Abstract

Reconstruction of the refractive index fluctuations in the atmosphere, or atmospheric tomography, is an underlying problem of many next generation adaptive optics (AO) systems, such as the multiconjugate adaptive optics or multiobject adaptive optics (MOAO). The dimension of the problem for the extremely large telescopes, such as the European Extremely Large Telescope (E-ELT), suggests the use of iterative schemes as an alternative to the matrix-vector multiply (MVM) methods. Recently, an algorithm based on the wavelet representation of the turbulence has been introduced in [Inverse Probl. 29, 085003 (2013)] by the authors to solve the atmospheric tomography using the conjugate gradient iteration. The authors also developed an efficient frequency-dependent preconditioner for the wavelet method in a later work. In this paper we study the computational aspects of the wavelet algorithm. We introduce three new techniques, the dual domain discretization strategy, a scale-dependent preconditioner, and a ground layer multiscale method, to derive a method that is globally O(n), parallelizable, and compact with respect to memory. We present the computational cost estimates and compare the theoretical numerical performance of the resulting finite element-wavelet hybrid algorithm with the MVM. The quality of the method is evaluated in terms of an MOAO simulation for the E-ELT on the European Southern Observatory (ESO) end-to-end simulation system OCTOPUS. The method is compared to the ESO version of the Fractal Iterative Method [Proc. SPIE 7736, 77360X (2010)] in terms of quality.

© 2014 Optical Society of America

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  1. A. Tokovinin and E. Viard, “Limiting precision of tomographic phase estimation,” J. Opt. Soc. Am. 18, 873–882 (2001).
    [CrossRef]
  2. A. Tokovinin, M. Le Louarn, and M. Sarazin, “Isoplanatism in a multiconjugate adaptive optics system,” J. Opt. Soc. Am. 17, 1819–1827 (2000).
    [CrossRef]
  3. B. Ellerbroek and C. R. Vogel, “Inverse problems in astronomical adaptive optics,” Inverse Probl. 25, 063001 (2009).
    [CrossRef]
  4. E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
    [CrossRef]
  5. L. Lessard, M. West, D. MacMynowski, and S. Lall, “Warm-started wavefront reconstruction for adaptive optics,” J. Opt. Soc. Am. A 25, 1147–1155 (2008).
    [CrossRef]
  6. L. Gilles, B. Ellerbroek, and C. R. Vogel, “Preconditioned conjugate gradient wave-front reconstructors for multiconjugate adaptive optics,” Appl. Opt. 42, 5233–5250 (2003).
    [CrossRef]
  7. Q. Yang, C. R. Vogel, and B. Ellerbroek, “Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography,” Appl. Opt. 45, 5281–5293 (2006).
    [CrossRef]
  8. M. Tallon, I. Tallon-Bosc, C. Béchet, F. Momey, M. Fradin, and É. Thiébaut, “Fractal iterative method for fast atmospheric tomography on extremely large telescopes,” Proc. SPIE 7736, 77360X (2010).
    [CrossRef]
  9. R. Ramlau and M. Rosensteiner, “An efficient solution to the atmospheric turbulence tomography problem using Kaczmarz iteration,” Inverse Probl. 28, 095004 (2012).
    [CrossRef]
  10. M. Rosensteiner and R. Ramlau, “Kaczmarz algorithm for multiconjugated adaptive optics with laser guide stars,” J. Opt. Soc. Am. A 30, 1680–1686 (2013).
    [CrossRef]
  11. T. Helin and M. Yudytskiy, “Wavelet methods in multi-conjugate adaptive optics,” Inverse Probl. 29, 085003 (2013).
    [CrossRef]
  12. M. Yudytskiy, T. Helin, and R. Ramlau, “A frequency dependent preconditioned wavelet method for atmospheric tomography,” in Third AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes (INAF—Osservatorio Astrosico di Arcetri, 2013).
  13. P. J. Hampton and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Sel. Top. Signal Process. 2, 781–792 (2008).
    [CrossRef]
  14. P. J. Hampton, P. Agathoklis, R. Conan, and C. Bradley, “Closed-loop control of a woofer–tweeter adaptive optics system using wavelet-based phase reconstruction,” J. Opt. Soc. Am. A 27, A145–A156 (2010).
    [CrossRef]
  15. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61 (SIAM, 1992).
  16. B. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. 19, 1803–1816 (2002).
    [CrossRef]
  17. M. Roggeman and B. Welsh, Imaging through Turbulence (Cambridge University, 1996).
  18. C. Béchet, M. Le Louarn, R. Clare, M. Tallon, I. Tallon-Bosc, and É. Thiébaut, “Closed-loop ground layer adaptive optics simulations with elongated spots: impact of modeling noise correlations,” in AO4ELT Proceedings (EDP Sciences, 2010), p. 03004.
  19. G. C. Papanicolaou and K. Sølna, “Wavelet based estimation of local Kolmogorov turbulence,” in Theory and Applications of Long-Range Dependence (Birkhäuser Boston, 2003), pp. 473–505.
  20. Y. Meyer and D. H. Salinger, Wavelets and Operators, Cambridge Studies in Advanced Mathematics (Cambridge University, 1992).
  21. W. Dahmen, “Multiscale analysis, approximation, and interpolation spaces,” in Approximation Theory VIII, Ser. Approx. Decompos. (World Scientific, 1995), Vol. 2.
  22. T. W. Nicholls, G. D. Boreman, and J. C. Dainty, “Use of a Shack–Hartmann wave-front sensor to measure deviations from a Kolmogorov phase spectrum,” Opt. Lett. 20, 2460–2462 (1995).
    [CrossRef]
  23. D. G. Pérez and L. Zunino, “Generalized wavefront phase for non-Kolmogorov turbulence,” Opt. Lett. 33, 572–574 (2008).
    [CrossRef]
  24. E. Klann, R. Ramlau, and L. Reichel, “Wavelet-based multilevel methods for linear ill-posed problems,” BIT Numer. Math. 51, 669–694 (2011).
    [CrossRef]
  25. Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. (SIAM, 2003).
  26. B. Ellerbroek and C. R. Vogel, “Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,” Proc. SPIE 5169, 206–217 (2003).
    [CrossRef]
  27. M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, 627234 (2006).
    [CrossRef]
  28. J. Franco, G. Bernabe, J. Fernandez, and M. E. Acacio, “A parallel implementation of the 2D wavelet transform using cuda,” in 17th Euromicro International Conference on Parallel, Distributed and Network-Based Processing, (2009), pp. 111–118.

