Iterated preconditioned LSQR method for inverse problems on unstructured grids

被引:30
作者
Arridge, S. R. [1 ]
Betcke, M. M.
Harhanen, L. [1 ]
机构
[1] Aalto Univ, Dept Engn Design & Prod, FI-00076 Aalto, Finland
基金
英国工程与自然科学研究理事会;
关键词
ill-posed inverse problems; inhomogeneous diffusion; priorconditioning; Krylov methods; LSQR; LEAST-SQUARES PROBLEMS; DIFFUSE OPTICAL TOMOGRAPHY; SPARSE LINEAR-EQUATIONS; ILL-POSED PROBLEMS; IN-VIVO; REGULARIZATION; ALGORITHM; SYSTEMS; BREAST; SPACE;
D O I
10.1088/0266-5611/30/7/075009
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
This article presents a method for solving large-scale linear inverse imaging problems regularized with a nonlinear, edge-preserving penalty term such as total variation or the Perona-Malik technique. Our method is aimed at problems defined on unstructured meshes, where such regularizers naturally arise in unfactorized form as a stiffness matrix of an anisotropic diffusion operator and factorization is prohibitively expensive. 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. Because the convergence of Krylov methods for problems with discontinuities is notoriously slow, we propose to accelerate it by means of priorconditioning (Bayesian preconditioning). Priorconditioning is a technique that, through transformation to the standard form, embeds the information contained in the prior (Bayesian interpretation of a regularizer) directly into the forward operator and thence into the solution space. We derive a factorization-free preconditioned LSQR algorithm (MLSQR), allowing implicit application of the preconditioner through efficient schemes such as multigrid. The resulting method is also matrix-free i.e. the forward map can be defined through its action on a vector. We illustrate the performance of the method on two numerical examples. Simple 1D-deblurring problem serves to visualize the discussion throughout the paper. The effectiveness of the proposed numerical scheme is demonstrated on a three-dimensional problem in fluorescence diffuse optical tomography with total variation regularization derived algebraic multigrid preconditioner, which is the type of large scale, unstructured mesh problem, requiring matrix-free and factorization-free approaches that motivated the work here.
引用
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页数:27
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