Sparse Matrix Techniques for MCAO

Transcription

1 Sparse Matrix Techniques for MCAO Luc Gilles Michigan Technological University, ECE Department Brent Ellerbroek Gemini Observatory Curt Vogel Montana State University, Math Sciences November 25, 2002 L. Gilles, CfAO Fall Retreat 2002 p.1/21

2 Wavefront Estimation from Idealized WFS Data - Open-loop WFS: s = Gϕ + η - η and ϕ random with known statistics C η = η η T and C ϕ = ϕ ϕ T ˆϕ = G s (noise-weighted pseudo-inverse) ˆϕ = arg min ϕ J(ϕ) J(ϕ) = 1 2 [ η 2 C 1 η + ϕ 2 C 1 ϕ ] = 1 2 [ η, C 1 η η + ϕ, Cϕ 1 ϕ ] ˆϕ = G s with G = ( G T C 1 η G + Cϕ 1 ) 1 G T Cη 1 - Block-layered structure for MCAO - (SNR) 2 = Gϕ 2 / η 2 = Gϕ 2 /(2 n z σ 2 ) L. Gilles, CfAO Fall Retreat 2002 p.2/21

3 Key Approximations to Inverse Turbulence Covariance Approximation #1 - C ϕ is block diagonal (different layers are statistically independent) - For each turbulent layer l, C ϕl is BTTB (phase structure function) - Approximate BBTB matrix by BCCB matrix (diagonalized by DFT) - C ϕl BCCB(τ l ) = F 1 diag(ˆτ l )F where ˆτ l = Fτ l (eigenvalues) - ˆτ l = c l κ 11/3 (Kolmogorov PSD) C 1 ϕ l BCCB(Λ l ) = F 1 diag(λ l ) F, λ l = ˆτ 1 l = c 1 l κ 11/3 L. Gilles, CfAO Fall Retreat 2002 p.3/21

4 Key Approximations to Inverse Turbulence Covariance - BCCB(Λ l ) is a full matrix. Approximation #2 - Entries Λ l rapidly decay to small values - Approximate BCCB(Λ l ) by Sl 2 where S l = c l S and S is the discrete Laplacian matrix with periodic boundary conditions S = BCCB(v) = F 1 diag(ˆv)f (sparse BCCB) ˆv = 4 [ sin 2 (πκ x x)/ x 2 + sin 2 (πκ y y)/ y 2] x, y 0 4π2 κ 2 - S = S y I x + I y S x where S y and S x are 1D versions L. Gilles, CfAO Fall Retreat 2002 p.4/21

5 Key Approximations to Inverse Turbulence Covariance Sparsity Patterns in 1D [S u] i = ( u i 1 + 2u i u i+1 ) / x 2 S = circ(v), v = (2, 1, 0,, 0, 1) T / x 2 = (2 e 0 e 1 e n 1 )/ x 2 S = F 1 diag(ˆv)f, ˆv = Fv = (2 ê 0 ê 1 ê n 1 )/ x 2 ˆv = 4 sin 2 (πκ x)/ x 2 (eigenvalues) x 0 4π 2 κ 2 L. Gilles, CfAO Fall Retreat 2002 p.5/21

6 Key Approximations to Inverse Turbulence Covariance Eigenvalues λ (1), λ (2) (biharmo) λ (2) (power 11/3) λ (1) c (1) = IDFT[ λ (1) ], c (2) = IDFT[ λ (2) ] C (1) = circ[ c (1) ] C (2) = circ[ c (2) ], Threshold = % Fill % Fill L. Gilles, CfAO Fall Retreat 2002 p.6/21

7 MCAO Minimum Variance Reconstructor Problem: find optimal actuator commands â minimizing MCAO wide-field error metric W â = Rs R = arg min J(R) R J(R) = ɛ 2 W = ɛ T Wɛ ɛ = H a â H ϕ ϕ (aperture-plane residual phase) R = H a }{{} F itting H ϕ G }{{} Estimation G = ( G T Cη 1 G }{{} + Cϕ 1 }{{} ) 1 G T Cη 1 Sparse+low rank (LGS) Sparse approx H a = ( H at W H a + low-rank ) 1 H at W L. Gilles, CfAO Fall Retreat 2002 p.7/21

8 Multigrid (MG) Methods - Can sometimes be used as stand-alone system solvers. - Can be used as preconditioners. - Rely on multiple scales (grid sizes) inherent in certain problems. - Need smoother which damps out high-frequency components of error on fine grids. - Classical Gauss-Seidel iteration works well for Laplace s equation. - Remaining low frequency error is well-represented on coarser grids. - Are recursive versions of the following 2-grid scheme. L. Gilles, CfAO Fall Retreat 2002 p.8/21

9 2-Grid Scheme x h S(x h, y h,...) r h A h x h y h x h x h + e h x h S(x h, y h,...) Restrict r H Ih Hr h Interpolate e h IH h e H Solve A H e H = r H - S(v, w,...) denotes application of smoother to solve Ax = w with initial guess x = v. - To obtain MG V-cycle, apply 2-grid scheme recursively. Carry out Solve step with (e H, r H ) in place of (x h, y h ). L. Gilles, CfAO Fall Retreat 2002 p.9/21

