Abstract

For the low accuracy of single-scale inversion method in photon correlation spectroscopy technology, a cascadic multigrid (CMG)- truncated singular value decomposition (TSVD) inversion method that combines the TSVD regularization with CMG technology is proposed. This method decomposes the original problem into several subproblems in different scale grid space. According to the particle sizes inverted from the coarsest scale to the finest scale, the solution of an original inversion problem can be obtained. For the inversion of each subproblem, TSVD method is used. The simulation and experimental data were respectively inverted by TSVD and CMG-TSVD methods. The inversion results demonstrate that the CMG-TSVD method has higher accuracy, more strong noise immunity and better smoothness than the TSVD method.

© 2013 Optical Society of America

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References

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  1. B. Chu and T. Liu, “Characterization of nanoparticles by scattering techniques,” J. Nanopart. Res. 2, 29–41 (2000).
    [CrossRef]
  2. R. S. Dias, J. Innerlohinger, and O. Glatter, “Coil-globule transition of DNA molecules induced by cationic surfactants: a dynamic light scattering study,” J. Phys. Chem. B 109, 10458–10463 (2005).
    [CrossRef]
  3. M. Alexander and D. G. Dalgleish, “Dynamic light scattering techniques and their applications in food science,” Food Biophys. 1, 2–13 (2006).
    [CrossRef]
  4. F. Krahl, V. Boyko, and K. F. Arndt, “Characterization of spatial inhomogeneities and dynamic properties of random cross-linked polystyrene networks by dynamic light scattering,” Polymer 51, 2576–2584 (2010).
    [CrossRef]
  5. D. E. Kopple, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
    [CrossRef]
  6. N. Ostrowsky, D. Sornette, P. Parker, and E. R. Pike, “Exponential sampling method for light scattering polydispersity analysis,” J. Mod. Opt. 28, 1059–1070 (1981).
  7. S. W. Provencher, “CONTIN: a general purpose constrained regularization program for inverting noisy linear algebraic and integral equations,” Comput. Phys. Commun. 27, 229–242 (1982).
    [CrossRef]
  8. B. E. Dahneke, Measurement of Suspended Particles by Quasi-Elastic Light Scattering (Wiley Interscience, 1983).
  9. I. D. Morrison and E. F. Grabowski, “Improved techniques for particle size determination by quasi-elastic light scattering,” Langmuir 1, 496–501 (1985).
    [CrossRef]
  10. J. G. McWhirter and E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” Phys. A 11, 1729–1745 (1978).
    [CrossRef]
  11. L. M. Gugliotta, G. S. Stegmayer, and L. A. Clementi, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
    [CrossRef]
  12. S. Li, “Inversion of particle size distribution from dynamic light scattering data with gray-code genetic algorithm,” Chin. J. Comput. Phys. 25, 323–329 (2008).
  13. X. J. Zhu, J. Shen, W. Liu, X. M. Sun, and Y. J. Wang, “Nonnegative least-squares truncated singular value decomposition to particle size distribution inversion from dynamic light scattering data,” Appl. Opt. 49, 6591–6596 (2010).
    [CrossRef]
  14. K. Stuben, “A review of algebraic multigrid,” J. Comput. Appl. Math. 128, 281–309 (2001).
    [CrossRef]
  15. F. A. Bornemann and P. Deuflhard, “The cascadic multigrid method for elliptic problems,” Numer. Math. 42, 917–924 (1996).
  16. F. A. Bornemann and R. Krause, “Classical and cascadic multigrid-A methodogical comparison,” Proceedings of the 9th International Conference on Domain Decomposition (Wiley, 1998), pp. 64–71.
  17. F. Bornemann and P. Deuflhard, “The cascadic multi-grid method,” The Eighth International Conference on Domain Decomposition Method for Partial Differential Equations(Wiley, 1997), pp. 205–212.
  18. P. C. Hansen, “Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems,” Numer. Algorithms 6, 1–35 (1994).
    [CrossRef]
  19. P. C. Hansen, “Analysis of discrete ill-posed problems by means of the l-curve,” SIAM Rev. 34, 561–580 (1992).
    [CrossRef]
  20. K. Miller, “Least squares methods for ill-posed problems with a prescribed bound,” SIAM J. Math. Anal. 1, 52–74 (1970).
    [CrossRef]
  21. A. B. Yu and N. Standish, “A study of particle size distribution,” Power Technol. 62, 101–118 (1990).
    [CrossRef]
  22. T. F. Coleman and Y. Li, “An interior trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim. 6, 418–445 (1996).
    [CrossRef]
  23. J. Wu, M. Li, and T. Yu, “Ill matrix and regularization method in surveying data processing,” J. Geodesy Geodyn. 30, 102–105 (2010).
  24. W. Liu, J. Shen, and X. Sun, “Design of multiple-tau photon correlation system implemented by FPGA,” in Proceedings of The International Conference on Embedded Software and Systems (IEEE, 2008), pp. 410–414.
  25. W. Liu, J. Shen, Y. Cheng, and W. Chen, “Novel photon correlator with less hardware resource,” Proc. SPIE 7283, 72833B (2009).
    [CrossRef]

