LSMR-SLIM: SLIM extensions of LSMR

Copyright (c) 2010-2011, David C.L. Fong, SOL, Stanford University
Copyright (c) 2013, Tim T.Y. Lin, SLIM, University of British Columbia
All rights reserved.

Purpose

LSMR is a least-squares solver that ensures monotonic decrease in the norm of the gradient of each iteration.

The main purpose of the SLIM fork of LSMR is to ensure compatibility with SPOT and pSPOT as well as the distributed array type introduced by the MATLAB Parallel Computing Toolbox. This means that a standard call to LSMR x = lsmr(A,b) should accept A of the following types:

• numeric
• distributed array
• function handle (see help text of lsmr.m)
• SPOT object
• pSPOT object

As well as b of the following types:

• numeric
• distributed array

Note that, if any of A or b is of type distrubted, the resulting x will also be of type distributed.

Note: warm-starting with a pervious estimate of the solution

Currently LSMR does not accept inputs of x as the initial solution. Any warm-start procedure involving LSMR must be done externally, as described in the help text of the FROTRAN version of LSMR:

If some initial estimate x0 is known and if damp = 0,
one could proceed as follows:

1. Compute a residual vector     r0 = b - A*x0.
2. Use LSMR to solve the system  A*dx = r0.
3. Add the correction dx to obtain a final solution x = x0 + dx.

This requires that x0 be available before and after the call
to LSMR.  To judge the benefits, suppose LSMR takes k1 iterations
to solve A*x = b and k2 iterations to solve A*dx = r0.
If x0 is "good", norm(r0) will be smaller than norm(b).
If the same stopping tolerances atol and btol are used for each
system, k1 and k2 will be similar, but the final solution x0 + dx
should be more accurate.  The only way to reduce the total work
is to use a larger stopping tolerance for the second system.
If some value btol is suitable for A*x = b, the larger value
btol*norm(b)/norm(r0)  should be suitable for A*dx = r0.