Abstract

We propose a self-regularized pseudo-time marching scheme to solve the ill-posed, nonlinear inverse problem associated with diffuse propagation of coherent light in a tissuelike object. In particular, in the context of diffuse correlation tomography (DCT), we consider the recovery of mechanical property distributions from partial and noisy boundary measurements of light intensity autocorrelation. We prove the existence of a minimizer for the Newton algorithm after establishing the existence of weak solutions for the forward equation of light amplitude autocorrelation and its Fréchet derivative and adjoint. The asymptotic stability of the solution of the ordinary differential equation obtained through the introduction of the pseudo-time is also analyzed. We show that the asymptotic solution obtained through the pseudo-time marching converges to that optimal solution provided the Hessian of the forward equation is positive definite in the neighborhood of optimal solution. The superior noise tolerance and regularization-insensitive nature of pseudo-dynamic strategy are proved through numerical simulations in the context of both DCT and diffuse optical tomography.

© 2010 Optical Society of America

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References

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  1. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1-R43 (2005).
    [CrossRef] [PubMed]
  2. D. A. Boas, D. H. Brooks, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang,“Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
    [CrossRef]
  3. C. Zhou, G. Yu, D. Furuya, J. H. Greenberg, A. G. Yodh and T. Durduran, “Diffuse correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 3, 1125-1144 (2006).
    [CrossRef]
  4. H. M. Varma, A. K. Nandakumaran, and R. M. Vasu, “Study of turbid media with light: recovery of mechanical and optical properties from boundary measurement of intensity autocorrelation of light,” J. Opt. Soc. Am. A 26, 1472-1483 (2009).
    [CrossRef]
  5. S. Sakadzic and L. V. Wang, “Correlation transfer and diffusion of ultrasound-modulated multiply scattered light,” Phys. Rev. Lett. 96, 163902 (2006).
    [CrossRef] [PubMed]
  6. A. N. Tikhonov and V. Y. Arsenin, Solution of Illposed Problems (Winston, 1997).
  7. K. Levenberg,“A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math. 2, 164-168 (1944).
  8. D. W. Marquardt,“An algorithm for least squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11, 431-441 (1963).
    [CrossRef]
  9. B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamical systems approach to a class of inverse problems in engineering,” Proc. R. Soc. London, Ser. A 465, 1561-1579 (2009).
    [CrossRef]
  10. V. Kolehmainen, S. Prince, S. R. Arridge, and J. P. Kaipio, “State-estimation approach to the nonstationary optical tomography problem,” J. Opt. Soc. Am. A 20, 876-889 (2003).
    [CrossRef]
  11. B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285-305 (2009).
    [CrossRef]
  12. D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffuse temporal field correlation,” Phys. Rev. Lett. 75, 1855-1858 (1995).
    [CrossRef] [PubMed]
  13. C. Usha Devi, R. S. Bharat Chandran, R. M. Vasu and A. K. Sood, “Measurement of visco-elastic properties of breast tissue mimicking materials using diffusing wave spectroscopy,” J. Biomed. Opt. 12, 034035 (2007).
    [CrossRef] [PubMed]
  14. S. Kesavan, Topics in Functional Analysis and Applications (New Age International, 2008).
  15. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer-Verlag, 1983).
  16. D. C. Dobson, “Convergence of a reconstruction method for the inverse conductivity problem,” SIAM J. Appl. Math. 52, 442-458 (1992).
    [CrossRef]
  17. J. Cea, Lectures on Optimization-Theory and Algorithms (Springer Verlag, 1978).
  18. D. Roy, “A numeric-analytic technique for non-linear deterministic and stochastic dynamical systems,” Proc. R. Soc. London, Ser. A 457, 539-566 (2001).
    [CrossRef]
  19. D. Roy, “Phase space linearization for non-linear oscillators: deterministic and stochastic systems,” J. Sound Vib. 231, 307-341 (2000).
    [CrossRef]
  20. M. C. Valsakumar, “Solution of Fokker-Planck equation using Trotter's formula,” J. Stat. Phys. 32, 545-553 (1983).
    [CrossRef]
  21. D. C. Dobson, “Phase reconstruction via nonlinear least-squares,” Inverse Probl. 8, 541-558 (1991).
    [CrossRef]

