991).
[CrossRef]

1984 (1)

P. E. Gill, W. Murray, M. A. Saunders, and M. H. Wright, “Procedures for optimization problems with a mixture of bounds and general linear constraints,” ACM Trans. Math. Software 10, 282–298 (1984).
[CrossRef]

1982 (1)

L. Eldén, “A weighted pseudoinverse, generalized singular values, and constrained least squares problems,” BIT Numer. Math 22, 487–502 (1982).
[CrossRef]

1981 (1)

J. G. Crump and J. H. Seinfeld, “A new algorithm for inversion of aerosol size distribution data,” Aerosol Sci. Technol. 1, 15–34 (1981).
[CrossRef]

1979 (1)

G. H. Golub, M. Heath, and H. Wahba, “Generalized cross validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–224 (1979).
[CrossRef]

1963 (1)

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101(1963).
[CrossRef]

1962 (1)

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

Alessandrini, J. L.

A. R. Roig and J. L. Alessandrini, “Particle size distributions from static light scattering with regularized non-negative least squares constraints,” Part. Part. Syst. Charact. 23, 431–437 (2006).
[CrossRef]

Arsenin, V. Y.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

Cai, X.

K. F. Ren, F. Xu, X. Cai, and J. M. Dorey, “Development of a precise and in situ turbidity measurement system,” Chem. Eng. Commun. 197, 250–259 (2010).
[CrossRef]

Christophersen, N.

O. C. Lingjearde and N. Christophersen, Regularization Principles: Solving Ill-Posed Inverse Problems, Lecture Notes to IN INVPAR, Department of Infomatics, University of Oslo (1998).

Crump, J. G.

J. G. Crump and J. H. Seinfeld, “A new algorithm for inversion of aerosol size distribution data,” Aerosol Sci. Technol. 1, 15–34 (1981).
[CrossRef]

Dorey, J. M.

K. F. Ren, F. Xu, X. Cai, and J. M. Dorey, “Development of a precise and in situ turbidity measurement system,” Chem. Eng. Commun. 197, 250–259 (2010).
[CrossRef]

Eldén, L.

L. Eldén, “A weighted pseudoinverse, generalized singular values, and constrained least squares problems,” BIT Numer. Math 22, 487–502 (1982).
[CrossRef]

Gill, P. E.

P. E. Gill, W. Murray, M. A. Saunders, and M. H. Wright, “Procedures for optimization problems with a mixture of bounds and general linear constraints,” ACM Trans. Math. Software 10, 282–298 (1984).
[CrossRef]

P. E. Gill, W. Murray, and M. H. Wright, Numerical Linear Algebra and Optimization (Addison-Wesley, 1991), Vol. 1.

Golub, G. H.

G. H. Golub and U. von Matt, “Quadratically constrained least squares and quadratic problems,” Numer. Math. 59, 561–580 (1991).
[CrossRef]

G. H. Golub, M. Heath, and H. Wahba, “Generalized cross validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–224 (1979).
[CrossRef]

Hansen, P. C.

P. C. Hansen, “Regularization tools: a MATLAB package for analysis and solution of discrete ill-posed problems,” Numer. Algorithms 6, 1–35 (1994).
[CrossRef]

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

Hanson, R. J.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974).

Heath, M.

G. H. Golub, M. Heath, and H. Wahba, “Generalized cross validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–224 (1979).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

Jones, A. R.

A. R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. 25, 1–53 (1999).
[CrossRef]

Kalivas, J. H.

F. Stout and J. H. Kalivas, “Tikhonov regularization in standardized and general form for multivariate calibration with application towards removing unwanted spectral artifacts,” J. Chemometrics 20, 22–33 (2006).
[CrossRef]

Kandlikar, M.

M. Kandlikar and G. Ramachandran, “Inverse methods for analysing aerosol spectrometer measurements: a critical review,” J. Aerosol Sci. 30, 413–437 (1999).
[CrossRef]

Lawson, C. L.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974).

Lingjearde, O. C.

O. C. Lingjearde and N. Christophersen, Regularization Principles: Solving Ill-Posed Inverse Problems, Lecture Notes to IN INVPAR, Department of Infomatics, University of Oslo (1998).

Martos, B.

B. Martos, Nonlinear Programming: Theory and Methods (Akadémiai Kiadó, 1975).

Mc Carthy, P. J.

P. J. Mc Carthy, “Direct analytic model of the L-curve for Tikhonov regularization parameter selection,” Inverse Probl. 19, 643–663 (2003).
[CrossRef]

Murray, W.

P. E. Gill, W. Murray, M. A. Saunders, and M. H. Wright, “Procedures for optimization problems with a mixture of bounds and general linear constraints,” ACM Trans. Math. Software 10, 282–298 (1984).
[CrossRef]

P. E. Gill, W. Murray, and M. H. Wright, Numerical Linear Algebra and Optimization (Addison-Wesley, 1991), Vol. 1.

O’Leary, D. P.