2013 (2)

M. Rosensteiner and R. Ramlau, “Kaczmarz algorithm for multiconjugated adaptive optics with laser guide stars,” J. Opt. Soc. Am. A 30, 1680–1686 (2013).
[CrossRef]

T. Helin and M. Yudytskiy, “Wavelet methods in multi-conjugate adaptive optics,” Inverse Probl. 29, 085003 (2013).
[CrossRef]

2012 (1)

R. Ramlau and M. Rosensteiner, “An efficient solution to the atmospheric turbulence tomography problem using Kaczmarz iteration,” Inverse Probl. 28, 095004 (2012).
[CrossRef]

2011 (1)

E. Klann, R. Ramlau, and L. Reichel, “Wavelet-based multilevel methods for linear ill-posed problems,” BIT Numer. Math. 51, 669–694 (2011).
[CrossRef]

2010 (2)

P. J. Hampton, P. Agathoklis, R. Conan, and C. Bradley, “Closed-loop control of a woofer–tweeter adaptive optics system using wavelet-based phase reconstruction,” J. Opt. Soc. Am. A 27, A145–A156 (2010).
[CrossRef]

M. Tallon, I. Tallon-Bosc, C. Béchet, F. Momey, M. Fradin, and É. Thiébaut, “Fractal iterative method for fast atmospheric tomography on extremely large telescopes,” Proc. SPIE 7736, 77360X (2010).
[CrossRef]

2009 (1)

B. Ellerbroek and C. R. Vogel, “Inverse problems in astronomical adaptive optics,” Inverse Probl. 25, 063001 (2009).
[CrossRef]

2008 (3)

2006 (2)

Q. Yang, C. R. Vogel, and B. Ellerbroek, “Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography,” Appl. Opt. 45, 5281–5293 (2006).
[CrossRef]

M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

2003 (3)

L. Gilles, B. Ellerbroek, and C. R. Vogel, “Preconditioned conjugate gradient wave-front reconstructors for multiconjugate adaptive optics,” Appl. Opt. 42, 5233–5250 (2003).
[CrossRef]

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

B. Ellerbroek and C. R. Vogel, “Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,” Proc. SPIE 5169, 206–217 (2003).
[CrossRef]

2002 (1)

B. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. 19, 1803–1816 (2002).
[CrossRef]

2001 (1)

A. Tokovinin and E. Viard, “Limiting precision of tomographic phase estimation,” J. Opt. Soc. Am. 18, 873–882 (2001).
[CrossRef]

2000 (1)

A. Tokovinin, M. Le Louarn, and M. Sarazin, “Isoplanatism in a multiconjugate adaptive optics system,” J. Opt. Soc. Am. 17, 1819–1827 (2000).
[CrossRef]

1995 (1)

Acacio, M. E.