2010

F. Krahl, V. Boyko, and K. F. Arndt, “Characterization of spatial inhomogeneities and dynamic properties of random cross-linked polystyrene networks by dynamic light scattering,” Polymer 51, 2576–2584 (2010).
[CrossRef]

X. J. Zhu, J. Shen, W. Liu, X. M. Sun, and Y. J. Wang, “Nonnegative least-squares truncated singular value decomposition to particle size distribution inversion from dynamic light scattering data,” Appl. Opt. 49, 6591–6596 (2010).
[CrossRef]

J. Wu, M. Li, and T. Yu, “Ill matrix and regularization method in surveying data processing,” J. Geodesy Geodyn. 30, 102–105 (2010).

2009

W. Liu, J. Shen, Y. Cheng, and W. Chen, “Novel photon correlator with less hardware resource,” Proc. SPIE 7283, 72833B (2009).
[CrossRef]

L. M. Gugliotta, G. S. Stegmayer, and L. A. Clementi, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
[CrossRef]

2008

S. Li, “Inversion of particle size distribution from dynamic light scattering data with gray-code genetic algorithm,” Chin. J. Comput. Phys. 25, 323–329 (2008).

2006

M. Alexander and D. G. Dalgleish, “Dynamic light scattering techniques and their applications in food science,” Food Biophys. 1, 2–13 (2006).
[CrossRef]

2005

R. S. Dias, J. Innerlohinger, and O. Glatter, “Coil-globule transition of DNA molecules induced by cationic surfactants: a dynamic light scattering study,” J. Phys. Chem. B 109, 10458–10463 (2005).
[CrossRef]

2001

K. Stuben, “A review of algebraic multigrid,” J. Comput. Appl. Math. 128, 281–309 (2001).
[CrossRef]

2000

B. Chu and T. Liu, “Characterization of nanoparticles by scattering techniques,” J. Nanopart. Res. 2, 29–41 (2000).
[CrossRef]

1996

F. A. Bornemann and P. Deuflhard, “The cascadic multigrid method for elliptic problems,” Numer. Math. 42, 917–924 (1996).

T. F. Coleman and Y. Li, “An interior trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim. 6, 418–445 (1996).
[CrossRef]

1994

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

1992

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the l-curve,” SIAM Rev. 34, 561–580 (1992).
[CrossRef]

1990

A. B. Yu and N. Standish, “A study of particle size distribution,” Power Technol. 62, 101–118 (1990).
[CrossRef]

1985

I. D. Morrison and E. F. Grabowski, “Improved techniques for particle size determination by quasi-elastic light scattering,” Langmuir 1, 496–501 (1985).
[CrossRef]

1982

S. W. Provencher, “CONTIN: a general purpose constrained regularization program for inverting noisy linear algebraic and integral equations,” Comput. Phys. Commun. 27, 229–242 (1982).
[CrossRef]

1981

N. Ostrowsky, D. Sornette, P. Parker, and E. R. Pike, “Exponential sampling method for light scattering polydispersity analysis,” J. Mod. Opt. 28, 1059–1070 (1981).

