Iterated preconditioned LSQR method for inverse problems on unstructured grids
Iterated preconditioned LSQR method for inverse problems on unstructured grids
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DOI：
10.1088/0266-5611/30/7/075009
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发表时间：
2014-06
期刊：
影响因子：
2.1
通讯作者：
S. Arridge;M. Betcke;Lauri Harhanen
中科院分区：
文献类型：
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作者：
S. Arridge;M. Betcke;Lauri Harhanen
本文提出了一种求解大规模线性逆成像问题的方法，该问题用非线性、边缘保持惩罚项（如全变分或Perona-Malik技术）正则化。我们的方法是针对非结构化网格上定义的问题，这种正则化自然出现在未因式分解的形式作为一个刚度矩阵的各向异性扩散算子和因式分解是昂贵的。在该方案中，非线性处理滞后扩散系数不动点迭代，这涉及到解决一个大规模的线性最小二乘问题，在每次迭代。由于Krylov方法的不连续性问题的收敛速度是出了名的慢，我们建议通过先验条件（贝叶斯预处理）来加速它。先验条件是一种技术，通过转换到标准形式，将先验（正则化子的贝叶斯解释）中包含的信息直接嵌入到前向算子中，从而进入解空间。我们推导出一个无因式分解的预处理LSQR算法（MLSQR），允许隐式应用的预处理器，通过有效的计划，如多重网格。所得到的方法也是无矩阵的，即前向映射可以通过其对向量的作用来定义。我们说明了两个数值例子的方法的性能。简单的1D去模糊问题用于可视化整个论文的讨论。所提出的数值方案的有效性证明在荧光漫射光学层析成像与总变分正则化推导的代数多重网格预处理器，这是大规模的，非结构化网格问题的类型，需要无矩阵和无因式分解的方法，激励这里的工作的三维问题。
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.