2009

H. M. Varma, A. K. Nandakumaran, and R. M. Vasu, “Study of turbid media with light: recovery of mechanical and optical properties from boundary measurement of intensity autocorrelation of light,” J. Opt. Soc. Am. A 26, 1472-1483 (2009).
[CrossRef]

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamical systems approach to a class of inverse problems in engineering,” Proc. R. Soc. London, Ser. A 465, 1561-1579 (2009).
[CrossRef]

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285-305 (2009).
[CrossRef]

2007

C. Usha Devi, R. S. Bharat Chandran, R. M. Vasu and A. K. Sood, “Measurement of visco-elastic properties of breast tissue mimicking materials using diffusing wave spectroscopy,” J. Biomed. Opt. 12, 034035 (2007).
[CrossRef] [PubMed]

2006

C. Zhou, G. Yu, D. Furuya, J. H. Greenberg, A. G. Yodh and T. Durduran, “Diffuse correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 3, 1125-1144 (2006).
[CrossRef]

S. Sakadzic and L. V. Wang, “Correlation transfer and diffusion of ultrasound-modulated multiply scattered light,” Phys. Rev. Lett. 96, 163902 (2006).
[CrossRef] [PubMed]

2005

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

2003

2001

D. A. Boas, D. H. Brooks, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang,“Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

D. Roy, “A numeric-analytic technique for non-linear deterministic and stochastic dynamical systems,” Proc. R. Soc. London, Ser. A 457, 539-566 (2001).
[CrossRef]

2000

D. Roy, “Phase space linearization for non-linear oscillators: deterministic and stochastic systems,” J. Sound Vib. 231, 307-341 (2000).
[CrossRef]

1995

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffuse temporal field correlation,” Phys. Rev. Lett. 75, 1855-1858 (1995).
[CrossRef] [PubMed]

1992

D. C. Dobson, “Convergence of a reconstruction method for the inverse conductivity problem,” SIAM J. Appl. Math. 52, 442-458 (1992).
[CrossRef]

1991

D. C. Dobson, “Phase reconstruction via nonlinear least-squares,” Inverse Probl. 8, 541-558 (1991).
[CrossRef]

1983

M. C. Valsakumar, “Solution of Fokker-Planck equation using Trotter's formula,” J. Stat. Phys. 32, 545-553 (1983).
[CrossRef]

1963

D. W. Marquardt,“An algorithm for least squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11, 431-441 (1963).
[CrossRef]

1944

K. Levenberg,“A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math. 2, 164-168 (1944).

Arridge, S. R.

Arsenin, V. Y.

A. N. Tikhonov and V. Y. Arsenin, Solution of Illposed Problems (Winston, 1997).

Banerjee, B.

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamical systems approach to a class of inverse problems in engineering,” Proc. R. Soc. London, Ser. A 465, 1561-1579 (2009).
[CrossRef]

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285-305 (2009).
[CrossRef]

Bharat Chandran, R. S.

C. Usha Devi, R. S. Bharat Chandran, R. M. Vasu and A. K. Sood, “Measurement of visco-elastic properties of breast tissue mimicking materials using diffusing wave spectroscopy,” J. Biomed. Opt. 12, 034035 (2007).
[CrossRef] [PubMed]

Boas, D. A.

D. A. Boas, D. H. Brooks, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang,“Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffuse temporal field correlation,” Phys. Rev. Lett. 75, 1855-1858 (1995).
[CrossRef] [PubMed]

Brooks, D. H.

D. A. Boas, D. H. Brooks, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang,“Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

Campbell, L. E.

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffuse temporal field correlation,” Phys. Rev. Lett. 75, 1855-1858 (1995).
[CrossRef] [PubMed]

Cea, J.

J. Cea, Lectures on Optimization-Theory and Algorithms (Springer Verlag, 1978).

Dimarzio, C. A.

D. A. Boas, D. H. Brooks, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang,“Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

Dobson, D. C.