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

Phillips, D. L.

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

Ramachandran, G.

M. Kandlikar and G. Ramachandran, “Inverse methods for analysing aerosol spectrometer measurements: a critical review,” J. Aerosol Sci. 30, 413–437 (1999).
[CrossRef]

Ren, K. F.

K. F. Ren, F. Xu, X. Cai, and J. M. Dorey, “Development of a precise and in situ turbidity measurement system,” Chem. Eng. Commun. 197, 250–259 (2010).
[CrossRef]

Roig, A. R.

A. R. Roig and J. L. Alessandrini, “Particle size distributions from static light scattering with regularized non-negative least squares constraints,” Part. Part. Syst. Charact. 23, 431–437 (2006).
[CrossRef]

Saunders, M. A.

P. E. Gill, W. Murray, M. A. Saunders, and M. H. Wright, “Procedures for optimization problems with a mixture of bounds and general linear constraints,” ACM Trans. Math. Software 10, 282–298 (1984).
[CrossRef]

Seinfeld, J. H.

J. G. Crump and J. H. Seinfeld, “A new algorithm for inversion of aerosol size distribution data,” Aerosol Sci. Technol. 1, 15–34 (1981).
[CrossRef]

Stout, F.

F. Stout and J. H. Kalivas, “Tikhonov regularization in standardized and general form for multivariate calibration with application towards removing unwanted spectral artifacts,” J. Chemometrics 20, 22–33 (2006).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).

Twomey, S.

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101(1963).
[CrossRef]

von Matt, U.

G. H. Golub and U. von Matt, “Quadratically constrained least squares and quadratic problems,” Numer. Math. 59, 561–580 (1991).
[CrossRef]

Wahba, H.

G. H. Golub, M. Heath, and H. Wahba, “Generalized cross validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–224 (1979).
[CrossRef]

Wright, M. H.

P. E. Gill, W. Murray, M. A. Saunders, and M. H. Wright, “Procedures for optimization problems with a mixture of bounds and general linear constraints,” ACM Trans. Math. Software 10, 282–298 (1984).
[CrossRef]

P. E. Gill, W. Murray, and M. H. Wright, Numerical Linear Algebra and Optimization (Addison-Wesley, 1991), Vol. 1.

Xu, F.

K. F. Ren, F. Xu, X. Cai, and J. M. Dorey, “Development of a precise and in situ turbidity measurement system,” Chem. Eng. Commun. 197, 250–259 (2010).
[CrossRef]

Xu, R.

R. Xu, Particle Characterization: Light Scattering Methods (Kluwer Academic, 2000).

Zangwill, W. I.

W. I. Zangwill, Nonlinear Programming: a Unified Approach (Prentice-Hall, 1969).

ACM Trans. Math. Software (1)

P. E. Gill, W. Murray, M. A. Saunders, and M. H. Wright, “Procedures for optimization problems with a mixture of bounds and general linear constraints,” ACM Trans. Math. Software 10, 282–298 (1984).
[CrossRef]

Aerosol Sci. Technol. (1)

J. G. Crump and J. H. Seinfeld, “A new algorithm for inversion of aerosol size distribution data,” Aerosol Sci. Technol. 1, 15–34 (1981).
[CrossRef]

BIT Numer. Math (1)

L. Eldén, “A weighted pseudoinverse, generalized singular values, and constrained least squares problems,” BIT Numer. Math 22, 487–502 (1982).
[CrossRef]

Chem. Eng. Commun. (1)

K. F. Ren, F. Xu, X. Cai, and J. M. Dorey, “Development of a precise and in situ turbidity measurement system,” Chem. Eng. Commun. 197, 250–259 (2010).
[CrossRef]

Inverse Probl. (1)

P. J. Mc Carthy, “Direct analytic model of the L-curve for Tikhonov regularization parameter selection,” Inverse Probl. 19, 643–663 (2003).
[CrossRef]

J. Aerosol Sci. (1)

M. Kandlikar and G. Ramachandran, “Inverse methods for analysing aerosol spectrometer measurements: a critical review,” J. Aerosol Sci. 30, 413–437 (1999).
[CrossRef]

J. Assoc. Comput. Mach. (2)

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101(1963).
[CrossRef]

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

J. Chemometrics (1)

F. Stout and J. H. Kalivas, “Tikhonov regularization in standardized and general form for multivariate calibration with application towards removing unwanted spectral artifacts,” J. Chemometrics 20, 22–33 (2006).
[CrossRef]

Numer. Algorithms (1)

P. C. Hansen, “Regularization tools: a MATLAB package for analysis and solution of discrete ill-posed problems,” Numer. Algorithms 6, 1–35 (1994).
[CrossRef]

