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A priorconditioned LSQR algorithm for linear ill-posed problems with edge-preserving regularization

Betcke, M (University College London)
Friday 07 February 2014, 11:45-12:15

Seminar Room 1, Newton Institute


Co-authors: Simon Arridge (University College London), Lauri Harhanen (Aalto University)

In this talk we present a method for solving large-scale linear inverse problems regularized with a nonlinear, edge-preserving penalty term such as e.g. total variation or Perona–Malik. In the proposed scheme, the nonlinearity is handled with lagged diffusivity fixed point iteration which involves solving a large-scale linear least squares problem in each iteration. The size of the linear problem calls for iterative methods e.g. Krylov methods which are matrix-free i.e. the forward map can be defined through its action on a vector. Because the convergence of Krylov methods for problems with discontinuities is notoriously slow, we propose to accelerate it by means of priorconditioning. Priorconditioning is a technique which embeds the information contained in the prior (expressed as a regularizer in Bayesian framework) directly into the forward operator and hence into the solution space. We derive a factorization-free priorconditioned LSQR algorithm, allowing implicit ap plication of the preconditioner through efficient schemes such as multigrid. We demonstrate the effectiveness of the proposed scheme on a three-dimensional problem in fluorescence diffuse optical tomography using algebraic multigrid preconditioner.


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