10 Multigrid (MG) Methods - Inter-grid transfers (restriction, or up-binning, and interpolation, or down-binning) are cheap. - Cost is typically dominated by smoother application on finest grid. - Choice of smoother is problem-dependent. - Block (i.e., layer-oriented) symmetric Gauss-Seidel (B-SGS) works well for MCAO estimation step. - FFT-based modified Richardson iteration works well for Ex-AO estimation. L. Gilles, CfAO Fall Retreat 2002 p.10/21

11 Block Gauss-Seidel Smoother Based on block L + D + U splitting A = A A A A n1... A n,n A nn }{{}}{{} L 0 A A 1n A n 1,n } {{ 0 } U D L. Gilles, CfAO Fall Retreat 2002 p.11/21

12 Block Gauss-Seidel Smoother Ax = b is equivalent to (L + D)x = b Ux. This motivates the block forward iteration (L + D)x k+1 = b Ux k, k = 0, 1,.... Similarly, we obtain the block backward interation (D + U)x k+1 = b Lx k, k = 0, 1,.... Block symmetric Gauss-Seidel (B-SGS) is obtained by interweaving forward and backward iterations. L. Gilles, CfAO Fall Retreat 2002 p.12/21

13 Efficient PCG Solver - MG preconditioned CG Estimation Step - Block SGS smoother: requires only inversion of diagonal blocks. Implemented using reordering + full Cholesky factorization. Fitting Step - Incomplete Cholesky preconditioned CG. Incomplete Cholesky applied to full sparse matrix without reordering. L. Gilles, CfAO Fall Retreat 2002 p.13/21

14 Preliminary Results Algorithm tested against conventional matrix multiply reconstructors on 8m class problems with degrees of freedom 32m class problems with degrees of freedom solvable in Matlab with 2-3Gb memory Convergence obtained in 2-10 iterations - Convergence rate a strong function of WFS noise level - Weak function of problem dimensionality, NGS vs. LGS MCAO L. Gilles, CfAO Fall Retreat 2002 p.14/21

15 Sample MCAO Problem Dimensionality Aperture diameter (m) WFS measurements Turbulence phase points DM actuators L. Gilles, CfAO Fall Retreat 2002 p.15/21

16 Sample MCAO Problem Dimensionality Estimation matrix to be inverted. 6 layers, 5 NGS s. D=16m. Phase screens L. Gilles, CfAO Fall Retreat 2002 p.16/21

17 Fig.1 Top layer diagonal block of estimation matrix. D=16m. Phase screens L. Gilles, CfAO Fall Retreat 2002 p.17/21

18 Fig.2 Fitting matrix (3 DM s, 932 actuators/dm). D=16m. L. Gilles, CfAO Fall Retreat 2002 p.18/21

19 Fig.3 Estimation, 6-layer profile, 5 WFSs using 5 NGS s. FoV diameter 100 arcsec, 1 V-cycle/CG iteration, 1 SGS iter/grid level, SNR = 20, r 0 = 25cm, x = r Estimation Error Norm Averaged over FoV 32m 16m 8m 10 0 CG Estimation Residual Norm 32m 16m 8m CG Estimation Iteration CG Estimation Iteration 10 0 Direct Solve vs CG (8m) RMS Estimation Error (32m) x CG Estimation Iteration 0 L. Gilles, CfAO Fall Retreat 2002 p.19/21

20 Fig.4 Fitting after 20 CG estimation iterations. Average over array of 5 5 observation directions. FoV diameter 100 arcsec. Incomplete Cholesky preconditioning, regularization parameter α = 1e-5. Error Norm (32m) Averaged over FoV 0Residual 10 3 DMs 2 DMs 1 DM Error Norm (16m) Averaged over FoV 0Residual 10 3 DMs 2 DMs 1 DM CG Fitting Iteration CG Fitting Iteration Error Norm (8m) Averaged over FoV 0Residual 10 3 DMs 2 DMs 1 DM 10 0 Residual Error Norm Averaged over FoV m 16m 8m CG Fitting Iteration CG Estimation Iteration L. Gilles, CfAO Fall Retreat 2002 p.20/21

21 Fig.5 Fitting after 20 CG estimation iterations. Average over array of 5 5 observation directions. FoV diameter 100 arcsec. Incomplete Cholesky preconditioning, regularization parameter α = 1e Direct Solve vs CG (8m) 10 0 CG Residual Norm (8m) DMs 2 DMs 1 DM DMs 2 DMs 1 DM CG Fitting Iteration CG Fitting Iteration 10 0 CG Residual Norm (16m) RMS Residual Error (32m, 2DMs) x DMs 2 DMs 1 DM CG Fitting Iteration 0 L. Gilles, CfAO Fall Retreat 2002 p.21/21