J. Franco, G. Bernabe, J. Fernandez, and M. E. Acacio, “A parallel implementation of the 2D wavelet transform using cuda,” in 17th Euromicro International Conference on Parallel, Distributed and Network-Based Processing, (2009), pp. 111–118.

Agathoklis, P.

Amorim, A.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Arcidiacono, C.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Baade, D.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Balestra, A.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Baruffolo, A.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Béchet, C.

M. Tallon, I. Tallon-Bosc, C. Béchet, F. Momey, M. Fradin, and É. Thiébaut, “Fractal iterative method for fast atmospheric tomography on extremely large telescopes,” Proc. SPIE 7736, 77360X (2010).
[CrossRef]

C. Béchet, M. Le Louarn, R. Clare, M. Tallon, I. Tallon-Bosc, and É. Thiébaut, “Closed-loop ground layer adaptive optics simulations with elongated spots: impact of modeling noise correlations,” in AO4ELT Proceedings (EDP Sciences, 2010), p. 03004.

Bernabe, G.

J. Franco, G. Bernabe, J. Fernandez, and M. E. Acacio, “A parallel implementation of the 2D wavelet transform using cuda,” in 17th Euromicro International Conference on Parallel, Distributed and Network-Based Processing, (2009), pp. 111–118.

Boreman, G. D.

Bradley, C.

P. J. Hampton, P. Agathoklis, R. Conan, and C. Bradley, “Closed-loop control of a woofer–tweeter adaptive optics system using wavelet-based phase reconstruction,” J. Opt. Soc. Am. A 27, A145–A156 (2010).
[CrossRef]

P. J. Hampton and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Sel. Top. Signal Process. 2, 781–792 (2008).
[CrossRef]

Brynnel, J.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Butler, D.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Cavadore, C.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Clare, R.

C. Béchet, M. Le Louarn, R. Clare, M. Tallon, I. Tallon-Bosc, and É. Thiébaut, “Closed-loop ground layer adaptive optics simulations with elongated spots: impact of modeling noise correlations,” in AO4ELT Proceedings (EDP Sciences, 2010), p. 03004.

Conan, R.

P. J. Hampton, P. Agathoklis, R. Conan, and C. Bradley, “Closed-loop control of a woofer–tweeter adaptive optics system using wavelet-based phase reconstruction,” J. Opt. Soc. Am. A 27, A145–A156 (2010).
[CrossRef]

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Dahmen, W.

W. Dahmen, “Multiscale analysis, approximation, and interpolation spaces,” in Approximation Theory VIII, Ser. Approx. Decompos. (World Scientific, 1995), Vol. 2.

Dainty, J. C.

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61 (SIAM, 1992).

Delabre, B.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Diolaiti, E.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Donaldson, R.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Ellerbroek, B.

B. Ellerbroek and C. R. Vogel, “Inverse problems in astronomical adaptive optics,” Inverse Probl. 25, 063001 (2009).
[CrossRef]

Q. Yang, C. R. Vogel, and B. Ellerbroek, “Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography,” Appl. Opt. 45, 5281–5293 (2006).
[CrossRef]

L. Gilles, B. Ellerbroek, and C. R. Vogel, “Preconditioned conjugate gradient wave-front reconstructors for multiconjugate adaptive optics,” Appl. Opt. 42, 5233–5250 (2003).
[CrossRef]

B. Ellerbroek and C. R. Vogel, “Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,” Proc. SPIE 5169, 206–217 (2003).
[CrossRef]

B. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. 19, 1803–1816 (2002).
[CrossRef]

Farinato, J.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Fedrigo, E.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Fernandez, J.

J. Franco, G. Bernabe, J. Fernandez, and M. E. Acacio, “A parallel implementation of the 2D wavelet transform using cuda,” in 17th Euromicro International Conference on Parallel, Distributed and Network-Based Processing, (2009), pp. 111–118.

Fradin, M.