1978

J. G. McWhirter and E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” Phys. A 11, 1729–1745 (1978).
[CrossRef]

1972

D. E. Kopple, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
[CrossRef]

1970

K. Miller, “Least squares methods for ill-posed problems with a prescribed bound,” SIAM J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

Alexander, M.

M. Alexander and D. G. Dalgleish, “Dynamic light scattering techniques and their applications in food science,” Food Biophys. 1, 2–13 (2006).
[CrossRef]

Arndt, K. F.

F. Krahl, V. Boyko, and K. F. Arndt, “Characterization of spatial inhomogeneities and dynamic properties of random cross-linked polystyrene networks by dynamic light scattering,” Polymer 51, 2576–2584 (2010).
[CrossRef]

Bornemann, F.

F. Bornemann and P. Deuflhard, “The cascadic multi-grid method,” The Eighth International Conference on Domain Decomposition Method for Partial Differential Equations(Wiley, 1997), pp. 205–212.

Bornemann, F. A.

F. A. Bornemann and P. Deuflhard, “The cascadic multigrid method for elliptic problems,” Numer. Math. 42, 917–924 (1996).

F. A. Bornemann and R. Krause, “Classical and cascadic multigrid-A methodogical comparison,” Proceedings of the 9th International Conference on Domain Decomposition (Wiley, 1998), pp. 64–71.

Boyko, V.

F. Krahl, V. Boyko, and K. F. Arndt, “Characterization of spatial inhomogeneities and dynamic properties of random cross-linked polystyrene networks by dynamic light scattering,” Polymer 51, 2576–2584 (2010).
[CrossRef]

Chen, W.

W. Liu, J. Shen, Y. Cheng, and W. Chen, “Novel photon correlator with less hardware resource,” Proc. SPIE 7283, 72833B (2009).
[CrossRef]

Cheng, Y.

W. Liu, J. Shen, Y. Cheng, and W. Chen, “Novel photon correlator with less hardware resource,” Proc. SPIE 7283, 72833B (2009).
[CrossRef]

Chu, B.

B. Chu and T. Liu, “Characterization of nanoparticles by scattering techniques,” J. Nanopart. Res. 2, 29–41 (2000).
[CrossRef]

Clementi, L. A.

L. M. Gugliotta, G. S. Stegmayer, and L. A. Clementi, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
[CrossRef]

Coleman, T. F.

T. F. Coleman and Y. Li, “An interior trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim. 6, 418–445 (1996).
[CrossRef]

Dahneke, B. E.

B. E. Dahneke, Measurement of Suspended Particles by Quasi-Elastic Light Scattering (Wiley Interscience, 1983).

Dalgleish, D. G.

M. Alexander and D. G. Dalgleish, “Dynamic light scattering techniques and their applications in food science,” Food Biophys. 1, 2–13 (2006).
[CrossRef]

Deuflhard, P.

F. A. Bornemann and P. Deuflhard, “The cascadic multigrid method for elliptic problems,” Numer. Math. 42, 917–924 (1996).

F. Bornemann and P. Deuflhard, “The cascadic multi-grid method,” The Eighth International Conference on Domain Decomposition Method for Partial Differential Equations(Wiley, 1997), pp. 205–212.

Dias, R. S.

R. S. Dias, J. Innerlohinger, and O. Glatter, “Coil-globule transition of DNA molecules induced by cationic surfactants: a dynamic light scattering study,” J. Phys. Chem. B 109, 10458–10463 (2005).
[CrossRef]

Glatter, O.

R. S. Dias, J. Innerlohinger, and O. Glatter, “Coil-globule transition of DNA molecules induced by cationic surfactants: a dynamic light scattering study,” J. Phys. Chem. B 109, 10458–10463 (2005).
[CrossRef]

Grabowski, E. F.

I. D. Morrison and E. F. Grabowski, “Improved techniques for particle size determination by quasi-elastic light scattering,” Langmuir 1, 496–501 (1985).
[CrossRef]

Gugliotta, L. M.

L. M. Gugliotta, G. S. Stegmayer, and L. A. Clementi, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
[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, “Analysis of discrete ill-posed problems by means of the l-curve,” SIAM Rev. 34, 561–580 (1992).
[CrossRef]

Innerlohinger, J.