D. C. Dobson, “Convergence of a reconstruction method for the inverse conductivity problem,” SIAM J. Appl. Math. 52, 442-458 (1992).
[CrossRef]

D. C. Dobson, “Phase reconstruction via nonlinear least-squares,” Inverse Probl. 8, 541-558 (1991).
[CrossRef]

Durduran, T.

C. Zhou, G. Yu, D. Furuya, J. H. Greenberg, A. G. Yodh and T. Durduran, “Diffuse correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 3, 1125-1144 (2006).
[CrossRef]

Furuya, D.

C. Zhou, G. Yu, D. Furuya, J. H. Greenberg, A. G. Yodh and T. Durduran, “Diffuse correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 3, 1125-1144 (2006).
[CrossRef]

Gaudette, R. J.

D. A. Boas, D. H. Brooks, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang,“Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

Gibson, A. P.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

Gilbarg, D.

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer-Verlag, 1983).

Greenberg, J. H.

C. Zhou, G. Yu, D. Furuya, J. H. Greenberg, A. G. Yodh and T. Durduran, “Diffuse correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 3, 1125-1144 (2006).
[CrossRef]

Hebden, J. C.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

Kaipio, J. P.

Kesavan, S.

S. Kesavan, Topics in Functional Analysis and Applications (New Age International, 2008).

Kilmer, M.

D. A. Boas, D. H. Brooks, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang,“Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

Kolehmainen, V.

Levenberg, K.

K. Levenberg,“A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math. 2, 164-168 (1944).

Marquardt, D. W.

D. W. Marquardt,“An algorithm for least squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11, 431-441 (1963).
[CrossRef]

Nandakumaran, A. K.

Prince, S.

Roy, D.

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamical systems approach to a class of inverse problems in engineering,” Proc. R. Soc. London, Ser. A 465, 1561-1579 (2009).
[CrossRef]

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285-305 (2009).
[CrossRef]

D. Roy, “A numeric-analytic technique for non-linear deterministic and stochastic dynamical systems,” Proc. R. Soc. London, Ser. A 457, 539-566 (2001).
[CrossRef]

D. Roy, “Phase space linearization for non-linear oscillators: deterministic and stochastic systems,” J. Sound Vib. 231, 307-341 (2000).
[CrossRef]

Sakadzic, S.

S. Sakadzic and L. V. Wang, “Correlation transfer and diffusion of ultrasound-modulated multiply scattered light,” Phys. Rev. Lett. 96, 163902 (2006).
[CrossRef] [PubMed]

Sood, A. K.

C. Usha Devi, R. S. Bharat Chandran, R. M. Vasu and A. K. Sood, “Measurement of visco-elastic properties of breast tissue mimicking materials using diffusing wave spectroscopy,” J. Biomed. Opt. 12, 034035 (2007).
[CrossRef] [PubMed]

Tikhonov, A. N.

A. N. Tikhonov and V. Y. Arsenin, Solution of Illposed Problems (Winston, 1997).

Trudinger, N. S.

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer-Verlag, 1983).

Usha Devi, C.

C. Usha Devi, R. S. Bharat Chandran, R. M. Vasu and A. K. Sood, “Measurement of visco-elastic properties of breast tissue mimicking materials using diffusing wave spectroscopy,” J. Biomed. Opt. 12, 034035 (2007).
[CrossRef] [PubMed]

Valsakumar, M. C.

M. C. Valsakumar, “Solution of Fokker-Planck equation using Trotter's formula,” J. Stat. Phys. 32, 545-553 (1983).
[CrossRef]

Varma, H. M.

Vasu, R. M.

H. M. Varma, A. K. Nandakumaran, and R. M. Vasu, “Study of turbid media with light: recovery of mechanical and optical properties from boundary measurement of intensity autocorrelation of light,” J. Opt. Soc. Am. A 26, 1472-1483 (2009).
[CrossRef]

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamical systems approach to a class of inverse problems in engineering,” Proc. R. Soc. London, Ser. A 465, 1561-1579 (2009).
[CrossRef]

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285-305 (2009).
[CrossRef]

C. Usha Devi, R. S. Bharat Chandran, R. M. Vasu and A. K. Sood, “Measurement of visco-elastic properties of breast tissue mimicking materials using diffusing wave spectroscopy,” J. Biomed. Opt. 12, 034035 (2007).
[CrossRef] [PubMed]

Wang, L. V.