M. Tallon, I. Tallon-Bosc, C. Béchet, F. Momey, M. Fradin, and É. Thiébaut, “Fractal iterative method for fast atmospheric tomography on extremely large telescopes,” Proc. SPIE 7736, 77360X (2010).
[CrossRef]

Franco, J.

J. Franco, G. Bernabe, J. Fernandez, and M. E. Acacio, “A parallel implementation of the 2D wavelet transform using cuda,” in 17th Euromicro International Conference on Parallel, Distributed and Network-Based Processing, (2009), pp. 111–118.

Franza, F.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Gilles, L.

Gilmozzi, R.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Hampton, P. J.

P. J. Hampton, P. Agathoklis, R. Conan, and C. Bradley, “Closed-loop control of a woofer–tweeter adaptive optics system using wavelet-based phase reconstruction,” J. Opt. Soc. Am. A 27, A145–A156 (2010).
[CrossRef]

P. J. Hampton and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Sel. Top. Signal Process. 2, 781–792 (2008).
[CrossRef]

Helin, T.

T. Helin and M. Yudytskiy, “Wavelet methods in multi-conjugate adaptive optics,” Inverse Probl. 29, 085003 (2013).
[CrossRef]

M. Yudytskiy, T. Helin, and R. Ramlau, “A frequency dependent preconditioned wavelet method for atmospheric tomography,” in Third AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes (INAF—Osservatorio Astrosico di Arcetri, 2013).

Hippler, S.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Hubin, N.

M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Klann, E.

E. Klann, R. Ramlau, and L. Reichel, “Wavelet-based multilevel methods for linear ill-posed problems,” BIT Numer. Math. 51, 669–694 (2011).
[CrossRef]

Korkiakoski, V.

M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

Lall, S.

Le Louarn, M.

M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

A. Tokovinin, M. Le Louarn, and M. Sarazin, “Isoplanatism in a multiconjugate adaptive optics system,” J. Opt. Soc. Am. 17, 1819–1827 (2000).
[CrossRef]

C. Béchet, M. Le Louarn, R. Clare, M. Tallon, I. Tallon-Bosc, and É. Thiébaut, “Closed-loop ground layer adaptive optics simulations with elongated spots: impact of modeling noise correlations,” in AO4ELT Proceedings (EDP Sciences, 2010), p. 03004.

Lessard, L.

Lizon, J.-Lo.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

MacMynowski, D.

Marchetti, E.

M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Meyer, Y.

Y. Meyer and D. H. Salinger, Wavelets and Operators, Cambridge Studies in Advanced Mathematics (Cambridge University, 1992).

Momey, F.

M. Tallon, I. Tallon-Bosc, C. Béchet, F. Momey, M. Fradin, and É. Thiébaut, “Fractal iterative method for fast atmospheric tomography on extremely large telescopes,” Proc. SPIE 7736, 77360X (2010).
[CrossRef]

Monnet, G.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Nicholls, T. W.

Papanicolaou, G. C.

G. C. Papanicolaou and K. Sølna, “Wavelet based estimation of local Kolmogorov turbulence,” in Theory and Applications of Long-Range Dependence (Birkhäuser Boston, 2003), pp. 473–505.

Pérez, D. G.

Ragazzoni, R.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Ramlau, R.

M. Rosensteiner and R. Ramlau, “Kaczmarz algorithm for multiconjugated adaptive optics with laser guide stars,” J. Opt. Soc. Am. A 30, 1680–1686 (2013).
[CrossRef]

R. Ramlau and M. Rosensteiner, “An efficient solution to the atmospheric turbulence tomography problem using Kaczmarz iteration,” Inverse Probl. 28, 095004 (2012).
[CrossRef]

E. Klann, R. Ramlau, and L. Reichel, “Wavelet-based multilevel methods for linear ill-posed problems,” BIT Numer. Math. 51, 669–694 (2011).
[CrossRef]

M. Yudytskiy, T. Helin, and R. Ramlau, “A frequency dependent preconditioned wavelet method for atmospheric tomography,” in Third AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes (INAF—Osservatorio Astrosico di Arcetri, 2013).

Reichel, L.

E. Klann, R. Ramlau, and L. Reichel, “Wavelet-based multilevel methods for linear ill-posed problems,” BIT Numer. Math. 51, 669–694 (2011).
[CrossRef]

Roggeman, M.

M. Roggeman and B. Welsh, Imaging through Turbulence (Cambridge University, 1996).

Rosensteiner, M.