R. S. Dias, J. Innerlohinger, and O. Glatter, “Coil-globule transition of DNA molecules induced by cationic surfactants: a dynamic light scattering study,” J. Phys. Chem. B 109, 10458–10463 (2005).
[CrossRef]

Kopple, D. E.

D. E. Kopple, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
[CrossRef]

Krahl, F.

F. Krahl, V. Boyko, and K. F. Arndt, “Characterization of spatial inhomogeneities and dynamic properties of random cross-linked polystyrene networks by dynamic light scattering,” Polymer 51, 2576–2584 (2010).
[CrossRef]

Krause, R.

F. A. Bornemann and R. Krause, “Classical and cascadic multigrid-A methodogical comparison,” Proceedings of the 9th International Conference on Domain Decomposition (Wiley, 1998), pp. 64–71.

Li, M.

J. Wu, M. Li, and T. Yu, “Ill matrix and regularization method in surveying data processing,” J. Geodesy Geodyn. 30, 102–105 (2010).

Li, S.

S. Li, “Inversion of particle size distribution from dynamic light scattering data with gray-code genetic algorithm,” Chin. J. Comput. Phys. 25, 323–329 (2008).

Li, Y.

T. F. Coleman and Y. Li, “An interior trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim. 6, 418–445 (1996).
[CrossRef]

Liu, T.

B. Chu and T. Liu, “Characterization of nanoparticles by scattering techniques,” J. Nanopart. Res. 2, 29–41 (2000).
[CrossRef]

Liu, W.

X. J. Zhu, J. Shen, W. Liu, X. M. Sun, and Y. J. Wang, “Nonnegative least-squares truncated singular value decomposition to particle size distribution inversion from dynamic light scattering data,” Appl. Opt. 49, 6591–6596 (2010).
[CrossRef]

W. Liu, J. Shen, Y. Cheng, and W. Chen, “Novel photon correlator with less hardware resource,” Proc. SPIE 7283, 72833B (2009).
[CrossRef]

W. Liu, J. Shen, and X. Sun, “Design of multiple-tau photon correlation system implemented by FPGA,” in Proceedings of The International Conference on Embedded Software and Systems (IEEE, 2008), pp. 410–414.

McWhirter, J. G.

J. G. McWhirter and E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” Phys. A 11, 1729–1745 (1978).
[CrossRef]

Miller, K.

K. Miller, “Least squares methods for ill-posed problems with a prescribed bound,” SIAM J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

Morrison, I. D.

I. D. Morrison and E. F. Grabowski, “Improved techniques for particle size determination by quasi-elastic light scattering,” Langmuir 1, 496–501 (1985).
[CrossRef]

Ostrowsky, N.

N. Ostrowsky, D. Sornette, P. Parker, and E. R. Pike, “Exponential sampling method for light scattering polydispersity analysis,” J. Mod. Opt. 28, 1059–1070 (1981).

Parker, P.

N. Ostrowsky, D. Sornette, P. Parker, and E. R. Pike, “Exponential sampling method for light scattering polydispersity analysis,” J. Mod. Opt. 28, 1059–1070 (1981).

Pike, E. R.

N. Ostrowsky, D. Sornette, P. Parker, and E. R. Pike, “Exponential sampling method for light scattering polydispersity analysis,” J. Mod. Opt. 28, 1059–1070 (1981).

J. G. McWhirter and E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” Phys. A 11, 1729–1745 (1978).
[CrossRef]

Provencher, S. W.

S. W. Provencher, “CONTIN: a general purpose constrained regularization program for inverting noisy linear algebraic and integral equations,” Comput. Phys. Commun. 27, 229–242 (1982).
[CrossRef]

Shen, J.

X. J. Zhu, J. Shen, W. Liu, X. M. Sun, and Y. J. Wang, “Nonnegative least-squares truncated singular value decomposition to particle size distribution inversion from dynamic light scattering data,” Appl. Opt. 49, 6591–6596 (2010).
[CrossRef]

W. Liu, J. Shen, Y. Cheng, and W. Chen, “Novel photon correlator with less hardware resource,” Proc. SPIE 7283, 72833B (2009).
[CrossRef]

W. Liu, J. Shen, and X. Sun, “Design of multiple-tau photon correlation system implemented by FPGA,” in Proceedings of The International Conference on Embedded Software and Systems (IEEE, 2008), pp. 410–414.

