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I am going to solve an linear iterative inverse problem. I have two functions in matlab which one of them play the forward and the other play the adjoint role. I am not familiar with inverse problems very much and I am going to test some algorithms to check which one can help me to obtain a sparse model. Since I have to use my functions to do the forward and inverse operator roles in Ax=b, so I think I am not able to use some algorithms which needs A as a matrix. As an example, I could use the linearized bregmanas follows:

$$ v^{k+1} = v^k + A^T(f-Au^k). $$

$$ u^{k+1}= \delta * shrink(v^{k+1},1/\mu). $$

in which, for A I use that function which transform the model u to the data space and for A^T the function which transform the data to the model space.

I want to know that is there any other iterative algorithm which can be used in this manner for sparsity?

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Yes, most solvers for sparsity regularized least squares problems have been designed to use matrix-vector multiplications rather than accessing the matrix directly. See for example SPGL1 at

http://www.math.ucdavis.edu/~mpf/spgl1/

A good list of software for these kinds of problems can be found (at the very bottom of the page) at the Rice web site on resources for compressive sensing:

http://dsp.rice.edu/cs

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  • $\begingroup$ I don't have the matrix A, instead I have a function as A(u) which gives f' $\endgroup$
    – user3482383
    Dec 31, 2014 at 9:00
  • $\begingroup$ That's the point- with SPGL1 you can supply a subroutines which compute $A(u)$ and $A^{T}(v)$ rather than supplying the matrix itself. $\endgroup$
    – Brian Borchers
    Dec 31, 2014 at 16:52
  • $\begingroup$ How can I use it when I have two function, one for forward, A, and one for adjoint, A', since SPGL1 needs one function handle? $\endgroup$
    – user3482383
    Jan 1, 2015 at 12:26
  • $\begingroup$ If you already have functions that compute $A(u)$ and $A^{T}(v)$, then you can write a wrapper function that simply checks the mode parameter and evaluates $A(u)$ if mode=1 and $A^{T}(v)$ if mode=2. $\endgroup$
    – Brian Borchers
    Jan 1, 2015 at 19:26

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