S. Sakadzic and L. V. Wang, “Correlation transfer and diffusion of ultrasound-modulated multiply scattered light,” Phys. Rev. Lett. 96, 163902 (2006).
[CrossRef] [PubMed]

Yodh, A. G.

C. Zhou, G. Yu, D. Furuya, J. H. Greenberg, A. G. Yodh and T. Durduran, “Diffuse correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 3, 1125-1144 (2006).
[CrossRef]

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffuse temporal field correlation,” Phys. Rev. Lett. 75, 1855-1858 (1995).
[CrossRef] [PubMed]

Yu, G.

C. Zhou, G. Yu, D. Furuya, J. H. Greenberg, A. G. Yodh and T. Durduran, “Diffuse correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 3, 1125-1144 (2006).
[CrossRef]

Zhang, Q.

D. A. Boas, D. H. Brooks, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang,“Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

Zhou, C.

C. Zhou, G. Yu, D. Furuya, J. H. Greenberg, A. G. Yodh and T. Durduran, “Diffuse correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 3, 1125-1144 (2006).
[CrossRef]

IEEE Signal Process. Mag.

D. A. Boas, D. H. Brooks, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang,“Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

Inverse Probl.

D. C. Dobson, “Phase reconstruction via nonlinear least-squares,” Inverse Probl. 8, 541-558 (1991).
[CrossRef]

J. Biomed. Opt.

C. Usha Devi, R. S. Bharat Chandran, R. M. Vasu and A. K. Sood, “Measurement of visco-elastic properties of breast tissue mimicking materials using diffusing wave spectroscopy,” J. Biomed. Opt. 12, 034035 (2007).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

J. Soc. Ind. Appl. Math.

D. W. Marquardt,“An algorithm for least squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11, 431-441 (1963).
[CrossRef]

J. Sound Vib.

D. Roy, “Phase space linearization for non-linear oscillators: deterministic and stochastic systems,” J. Sound Vib. 231, 307-341 (2000).
[CrossRef]

J. Stat. Phys.

M. C. Valsakumar, “Solution of Fokker-Planck equation using Trotter's formula,” J. Stat. Phys. 32, 545-553 (1983).
[CrossRef]

Opt. Express

C. Zhou, G. Yu, D. Furuya, J. H. Greenberg, A. G. Yodh and T. Durduran, “Diffuse correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 3, 1125-1144 (2006).
[CrossRef]

Phys. Med. Biol.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285-305 (2009).
[CrossRef]

Phys. Rev. Lett.

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffuse temporal field correlation,” Phys. Rev. Lett. 75, 1855-1858 (1995).
[CrossRef] [PubMed]

S. Sakadzic and L. V. Wang, “Correlation transfer and diffusion of ultrasound-modulated multiply scattered light,” Phys. Rev. Lett. 96, 163902 (2006).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamical systems approach to a class of inverse problems in engineering,” Proc. R. Soc. London, Ser. A 465, 1561-1579 (2009).
[CrossRef]

D. Roy, “A numeric-analytic technique for non-linear deterministic and stochastic dynamical systems,” Proc. R. Soc. London, Ser. A 457, 539-566 (2001).
[CrossRef]

Q. Appl. Math.

K. Levenberg,“A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math. 2, 164-168 (1944).

SIAM J. Appl. Math.

D. C. Dobson, “Convergence of a reconstruction method for the inverse conductivity problem,” SIAM J. Appl. Math. 52, 442-458 (1992).
[CrossRef]

Other

J. Cea, Lectures on Optimization-Theory and Algorithms (Springer Verlag, 1978).

S. Kesavan, Topics in Functional Analysis and Applications (New Age International, 2008).

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer-Verlag, 1983).

