Abstract

We present further results of the simulation research on the constrained regularized least squares (CRLS) solution of the ill-conditioned inverse problem in spectral extinction (turbidimetric) measurements, which we originally presented in this journal [Appl. Opt. 49, 4591 (2010)]. The inverse problem consists of determining the particle size distribution (PSD) function of a particulate system on the basis of a measured extinction coefficient as a function of wavelength. In our previous paper, it was shown that under assumed conditions the problem can be formulated in terms of the discretized Fredholm integral equation of the first kind. The CRLS method incorporates two constraints, which the PSD sought will satisfy: nonnegativity of the PSD values and normalization of the PSD to unity when integrated over the whole range of particle size, into the regularized least squares (RLS) method. This leads to the quadratic programming problem, which is solved by means of the active set algorithm within the research. The simulation research that is the subject of the present paper is a continuation and extension of the research described in our previous paper. In the present research, the performance of the CRLS method variants is compared not only to the corresponding RLS method variants but also to other regularization techniques: the truncated generalized singular value decomposition and the filtered generalized singular value decomposition, as well as nonlinear iterative algorithms: The Twomey algorithm and the Twomey—Markowski algorithm. Moreover, two methods of selecting the optimum value of the regularization parameter are considered: The L-curve method and the generalized cross validation method. The results of our simulation research provide even stronger proof that the CRLS method performs considerably better with reconstruction of PSD than other inversing methods, in terms of better fidelity and smaller uncertainty.

© 2012 Optical Society of America

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References

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  1. J. Mroczka and D. Szczuczyński, “Improved regularized solution of the inverse problem in turbidimetric measurements,” Appl. Opt. 49, 4591–4603 (2010).
    [CrossRef]
  2. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).
  3. A. R. Jones, “Light scattering for particle characterization,” Progr. Energy Combust. Sci. 25, 1–53 (1992).
  4. P. C. Hansen, “Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems,” Numerical Algorithms 6, 1–35 (1994).
  5. 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]
  6. M. Kandlikar and G. Ramachandran, “Inverse methods for analysing aerosol spectrometer measurements: A Critical Review,” J. Aerosol Sci. 30, 413–437 (1999).
    [CrossRef]
  7. F. Stout and J. H. Kalivas, “Tikhonov regularization in standardized and general form for multivariate calibration with application towards removing unwanted spectral artifacts,” J. Chemom. 20, 22–33 (2006).
    [CrossRef]
  8. 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]
  9. 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]
  10. C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice Hall, 1974).
  11. 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]
  12. G. H. Golub and U. von Matt, “Quadratically constrained least squares and quadratic problems,” Numer. Math. 59, 561–580 (1991).
    [CrossRef]
  13. 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. Softw. 10, 282–298 (1984).
    [CrossRef]
  14. P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral equations of the first kind,” Inverse Probl. 8, 849–872 (1992).
    [CrossRef]
  15. P. C. Hansen, “Regularization, GSVD and truncated GSVD,” BIT (Nord. Tidskr. Inf.-behandl.) 29, 491–504 (1989).
    [CrossRef]
  16. P. C. Hansen, “Relations between SVD and GSVD of discrete regularization problems in standard and general form,” Linear Algebra Appl. 141, 165–176 (1990).
    [CrossRef]
  17. P. C. Hansen, T. Sekii, and H. Shibahashi, “The modified truncated SVD method for regularization in general form,” SIAM J. Sci. Stat. Comput. 13, 1142–1150 (1992).
    [CrossRef]
  18. 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]
  19. P. J. McCarthy, “Direct analytic model of the L-curve for Tikhonov regularization parameter selection,” Inverse Probl. 19, 643–663 (2003).
    [CrossRef]
  20. 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]
  21. S. Twomey, “Comparison of constrained linear inversion and an iterative non-linear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975).
    [CrossRef]
  22. W. Winklmayr, H. Wang, and W. John, “Adaptation of the Twomey Algorithm to the Inversion of Cascade Impactor Data,” Aerosol Sci. Technol. 13, 322–331 (1990).
    [CrossRef]
  23. G. R. Markowski, “Improving Twomey’s Algorithm for Inversion of Aerosol Measurement Data,” Aerosol Sci. Technol 7, 127–141 (1987).
    [CrossRef]
  24. Index of Refraction, Technical Note-007 (Duke Scientific Corporation, December 1, 1996).
  25. “Release on the refractive index of ordinary water substance as a function of wavelength, temperature and pressure” (International Association for the Properties of Water and Steam, 1997).

