A multilevel algorithm for inverse problems with elliptic PDE constraints
A multilevel algorithm for inverse problems with elliptic PDE constraints
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DOI：
10.1088/0266-5611/24/3/034010
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发表时间：
2008-06-01
期刊：
影响因子：
2.1
通讯作者：
Dogan, Guenay
中科院分区：
文献类型：
--
作者：
Biros, George;Dogan, Guenay
我们提出了一种用于解决源识别问题的多级算法，其中前向问题是二维单元盒上的椭圆偏微分方程。 Hessian 对应于 Tikhonov 正则化的第一类 Fredholm 方程。我们的方法使用近似 Hessian 算子，首先，谱分解是已知的，其次，存在可以执行谱变换的快速算法。基于这种分解，我们提出了一种共轭梯度求解器，我们以多级子空间投影方案为前提条件。粗级预处理器是精确求解，而精细级预处理器是缩放理查森迭代的一个步骤。作为模型问题，我们考虑具有可变系数和部分观测值的 2D-Neumann Poisson 问题。这种情况的近似 Hessian 矩阵是与具有常数系数和完整观测值的问题相关的 Hessian 矩阵。我们可以使用快速余弦变换来计算频谱变换。我们研究了使用 Galerkin 或水平离散 Hessian 算子的效果，并提供了数值实验的结果，表明该方法对于完整和部分观测的有效性。
We present a multilevel algorithm for the solution of a source identification problem in which the forward problem is an elliptic partial differential equation on the 2D unit box. The Hessian corresponds to a Tikhonov-regularized first-kind Fredholm equation. Our method uses an approximate Hessian operator for which, first, the spectral decomposition is known, and second, there exists a fast algorithm that can perform the spectral transform. Based on this decomposition we propose a conjugate gradients solver which we precondition with a multilevel subspace projection scheme. The coarse-level preconditioner is an exact solve and the finer-levels preconditioner is one step of the scaled Richardson iteration. As a model problem, we consider the 2D-Neumann Poisson problem with variable coefficients and partial observations. The approximate Hessian for this case is the Hessian related to a problem with constant coefficients and full observations. We can use a fast cosine transform to compute the spectral transforms. We examine the effect of using Galerkin or level-discretized Hessian operators and we provide results from numerical experiments that indicate the effectiveness of the method for full and partial observations.