Abstrakt
Large-scale discrete inverse problems of the form Ax ALMOST EQUAL TO b arise in many practical applications. Iterative regularization methods such as LSQR are commonly used for the solution of these problems.
Preconditioning is often applied to Krylov subspace methods to accelerate their convergence and improve efficiency. It can be also used to impose additional conditions such as non-negativity on the computed approximates. In such cases, iteration-dependent preconditioners are often required. Incorporating iteration-dependent preconditioners to Krylov subspace methods typically affects orthogonality properties of the computed bases and re-orthogonalization strategies need to be considered.
Here, we first overview approaches to preconditioning of LSQR with fixed preconditioner. Then we focus on a variant of preconditioned LSQR with an iteration-dependent preconditioner. We explain the orthogonalization strategy applied in the algorithm, describe its key properties and relations to CGLS and other algorithms. Numerical experiments will be used for illustration and comparison.