A. N. Tikhonov and V. Y. Arsenin, Solution of Illposed Problems (Winston, 1997).

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Figures (7)

Fig. 1
Fig. 1

Original particle diffusion coefficient distribution ( cm 2 s ) used in the simulations

Fig. 2
Fig. 2

Reconstructed particle diffusion coefficient distribution ( cm 2 s ) from data with 1% Gaussian noise for the Gauss–Newton method: (a) gray-level-plot and (b) cross-sectional plots through the centers of the inhomogeneities in (a) (dashed curve) as well as the original inhomogeneous object.

Fig. 3
Fig. 3

Reconstructed particle diffusion coefficient distribution ( cm 2 s ) from data with 2% Gaussian noise for the Gauss–Newton method: (a) gray-level plot and (b) cross-sectional plots through the centers of the inhomogeneities in (a) (dashed curve) as well as the original inhomogeneous object.

Fig. 4
Fig. 4

Reconstructed particle diffusion coefficient distribution ( cm 2 s ) from data with 2% Gaussian noise for the pseudo-time marching scheme: (a) gray-level plot and (b) cross-sectional plots through the centers of the inhomogeneities in (a) (dashed curve) as well as the original inhomogeneous object.

Fig. 5
Fig. 5

Reconstructed particle diffusion coefficient distribution ( cm 2 s ) from data with 5% Gaussian noise for the pseudo-time marching scheme: (a) gray-level plot and (b) cross-sectional plots through the centers of the inhomogeneities in (a) (dashed curve) as well as the original inhomogeneous object.

Fig. 6
Fig. 6

Reconstructed absorption coefficient distribution ( cm 1 ) from data with 5% Gaussian noise for the ordinary Gauss–Newton method: (a) gray-level plot and (b) cross-sectional plots through the centers of the inhomogeneities in (a) (dashed curve) as well as the original inhomogeneous object.

Fig. 7
Fig. 7

Reconstructed absorption coefficient distribution ( cm 1 ) from data with 5% Gaussian noise for the pseudo-time marching scheme (a) gray-level plot and (b) cross-sectional plots through the centers of the inhomogeneities in (a) (dashed curve) as well as the original inhomogeneous object.

Equations (47)

