Module oc_lsqrreg_mod

1.

Purpose: regularized LSQR solver

$$\fbox {$ \displaystyle \left [\mathcal W\mathcal L\atop\epsi... ...ht ] \bold m\approx \left [\mathcal W\bold D\atop0\right ] $}$$

		 $\bold m= 0$ 
		 $\left [\Ud\atop\Um\right ] = \left [\mathcal W\bold D\atop0\right ]$ $\b = \sqrt{\parallel \bold U\parallel^2 }$		$\bold U=\frac{1}{\b}\bold U$ 
		 $\bold v= \left [\mathcal L^*\mathcal W^*\quad\epsilon\mathcal A^*\right ] \left [\Ud\atop\Um\right ]$				$\alpha= \sqrt{\parallel \bold v\parallel^2 }$		$\bold v=\frac{1}{\alpha}\bold v$ 
		 $\bold w= \bold v$ 
		 $\bar\rho =\alpha$		$\bar\phi = \b$ 
		 iterate { 
		 		  $\bold U= -\alpha\bold U$ $\left [\Ud\atop\Um\right ] =\left [\Ud\atop\Um\right ] +\left [\mathcal W\mathcal L\atop\epsilon\mathcal A\right ]\bold v$		 		  $\b = \sqrt{\parallel \bold U\parallel^2 }$ $\bold U=\frac{1}{\b}\bold U$
		 		  $\bold v= -\b \bold v$ $\bold v=\bold v+\left [\mathcal L^*\mathcal W^*\quad\epsilon\mathcal A^*\right ]\left [\Ud\atop\Um\right ]$		 		  $\alpha= \sqrt{\parallel \bold v\parallel^2 }$ $\bold v=\frac{1}{\alpha}\bold v$
		 		  $\rho = \sqrt{\bar\rho^2 + \b^2}$ 
		 		  $c=\frac{\bar\rho}{\rho}$ $\bar\rho=-c\alpha$ 
		 		  $s=\frac{\b}{\rho}$ $\theta = s\alpha$ 
		 		  $\phi = c\bar\phi$ $\bar\phi=s\bar\phi$ 
		 		  $t_1=\frac{\phi}{\rho}$ $t_2=-\frac{\theta}{\rho}$
		 		  $\bold m= \bold m+ t_1 \bold w$ 
		 		  $\bold w= \bold v+ t_2 \bold w$ 
		 		 }

2.

Functions and subroutines

(a)

subroutine oc_lsqrreg_init(niter,eps,maxmem,verb,movie)

Purpose: initialize the regularized LSQR solver

• niter: iterations number
• eps: scaling factor
• verb: verbose flag (optional)
• movie: movie output flag (optional)

(b)

subroutine oc_lsqrreg( L,A, x_,yy_,nreg,W,op1 ...op9)

Purpose: regularized LSQR solver

• L: out-of-core linear operator
• A: out-of-core regularization operator
• nreg: dimension of the regularization output
• W: out-of-core weighting operator