M. Rosensteiner and R. Ramlau, “Kaczmarz algorithm for multiconjugated adaptive optics with laser guide stars,” J. Opt. Soc. Am. A 30, 1680–1686 (2013).
[CrossRef]

R. Ramlau and M. Rosensteiner, “An efficient solution to the atmospheric turbulence tomography problem using Kaczmarz iteration,” Inverse Probl. 28, 095004 (2012).
[CrossRef]

Saad, Y.

Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. (SIAM, 2003).

Salinger, D. H.

Y. Meyer and D. H. Salinger, Wavelets and Operators, Cambridge Studies in Advanced Mathematics (Cambridge University, 1992).

Sarazin, M.

A. Tokovinin, M. Le Louarn, and M. Sarazin, “Isoplanatism in a multiconjugate adaptive optics system,” J. Opt. Soc. Am. 17, 1819–1827 (2000).
[CrossRef]

Sølna, K.

G. C. Papanicolaou and K. Sølna, “Wavelet based estimation of local Kolmogorov turbulence,” in Theory and Applications of Long-Range Dependence (Birkhäuser Boston, 2003), pp. 473–505.

Tallon, M.

M. Tallon, I. Tallon-Bosc, C. Béchet, F. Momey, M. Fradin, and É. Thiébaut, “Fractal iterative method for fast atmospheric tomography on extremely large telescopes,” Proc. SPIE 7736, 77360X (2010).
[CrossRef]

C. Béchet, M. Le Louarn, R. Clare, M. Tallon, I. Tallon-Bosc, and É. Thiébaut, “Closed-loop ground layer adaptive optics simulations with elongated spots: impact of modeling noise correlations,” in AO4ELT Proceedings (EDP Sciences, 2010), p. 03004.

Tallon-Bosc, I.

M. Tallon, I. Tallon-Bosc, C. Béchet, F. Momey, M. Fradin, and É. Thiébaut, “Fractal iterative method for fast atmospheric tomography on extremely large telescopes,” Proc. SPIE 7736, 77360X (2010).
[CrossRef]

C. Béchet, M. Le Louarn, R. Clare, M. Tallon, I. Tallon-Bosc, and É. Thiébaut, “Closed-loop ground layer adaptive optics simulations with elongated spots: impact of modeling noise correlations,” in AO4ELT Proceedings (EDP Sciences, 2010), p. 03004.

Thiébaut, É.

M. Tallon, I. Tallon-Bosc, C. Béchet, F. Momey, M. Fradin, and É. Thiébaut, “Fractal iterative method for fast atmospheric tomography on extremely large telescopes,” Proc. SPIE 7736, 77360X (2010).
[CrossRef]

C. Béchet, M. Le Louarn, R. Clare, M. Tallon, I. Tallon-Bosc, and É. Thiébaut, “Closed-loop ground layer adaptive optics simulations with elongated spots: impact of modeling noise correlations,” in AO4ELT Proceedings (EDP Sciences, 2010), p. 03004.

Tokovinin, A.

A. Tokovinin and E. Viard, “Limiting precision of tomographic phase estimation,” J. Opt. Soc. Am. 18, 873–882 (2001).
[CrossRef]

A. Tokovinin, M. Le Louarn, and M. Sarazin, “Isoplanatism in a multiconjugate adaptive optics system,” J. Opt. Soc. Am. 17, 1819–1827 (2000).
[CrossRef]

Vérinaud, C.

M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

Viard, E.

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

A. Tokovinin and E. Viard, “Limiting precision of tomographic phase estimation,” J. Opt. Soc. Am. 18, 873–882 (2001).
[CrossRef]

Vogel, C. R.

B. Ellerbroek and C. R. Vogel, “Inverse problems in astronomical adaptive optics,” Inverse Probl. 25, 063001 (2009).
[CrossRef]

Q. Yang, C. R. Vogel, and B. Ellerbroek, “Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography,” Appl. Opt. 45, 5281–5293 (2006).
[CrossRef]

L. Gilles, B. Ellerbroek, and C. R. Vogel, “Preconditioned conjugate gradient wave-front reconstructors for multiconjugate adaptive optics,” Appl. Opt. 42, 5233–5250 (2003).
[CrossRef]

B. Ellerbroek and C. R. Vogel, “Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,” Proc. SPIE 5169, 206–217 (2003).
[CrossRef]

Welsh, B.