Sornette, D.

N. Ostrowsky, D. Sornette, P. Parker, and E. R. Pike, “Exponential sampling method for light scattering polydispersity analysis,” J. Mod. Opt. 28, 1059–1070 (1981).

Standish, N.

A. B. Yu and N. Standish, “A study of particle size distribution,” Power Technol. 62, 101–118 (1990).
[CrossRef]

Stegmayer, G. S.

L. M. Gugliotta, G. S. Stegmayer, and L. A. Clementi, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
[CrossRef]

Stuben, K.

K. Stuben, “A review of algebraic multigrid,” J. Comput. Appl. Math. 128, 281–309 (2001).
[CrossRef]

Sun, X.

W. Liu, J. Shen, and X. Sun, “Design of multiple-tau photon correlation system implemented by FPGA,” in Proceedings of The International Conference on Embedded Software and Systems (IEEE, 2008), pp. 410–414.

Sun, X. M.

Wang, Y. J.

Wu, J.

J. Wu, M. Li, and T. Yu, “Ill matrix and regularization method in surveying data processing,” J. Geodesy Geodyn. 30, 102–105 (2010).

Yu, A. B.

A. B. Yu and N. Standish, “A study of particle size distribution,” Power Technol. 62, 101–118 (1990).
[CrossRef]

Yu, T.

J. Wu, M. Li, and T. Yu, “Ill matrix and regularization method in surveying data processing,” J. Geodesy Geodyn. 30, 102–105 (2010).

Zhu, X. J.

Appl. Opt.

Chin. J. Comput. Phys.

S. Li, “Inversion of particle size distribution from dynamic light scattering data with gray-code genetic algorithm,” Chin. J. Comput. Phys. 25, 323–329 (2008).

Comput. Phys. Commun.

S. W. Provencher, “CONTIN: a general purpose constrained regularization program for inverting noisy linear algebraic and integral equations,” Comput. Phys. Commun. 27, 229–242 (1982).
[CrossRef]

Food Biophys.

M. Alexander and D. G. Dalgleish, “Dynamic light scattering techniques and their applications in food science,” Food Biophys. 1, 2–13 (2006).
[CrossRef]

J. Chem. Phys.

D. E. Kopple, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
[CrossRef]

J. Comput. Appl. Math.

K. Stuben, “A review of algebraic multigrid,” J. Comput. Appl. Math. 128, 281–309 (2001).
[CrossRef]

J. Geodesy Geodyn.

J. Wu, M. Li, and T. Yu, “Ill matrix and regularization method in surveying data processing,” J. Geodesy Geodyn. 30, 102–105 (2010).

J. Mod. Opt.

N. Ostrowsky, D. Sornette, P. Parker, and E. R. Pike, “Exponential sampling method for light scattering polydispersity analysis,” J. Mod. Opt. 28, 1059–1070 (1981).

J. Nanopart. Res.

B. Chu and T. Liu, “Characterization of nanoparticles by scattering techniques,” J. Nanopart. Res. 2, 29–41 (2000).
[CrossRef]

J. Phys. Chem. B

R. S. Dias, J. Innerlohinger, and O. Glatter, “Coil-globule transition of DNA molecules induced by cationic surfactants: a dynamic light scattering study,” J. Phys. Chem. B 109, 10458–10463 (2005).
[CrossRef]

Langmuir

I. D. Morrison and E. F. Grabowski, “Improved techniques for particle size determination by quasi-elastic light scattering,” Langmuir 1, 496–501 (1985).
[CrossRef]

Numer. Algorithms

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.

F. A. Bornemann and P. Deuflhard, “The cascadic multigrid method for elliptic problems,” Numer. Math. 42, 917–924 (1996).

Part. Part. Syst. Charact.