Numer. Math. (1)

G. H. Golub and U. von Matt, “Quadratically constrained least squares and quadratic problems,” Numer. Math. 59, 561–580 (1991).
[CrossRef]

Part. Part. Syst. Charact. (1)

A. R. Roig and J. L. Alessandrini, “Particle size distributions from static light scattering with regularized non-negative least squares constraints,” Part. Part. Syst. Charact. 23, 431–437 (2006).
[CrossRef]

Prog. Energy Combust. Sci. (1)

A. R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. 25, 1–53 (1999).
[CrossRef]

SIAM J. Sci. Comput. (1)

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

Technometrics (1)

G. H. Golub, M. Heath, and H. Wahba, “Generalized cross validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–224 (1979).
[CrossRef]

Other (10)

“Index of refraction,” Technical note 007 (Duke Scientific Corporation, 1 December 1996).

“Release on the refractive index of ordinary water substance as a function of wavelength, temperature and pressure” (The International Association for the Properties of Water and Steam, September 1997).

P. E. Gill, W. Murray, and M. H. Wright, Numerical Linear Algebra and Optimization (Addison-Wesley, 1991), Vol. 1.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974).

B. Martos, Nonlinear Programming: Theory and Methods (Akadémiai Kiadó, 1975).

W. I. Zangwill, Nonlinear Programming: a Unified Approach (Prentice-Hall, 1969).

R. Xu, Particle Characterization: Light Scattering Methods (Kluwer Academic, 2000).

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).

O. C. Lingjearde and N. Christophersen, Regularization Principles: Solving Ill-Posed Inverse Problems, Lecture Notes to IN INVPAR, Department of Infomatics, University of Oslo (1998).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

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Figures (8)

Fig. 1
Fig. 1

Unimodal PSD reconstructed using the RLS method for various L matrices: (a) identity matrix, (b) discrete approximation of the first derivative operator, (c) discrete approximation of the second derivative operator, and (d) discrete approximation of the third derivative operator.

Fig. 2
Fig. 2

Unimodal PSD reconstructed using the CRLS method for various L matrices: (a) identity matrix, (b) discrete approximation of the first derivative operator, (c) discrete approximation of the second derivative operator, and (d) discrete approximation of the third derivative operator.

Fig. 3
Fig. 3

Comparison of the uncertainty Σ f , p 1 of the unimodal PSD reconstructed with use of the RLS and CRLS methods for various p values and for various L matrices: (a) identity matrix, (b) discrete approximation of the first derivative operator, (c) discrete approximation of the second derivative operator, and (d) discrete approximation of the third derivative operator.

Fig. 4
Fig. 4

Comparison of the deviation Δ f , p 1 of the unimodal PSD, reconstructed with use of the RLS and CRLS methods, from the test PSD for various p values and for various L matrices: (a) identity matrix, (b) discrete approximation of the first derivative operator, (c) discrete approximation of the second derivative operator, and (d) discrete approximation of the third derivative operator.

Fig. 5
Fig. 5

Bimodal PSD reconstructed using the RLS method for various L matrices: (a) identity matrix, (b) discrete approximation of the first derivative operator, (c) discrete approximation of the second derivative operator, and (d) discrete approximation of the third derivative operator.

Fig. 6
Fig. 6

Bimodal PSD reconstructed using the CRLS method for various L matrices: (a) identity matrix, (b) discrete approximation of the first derivative operator, (c) discrete approximation of the second derivative operator, and (d) discrete approximation of the third derivative operator.

Fig. 7
Fig. 7

Comparison of the uncertainty Σ f , p 2 of the bimodal PSD reconstructed with use of the RLS and CRLS methods for various p values and for various L matrices: (a) identity matrix, (b) discrete approximation of the first derivative operator, (c) discrete approximation of the second derivative operator, and (d) discrete approximation of the third derivative operator.

Fig. 8
Fig. 8

Comparison of the deviation Δ f , p 2 of the bimodal PSD, reconstructed with use of the RLS and CRLS methods, from the test PSD for various p values and for various L matrices: (a) identity matrix, (b) discrete approximation of the first derivative operator, (c) discrete approximation of the second derivative operator, and (d) discrete approximation of the third derivative operator.