M. Roggeman and B. Welsh, Imaging through Turbulence (Cambridge University, 1996).

West, M.

Yang, Q.

Yudytskiy, M.

T. Helin and M. Yudytskiy, “Wavelet methods in multi-conjugate adaptive optics,” Inverse Probl. 29, 085003 (2013).
[CrossRef]

M. Yudytskiy, T. Helin, and R. Ramlau, “A frequency dependent preconditioned wavelet method for atmospheric tomography,” in Third AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes (INAF—Osservatorio Astrosico di Arcetri, 2013).

Zunino, L.

Appl. Opt. (2)

BIT Numer. Math. (1)

E. Klann, R. Ramlau, and L. Reichel, “Wavelet-based multilevel methods for linear ill-posed problems,” BIT Numer. Math. 51, 669–694 (2011).
[CrossRef]

IEEE J. Sel. Top. Signal Process. (1)

P. J. Hampton and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Sel. Top. Signal Process. 2, 781–792 (2008).
[CrossRef]

Inverse Probl. (3)

T. Helin and M. Yudytskiy, “Wavelet methods in multi-conjugate adaptive optics,” Inverse Probl. 29, 085003 (2013).
[CrossRef]

R. Ramlau and M. Rosensteiner, “An efficient solution to the atmospheric turbulence tomography problem using Kaczmarz iteration,” Inverse Probl. 28, 095004 (2012).
[CrossRef]

B. Ellerbroek and C. R. Vogel, “Inverse problems in astronomical adaptive optics,” Inverse Probl. 25, 063001 (2009).
[CrossRef]

J. Opt. Soc. Am. (3)

A. Tokovinin and E. Viard, “Limiting precision of tomographic phase estimation,” J. Opt. Soc. Am. 18, 873–882 (2001).
[CrossRef]

A. Tokovinin, M. Le Louarn, and M. Sarazin, “Isoplanatism in a multiconjugate adaptive optics system,” J. Opt. Soc. Am. 17, 1819–1827 (2000).
[CrossRef]

B. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. 19, 1803–1816 (2002).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Lett. (2)

Proc. SPIE (4)

B. Ellerbroek and C. R. Vogel, “Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,” Proc. SPIE 5169, 206–217 (2003).
[CrossRef]

M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

M. Tallon, I. Tallon-Bosc, C. Béchet, F. Momey, M. Fradin, and É. Thiébaut, “Fractal iterative method for fast atmospheric tomography on extremely large telescopes,” Proc. SPIE 7736, 77360X (2010).
[CrossRef]

E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-Lo. Lizon, R. Gilmozzi, G. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Viard, D. Butler, S. Hippler, and A. Amorim, “MAD the ESO multi-conjugate adaptive optics demonstrator,” Proc. SPIE 4839, 317–328 (2003).
[CrossRef]

Other (9)

I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61 (SIAM, 1992).

M. Yudytskiy, T. Helin, and R. Ramlau, “A frequency dependent preconditioned wavelet method for atmospheric tomography,” in Third AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes (INAF—Osservatorio Astrosico di Arcetri, 2013).

M. Roggeman and B. Welsh, Imaging through Turbulence (Cambridge University, 1996).

C. Béchet, M. Le Louarn, R. Clare, M. Tallon, I. Tallon-Bosc, and É. Thiébaut, “Closed-loop ground layer adaptive optics simulations with elongated spots: impact of modeling noise correlations,” in AO4ELT Proceedings (EDP Sciences, 2010), p. 03004.

G. C. Papanicolaou and K. Sølna, “Wavelet based estimation of local Kolmogorov turbulence,” in Theory and Applications of Long-Range Dependence (Birkhäuser Boston, 2003), pp. 473–505.

Y. Meyer and D. H. Salinger, Wavelets and Operators, Cambridge Studies in Advanced Mathematics (Cambridge University, 1992).

W. Dahmen, “Multiscale analysis, approximation, and interpolation spaces,” in Approximation Theory VIII, Ser. Approx. Decompos. (World Scientific, 1995), Vol. 2.

J. Franco, G. Bernabe, J. Fernandez, and M. E. Acacio, “A parallel implementation of the 2D wavelet transform using cuda,” in 17th Euromicro International Conference on Parallel, Distributed and Network-Based Processing, (2009), pp. 111–118.

Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. (SIAM, 2003).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic representation of a DWT matrix W4 of Daubechies 3 wavelets. Nonzero entries of the matrix are marked. Daubechies 3 filters l and h have six elements. Low-pass and high-pass filters are repeated in rows 0 to 241 and 24 to 251, respectively. A horizontal-shift of two elements in each row relates to the downsampling of the signal. Periodic extension of the wavelets on the boundary is represented by wrapping around the filter coefficients.

Fig. 2.
Fig. 2.

Left: operator Im produces a bilinear interpolation of the layer grid (circles) to the incoming wavefront grid (crosses). Right: operation is factorized into a linear interpolation in the x direction (orange squares) followed by a linear interpolation in the y direction (blue circles).

Fig. 3.
Fig. 3.

Effect of thresholding at scale j to the preconditioner is illustrated. The values on the diagonal of Λj are plotted for three thresholding scales j=J3, J2, and J1. The y axis corresponds to the values on a logarithmic scale. The x axis stands for the global index of the wavelet coefficients. These are ordered from left to right starting from the ground layer up to the layer with highest altitude; for each layer the values are plotted from left to right starting with the coarsest scale.

Fig. 4.
Fig. 4.

LE Strehl versus detected number of photons per subaperture and frame.

Fig. 5.
Fig. 5.

Performance of the algorithm with different number of iterations.

Fig. 6.
Fig. 6.

Sensitivity of the method with different GLMS scales.

Fig. 7.
Fig. 7.

Sensitivity of the method with respect to the preconditioner thresholding scale j. Thresholding is at three last scales (solid curve), two last scales (dashed curve), and last scale (dotted curve).

Tables (7)

Tables Icon

Algorithm 3.1 (GLMS method)

Tables Icon

Algorithm 3.2 (Open loop reconstructor)

Tables Icon

Table 1. Computational Cost and Memory Usage Estimates of the Operator Blocksh

Tables Icon

Table 2. Computational Cost Estimates of Components of M

Tables Icon

Table 3. Computational Cost and Memory Usage Estimates of a GLMS Step

Tables Icon

Table 4. Computational Cost Estimates

Tables Icon

Table 5. Parameter Settings of the Wavelet Method for Varying LGS and NGS Flux

Equations (25)

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

sm=ΓmPmLGSϕandsm=ΓmPmNGSϕ,
(PmLGSϕ)(r)==1Lϕ((1hH)r+θmh)
(PmNGSϕ)(r)==1Lϕ(r+θmh),
s=(sm)m=1M=Aϕ,
argminϕ(CΦ1/2ϕ22+CE1/2(sAϕ)22).
CE1=diag((TC^m1T)m=1Mlgs,(C^m1)m=Mlgs+1M),
φ(x)=φ(x1)φ(x2),x=(x1,x2),
ψ1(x)=φ(x1)ψ(x2),ψ2(x)=ψ(x1)φ(x2),ψ3(x)=ψ(x1)ψ(x2),
φjk(x)=2j/2φ(2jxk)andψjkt(x)=2j/2ψt(2jxk).
f(x)=kZ2a0kφ0k(x)+j=0J1kZ2t=13djktψjkt(x).
(Wj)i,(2i+pmod2j+1)=0q<|l|(qmod2j+1)=plq,
(Wj)2j+i,(2i+pmod2j+1)=0q<|l|(qmod2j+1)=phq,
(Wj(Wjaj+1)T)T=Wjaj+1WjT=(ajdj1dj2dj3).
WjT(ajdj1dj2dj3)Wj=aj+1.
m(κ)=cρ(h)(1Kout2+|κ|2)11/6,
Cϕ1/2fL22=1cρ(h)(Kout2+|κ|2)1112FfL221cρ(h)(Kout113fL22+(Δ)1112fL22),
(Δ)1112fL22jZkZ2t=1,2,322·116j|f,ψjkt|2,
D=diag(1cρ(h)(Kout113+2113j))j,k1,k2,t,
CΦ1/2ϕ(L2)L2==1LC1/2ϕL22=1L(Dc,c)2=(Dc,c)2,
W=diag(δ1W1,,δLWL)andW1=diag(δ11W11,,δL1WL1),
Pm=(Im1ImL).
((W1)TATCE1AW1+αD)cM=(W1)TATCE1sb.
Λj=diag((W1)TATCE1AW1)+αDj,
Dj=diag(max(τ1,jI,D1),,max(τL,jI,DL)),
τ,j=cρ(h)1(Kout113+2113j).

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