2010 (1)

2006 (2)

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

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]

2003 (1)

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

1999 (1)

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

1994 (1)

P. C. Hansen, “Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems,” Numerical Algorithms 6, 1–35 (1994).

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]

1992 (3)

A. R. Jones, “Light scattering for particle characterization,” Progr. Energy Combust. Sci. 25, 1–53 (1992).

P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral equations of the first kind,” Inverse Probl. 8, 849–872 (1992).
[CrossRef]

P. C. Hansen, T. Sekii, and H. Shibahashi, “The modified truncated SVD method for regularization in general form,” SIAM J. Sci. Stat. Comput. 13, 1142–1150 (1992).
[CrossRef]

1991 (1)

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

1990 (2)

P. C. Hansen, “Relations between SVD and GSVD of discrete regularization problems in standard and general form,” Linear Algebra Appl. 141, 165–176 (1990).
[CrossRef]

W. Winklmayr, H. Wang, and W. John, “Adaptation of the Twomey Algorithm to the Inversion of Cascade Impactor Data,” Aerosol Sci. Technol. 13, 322–331 (1990).
[CrossRef]

1989 (1)

P. C. Hansen, “Regularization, GSVD and truncated GSVD,” BIT (Nord. Tidskr. Inf.-behandl.) 29, 491–504 (1989).
[CrossRef]

1987 (1)

G. R. Markowski, “Improving Twomey’s Algorithm for Inversion of Aerosol Measurement Data,” Aerosol Sci. Technol 7, 127–141 (1987).
[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. Softw. 10, 282–298 (1984).
[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]

1975 (1)

S. Twomey, “Comparison of constrained linear inversion and an iterative non-linear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975).
[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]

Bohren, C. F.

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

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]

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. Softw. 10, 282–298 (1984).
[CrossRef]

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,” Numerical Algorithms 6, 1–35 (1994).

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]

P. C. Hansen, T. Sekii, and H. Shibahashi, “The modified truncated SVD method for regularization in general form,” SIAM J. Sci. Stat. Comput. 13, 1142–1150 (1992).
[CrossRef]

P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral equations of the first kind,” Inverse Probl. 8, 849–872 (1992).
[CrossRef]

P. C. Hansen, “Relations between SVD and GSVD of discrete regularization problems in standard and general form,” Linear Algebra Appl. 141, 165–176 (1990).
[CrossRef]

P. C. Hansen, “Regularization, GSVD and truncated GSVD,” BIT (Nord. Tidskr. Inf.-behandl.) 29, 491–504 (1989).
[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).

John, W.

W. Winklmayr, H. Wang, and W. John, “Adaptation of the Twomey Algorithm to the Inversion of Cascade Impactor Data,” Aerosol Sci. Technol. 13, 322–331 (1990).
[CrossRef]

Jones, A. R.

A. R. Jones, “Light scattering for particle characterization,” Progr. Energy Combust. Sci. 25, 1–53 (1992).

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. Chemom. 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).

Markowski, G. R.

G. R. Markowski, “Improving Twomey’s Algorithm for Inversion of Aerosol Measurement Data,” Aerosol Sci. Technol 7, 127–141 (1987).
[CrossRef]

McCarthy, P. J.

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

Mroczka, J.

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. Softw. 10, 282–298 (1984).
[CrossRef]

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]

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. Softw. 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]

Sekii, T.

P. C. Hansen, T. Sekii, and H. Shibahashi, “The modified truncated SVD method for regularization in general form,” SIAM J. Sci. Stat. Comput. 13, 1142–1150 (1992).
[CrossRef]

Shibahashi, H.

P. C. Hansen, T. Sekii, and H. Shibahashi, “The modified truncated SVD method for regularization in general form,” SIAM J. Sci. Stat. Comput. 13, 1142–1150 (1992).
[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. Chemom. 20, 22–33 (2006).
[CrossRef]

Szczuczynski, D.

Twomey, S.

S. Twomey, “Comparison of constrained linear inversion and an iterative non-linear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975).
[CrossRef]

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]

Wang, H.

W. Winklmayr, H. Wang, and W. John, “Adaptation of the Twomey Algorithm to the Inversion of Cascade Impactor Data,” Aerosol Sci. Technol. 13, 322–331 (1990).
[CrossRef]

Winklmayr, W.

W. Winklmayr, H. Wang, and W. John, “Adaptation of the Twomey Algorithm to the Inversion of Cascade Impactor Data,” Aerosol Sci. Technol. 13, 322–331 (1990).
[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. Softw. 10, 282–298 (1984).
[CrossRef]

ACM Trans. Math. Softw. (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. Softw. 10, 282–298 (1984).
[CrossRef]

Aerosol Sci. Technol (1)

G. R. Markowski, “Improving Twomey’s Algorithm for Inversion of Aerosol Measurement Data,” Aerosol Sci. Technol 7, 127–141 (1987).
[CrossRef]

Aerosol Sci. Technol. (2)

W. Winklmayr, H. Wang, and W. John, “Adaptation of the Twomey Algorithm to the Inversion of Cascade Impactor Data,” Aerosol Sci. Technol. 13, 322–331 (1990).
[CrossRef]

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]

Appl. Opt. (1)

BIT (Nord. Tidskr. Inf.-behandl.) (1)

P. C. Hansen, “Regularization, GSVD and truncated GSVD,” BIT (Nord. Tidskr. Inf.-behandl.) 29, 491–504 (1989).
[CrossRef]

Inverse Probl. (2)

P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral equations of the first kind,” Inverse Probl. 8, 849–872 (1992).
[CrossRef]

P. J. McCarthy, “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. Chemom. (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. Chemom. 20, 22–33 (2006).
[CrossRef]