Tables (4)

Tables Icon

Table 1 Comparison of Uncertainty Σ f , p 1 ( nm 1 ) of Unimodal Particle Size Distribution Reconstructed with use of Regularized Least-Squares and Constrained Regularized Least-Squares Methods for Various p Values and for Various L Matrices

Tables Icon

Table 2 Comparison of Deviation Δ f , p 1 ( nm 1 ) of Unimodal Particle Size Distribution, Reconstructed with Use of Regularized Least-Squares and Constrained Regularized Least-Squares Methods, from Test Particle Size Distribution for Various p Values and for Various L Matrices

Tables Icon

Table 3 Comparison of Uncertainty Σ f , p 2 ( nm 1 ) of Bimodal Particle Size Distribution Reconstructed with Use of Regularized Least-Squares and Constrained Regularized Least-Squares Methods for Various p Values and for Various L Matrices

Tables Icon

Table 4 Comparison of Deviation Δ f , p 2 ( nm 1 ) of Bimodal Particle Size Distribution, Reconstructed with use of Regularized Least-Squares and Constrained Regularized Least-Squares Methods, from Test Particle Size Distribution for Various p Values and for Various L Matrices

Equations (43)

Equations on this page are rendered with MathJax. Learn more.

c ( λ ) = 0 K ( λ , a ) f ( a ) d a .
K ( λ , a ) = N v C ext ( a , λ ) ,
C ext ( a , λ ) = 2 π k 2 Re { n = 1 ( 2 n + 1 ) ( a n + b n ) } ,
a n = m ψ n ( m x ) ψ n ( x ) ψ n ( x ) ψ n ( m x ) m ψ n ( m x ) ξ n ( x ) ξ n ( x ) ψ n ( m x ) , b n = ψ n ( m x ) ψ n ( x ) m ψ n ( x ) ψ n ( m x ) ψ n ( m x ) ξ n ( x ) m ξ n ( x ) ψ n ( m x ) ,
m = N 1 N 2 ,
x = 2 π a N 2 λ .
c = Kf ,
c = [ c ( λ 1 ) c ( λ 2 ) c ( λ p ) ] T ,
f = [ f ( a 1 ) f ( a 2 ) f ( a q ) ] T ,
( K ) i j = K ( λ i , a j ) Δ a , i = 1 , , p , j = 1 , , q ,
a j = ( j 1 2 ) Δ a ,
Δ a = a final a initial q .
f ( a ) 0 for each     a ( 0 , ) ,
0 f ( a ) d a = 1 .
f 0 q ,
[ Δ a Δ a Δ a ] 1 × q f = 1.
f ^ RLS = arg min f { c Kf 2 2 + γ 2 L ( f f * ) 2 2 } ,
L = I q ,
L ( q 1 ) × q ( 1 ) = [ 1 1 0 0 0 1 1 0 0 0 1 1 ] ,
L ( q 2 ) × q ( 2 ) = L ( q 2 ) × ( q 1 ) ( 1 ) L ( q 1 ) × q ( 1 ) = [ 1 2 1 0 0 0 1 2 1 0 0 0 1 2 1 ] ,
L ( q 3 ) × q ( 3 ) = L ( q 3 ) × ( q 2 ) ( 1 ) L ( q 2 ) × q ( 2 ) = [ 1 3 3 1 0 0 0 1 3 3 1 0 0 1 3 3 3 1 ] .
f ^ RLS = ( K T K + γ L ) 1 ( K T c + γ Lf * ) .
Af = b ,
f 0 q ,
A = [ Δ a Δ a Δ a ] 1 × q
f ^ RLS = arg min f { c Kf 2 2 + γ 2 Lf 2 2 } = arg min f { ( c Kf ) T ( c Kf ) + γ 2 ( Lf ) T ( Lf ) } = arg min f { f T ( K T K + γ 2 L T L ) f 2 c T Kf } .
Q = 2 ( K T K + γ 2 L T L ) ,
q = 2 K T c ,
f ^ RLS = arg min f { 1 2 f T Qf + q T f } .
c = K k f k ,
f k = [ f k f k + q K f k + ( K 1 ) q K ] T ,
K k = [ K , k K , k + q K K , k + ( K 1 ) q K ] ,
k = 1 , , q K ,
Cf d ,
f l f f u .
f test 1 ( a ) = 0.60 f n ( a , 500 nm , 75 nm ) + 0.40 f l ( a , 6.15 , 0.10 ) ,
f test 2 ( a ) = 0.375 f n ( a , 450 nm , 40 nm ) + 0.292 f n ( a , 550 nm , 22.5 nm ) + 0.333 f l ( a , 6.135 , 0.150 ) ,
σ p = p 100 max ( c ) ,
c p ( i ) = c + e p ( i ) , i = 1 , , N simul .
f ¯ p , j = 1 N simul i = 1 N simul f p , j ( i ) , j = 1 , , 40 ,
σ f , p , j = [ 1 N simul 1 i = 1 N simul ( f p , j ( i ) f ¯ p , j ) 2 ] 1 / 2 , j = 1 , , 40.
Σ f , p = [ 1 40 j = 1 40 σ f , p , j 2 ] 1 / 2 .
Δ f , p = { 1 40 j = 1 40 [ f ¯ p , j f test ( a sol , j ) ] 2 } 1 / 2 .

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