L. M. Gugliotta, G. S. Stegmayer, and L. A. Clementi, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
[CrossRef]

Phys. A

J. G. McWhirter and E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” Phys. A 11, 1729–1745 (1978).
[CrossRef]

Polymer

F. Krahl, V. Boyko, and K. F. Arndt, “Characterization of spatial inhomogeneities and dynamic properties of random cross-linked polystyrene networks by dynamic light scattering,” Polymer 51, 2576–2584 (2010).
[CrossRef]

Power Technol.

A. B. Yu and N. Standish, “A study of particle size distribution,” Power Technol. 62, 101–118 (1990).
[CrossRef]

Proc. SPIE

W. Liu, J. Shen, Y. Cheng, and W. Chen, “Novel photon correlator with less hardware resource,” Proc. SPIE 7283, 72833B (2009).
[CrossRef]

SIAM J. Math. Anal.

K. Miller, “Least squares methods for ill-posed problems with a prescribed bound,” SIAM J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

SIAM J. Optim.

T. F. Coleman and Y. Li, “An interior trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim. 6, 418–445 (1996).
[CrossRef]

SIAM Rev.

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the l-curve,” SIAM Rev. 34, 561–580 (1992).
[CrossRef]

Other

W. Liu, J. Shen, and X. Sun, “Design of multiple-tau photon correlation system implemented by FPGA,” in Proceedings of The International Conference on Embedded Software and Systems (IEEE, 2008), pp. 410–414.

B. E. Dahneke, Measurement of Suspended Particles by Quasi-Elastic Light Scattering (Wiley Interscience, 1983).

F. A. Bornemann and R. Krause, “Classical and cascadic multigrid-A methodogical comparison,” Proceedings of the 9th International Conference on Domain Decomposition (Wiley, 1998), pp. 64–71.

F. Bornemann and P. Deuflhard, “The cascadic multi-grid method,” The Eighth International Conference on Domain Decomposition Method for Partial Differential Equations(Wiley, 1997), pp. 205–212.

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Figures (8)

Fig. 1.
Fig. 1.

Cascadic multigrid diagram.

Fig. 2.
Fig. 2.

Inversion PSD of noise-free ACF on different scale (a) scale=3, (b) scale=2, and (c) scale=1.

Fig. 3.
Fig. 3.

Inversion PSD of noise-free ACF on different scale (a) scale=3, (b) scale=2, and (c) scale=0.

Fig. 4.
Fig. 4.

Inversion PSD comparisons of unimodal distribution particles at different noises levels (a) noise level=0, (b) noise level=0.005, and (c) noise level=0.01.

Fig. 5.
Fig. 5.

Inversion PSD comparisons of bimodal distribution particles at different noises levels (a) noise level=0, (b) noise level=0.005, and (c) noise level=0.01.

Fig. 6.
Fig. 6.

Singular values of each subproblem (a) unimodal distribution particles and (b) bimodal distribution particles.

Fig. 7.
Fig. 7.

Inversion PSD of 300 nm unimodal particles.

Fig. 8.
Fig. 8.

Inversion PSD of 60–200 nm bimodal particles.

Tables (3)

Tables Icon

Table 1. Inversion Data of 200–400 nm Unimodal Distribution Particles

Tables Icon

Table 2. Inversion Data of 100–850 nm Bimodal Distribution Particles

Tables Icon

Table 3. Inversion PSD Data of 300 nm Unimodal Particles and 60–200 nm Bimodal Particles

Equations (20)

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

Ax=b,
Aixi=bi,
g(τ)=0G(Γ)exp(2Γτ)dΓ,
Γ=Dq2,q=4πmλ0sin(θ2),D=kBT3πηd,
g(τ)=i=1NG(Γi)exp(2Γiτ).
Ax=b,
Axb22=min=xLS.
A=UVT=i=1nuiσiviT,
xLS=i=1nui,bσivi.
Ak=i=1kuiσiviT,
Akx=b.
xTSVD=i=1kui,bσivi.
fi={1σi>λ0σi<λ,
minAkxb22.
Ai+1=AiIi+1i,
Iii+1=[10000000001000000000100000000010].
{x2ji=xji+1x2j+1i=12(xji+1+xj+1i+1).
f(α)=σ2π(αmaxαmin)[t(1t)]1exp{0.5[u+σln(t1t)]2},
t=ααminαmaxαmin.
relative error=xxtheory2xtheory2.

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