J. Comput. Phys. (1)

S. Twomey, “Comparison of constrained linear inversion and an iterative non-linear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975).
[CrossRef]

Linear Algebra Appl. (1)

P. C. Hansen, “Relations between SVD and GSVD of discrete regularization problems in standard and general form,” Linear Algebra Appl. 141, 165–176 (1990).
[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]

Numerical Algorithms (1)

P. C. Hansen, “Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems,” Numerical Algorithms 6, 1–35 (1994).

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]

Progr. Energy Combust. Sci. (1)

A. R. Jones, “Light scattering for particle characterization,” Progr. Energy Combust. Sci. 25, 1–53 (1992).

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]

SIAM J. Sci. Stat. Comput. (1)

P. C. Hansen, T. Sekii, and H. Shibahashi, “The modified truncated SVD method for regularization in general form,” SIAM J. Sci. Stat. Comput. 13, 1142–1150 (1992).
[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 (4)

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

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

Index of Refraction, Technical Note-007 (Duke Scientific Corporation, December 1, 1996).

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

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Figures (8)

Fig. 1.
Fig. 1.

Values of the measure Σrec,δ of the uncertainty of the solutions of the considered inverse problem in spectral extinction obtained during the simulation research for test PSD functions ftest1(a) and ftest2(a) and the measurement data uncertainty levels δ: 0.50% and 2.50%. a. linear scale of the Σrec,δ axis, and b. logarithmic scale of the Σrec,δ axis; mapping of the inversion methods to their indices is given in the text.

Fig. 2.
Fig. 2.

Values of the measure Δrec,δ of the discrepancy from the true (test) PSD function of the solutions of the considered inverse problem in spectral extinction obtained during the simulation research for test PSD functions ftest1(a) and ftest2(a) and the measurement data uncertainty levels δ: 0.50% and 2.50%. a. linear scale of the Δrec,δ axis, b. logarithmic scale of the Δrec,δ axis; mapping of the inversion methods to their indices is given in the text.

Fig. 3.
Fig. 3.

Results of the reconstruction of the unimodal test PSD ftest1(a) based on the simulated measurement data with uncertainty level δ=2.50% with use of regularization techniques A. RLS (Tikhonov regularization), B. CRLS, C. TGSVD, and D. FGSVD, with the following regularizing matrices L: a. identity matrix Iq, b. discrete approximation of the differentiation operator of the 1st order L(q1)×q(1), c. discrete approximation of the differentiation operator of the 2nd order L(q2)×q(2), and d. discrete approximation of the differentiation operator of the 3rd order L(q3)×q(3), with the L-curve criterion applied for selecting the optimal value of the regularization parameter.

Fig. 4.
Fig. 4.

Results of the reconstruction of the unimodal test PSD ftest1(a) based on the simulated measurement data with uncertainty level δ=2.50% with use of regularization techniques A. RLS (Tikhonov regularization), B. CRLS, C. TGSVD, D. FGSVD, with the following regularizing matrices L: a. identity matrix Iq, b. discrete approximation of the differentiation operator of the 1st order L(q1)×q(1), c. discrete approximation of the differentiation operator of the 2nd order L(q2)×q(2), and d. discrete approximation of the differentiation operator of the 3rd order L(q3)×q(3), with the GCV criterion applied for selecting the optimal value of the regularization parameter.

Fig. 5.
Fig. 5.

Results of the reconstruction of the unimodal test PSD ftest1(a) based on the simulated measurement data with uncertainty level δ=2.50% with use of nonlinear iterative inversion techniques A. Twomey method and B. Twomey—Markowski method.

Fig. 6.
Fig. 6.

Results of the reconstruction of the bimodal test PSD ftest2(a) based on the simulated measurement data with uncertainty level δ=2.50% with use of regularization techniques A. RLS (Tikhonov regularization), B. CRLS, C. TGSVD, D. FGSVD, with the following regularizing matrices L: a. identity matrix Iq, b. discrete approximation of the differentiation operator of the 1st order L(q1)×q(1), c. discrete approximation of the differentiation operator of the 2nd order L(q2)×q(2), and d. discrete approximation of the differentiation operator of the 3rd order L(q3)×q(3), with the L-curve criterion applied for selecting the optimal value of the regularization parameter.

Fig. 7.
Fig. 7.

Results of the reconstruction of the bimodal test PSD ftest2(a) based on the simulated measurement data with uncertainty level δ=2.50% with use of regularization techniques A. RLS (Tikhonov regularization), B. CRLS, C. TGSVD, D. FGSVD, with the following regularizing matrices L: a. identity matrix Iq, b. discrete approximation of the differentiation operator of the 1st order L(q1)×q(1), c. discrete approximation of the differentiation operator of the 2nd order L(q2)×q(2), and d. discrete approximation of the differentiation operator of the 3rd order L(q3)×q(3), with the GCV criterion applied for selecting the optimal value of the regularization parameter.

Fig. 8.
Fig. 8.

Results of the reconstruction of the bimodal test PSD ftest2(a) based on the simulated measurement data with uncertainty level δ=2.50% with use of nonlinear iterative inversion techniques A. Twomey method and B. Twomey—Markowski method.

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