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

D G ( r , τ ) ( μ a + 1 3 Δ r 2 ( r , τ ) α k 0 2 μ s ) G ( r , τ ) = q 0 ( r r 0 ) ,
D ( r ) G ( r , τ ) n = G ( r , τ ) ,
I ( r , τ ) I ( r , t + τ ) g 2 ( r , τ ) = 1 + γ | g 1 ( r , τ ) | 2 .
G , ψ = G , ψ L 2 ( Ω ) + | G | Ω , | ψ | Ω L 2 ( Ω )
G V = [ G L 2 ( Ω ) 2 + | G | Ω L 2 ( Ω ) 2 ] 1 2 .
Ω D G ψ + Ω G ψ + Ω ( μ a + A f τ ) G ψ = Ω q 0 ψ ,
B ( G , ψ ) = Ω D G ψ + Ω G ψ + Ω ( μ a + A f τ ) G ψ ,
L ( ψ ) = Ω q 0 ψ .
G V C q 0 L 2 ( Ω )
G H 2 ( Ω ) C q 0 L 2 ( Ω ) .
D G δ ( r , τ ) ( μ a + A f τ ) G δ ( r , τ ) = A f δ τ G
G δ ( r , τ ) + D G δ ( r , τ ) n = 0 .
G δ H 2 ( Ω ) C G C ( Ω ) f δ L 2 ( Ω )
D G * ( μ a + A f τ ) G * = 0
G * + D G * n = ϕ ,
G * H 1 ( Ω ) C ϕ L 2 ( Ω ) .
G * H 2 ( Ω ) C ϕ L 2 ( Ω ) .
G * L ( Ω ) K 1 ϕ L 2 ( Ω ) .
D F ( f ) ( f δ ) = | G δ ( r , τ ) | Ω .
D F * ( f ) ( ϕ ) = | A τ G G * | Ω .
F ( f ) = | G ( r , τ ) | Ω .
Θ ( ρ ) = 1 2 F ( ρ ̃ + f ) m e L 2 ( Ω ) 2 + β 2 ρ L 2 ( Ω ) 2 .
Θ ( ρ + δ ρ ) = Θ ( ρ ) + D Θ ( ρ ) ( δ ρ ) + 1 2 D 2 Θ ( ρ ) ( δ ρ , δ ρ ) .
D Θ ( ρ ) ( δ ρ ) = ϕ , D F ( ρ + f ) ( δ ρ ) + β ρ , δ ρ ,
ϕ , D F ( ρ + f ) ( δ ρ ) = A τ δ ρ , D F * ( ρ + f ) ( ϕ ) .
D Θ ( ρ ) ( δ ρ ) = A τ D F * ( ρ + f ) ( ϕ ) + β ρ , δ ρ .
Gr ( ρ ) = D F * ( ρ + f ) [ F ( ρ + f ) m e ] + β ρ .
D 2 Θ ( ρ ) ( δ ρ , δ ρ ) = D F * ( ρ + f ) D F ( ρ + f ) ( δ ρ ) + β δ ρ , δ ρ .
H ( ρ ) ( δ ρ ) = D F * ( ρ + f ) D F ( ρ + f ) ( δ ρ ) + β δ ρ .
H ( ρ ) ( δ ρ ) = Gr ( ρ ) .
ρ i + 1 = ρ i H ( ρ i ) 1 Gr ( ρ i ) .
Θ ( ρ i n ) M β + 1 n M β + 1 .
β 2 ρ i n L 2 ( Ω ) Θ ( ρ i n ) M β + 1 .
Θ ( ρ β ) lim inf Θ ( ρ i n ) M β + 1 n .
D 2 Θ ( ρ ) ( δ ρ , δ ρ ) 0 ρ , δ ρ A , δ ρ 0 ,
D 2 Θ ( ρ ) ( δ ρ , δ ρ ) = H ( ρ ) ( δ ρ ) , δ ρ 0 ρ , δ ρ V , δ ρ 0 .
D B i + 1 = D B i ( J T J ( D B i ) + λ I ) 1 J T ( D B i ) Δ m i .
M { G ( r , τ ) } G 2 ( r , 0 ) ( m 1 γ ) Γ ( r , τ )
{ Γ ( r i , τ ) D B δ ( r j ) } = Re { 2 μ s k 0 2 τ G ¯ ( r i , τ ) G R ϕ ( r j , r i , τ ) G ( r j , τ ) } .
D ̇ B + M ( D B 0 ) ( D B ( t ) D B 0 ) + V ( D B 0 ) = 0 ,
D B ( t ) = exp ( [ M ( D B 0 ) ] ( t ) ) ( D B 0 ( t ) ) + 0 t exp ( M ( D B 0 ) ( t s ) ) f ( s ) d s ,
D B ( t ) = exp ( [ M ( D B 0 ) ] ( t ) ) ( D B 0 ( t ) ) + M ( D B 0 ) 1 ( M ( D B 0 ) D B 0 V ( D B 0 ) ) .
D B ( t ) = exp ( [ M ( D B 0 ) ] ( t ) ) ( D B 0 ( t ) ) + D B * ,
Δ D B i + 1 = exp ( [ ( M ( D B i ) + λ I ) Δ t ] ) D B i + t i t i + Δ t exp ( [ ( M ( D B i ) + λ I ) ] ) ( t i + Δ t s ) f ( s ) d s ,
D B i + 1 = D B i + Δ D B i + 1
D B ( x , y ) = { 0.25 × 10 8 cm 2 s if ( x + 2.5 ) 2 + ( y ) 2 0.7 0.75 × 10 8 cm 2 s if ( x 2.5 ) 2 + ( y ) 2 0.7 } .
μ a ( x , y ) = { 3 × 10 2 cm 1 if ( x + 2.5 ) 2 + ( y ) 2 0.9 2 × 10 2 cm 1 if ( x 2.5 ) 2 + ( y ) 2 0.9 } .

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