Scalable Matrix-Free Adaptive Product-Convolution Approximation for Locally Translation-Invariant Operators
Scalable Matrix-Free Adaptive Product-Convolution Approximation for Locally Translation-Invariant Operators
复制标题
局部平移不变算子的可扩展无矩阵自适应乘积卷积近似
DOI：
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复制
发表时间：
2018
影响因子：
3.1
通讯作者：
O. Ghattas
中科院分区：
文献类型：
--
作者：
Nick Alger;Vishwas Rao;Aaron Myers;T. Bui;O. Ghattas
我们提出了基于卷积操作员的“产品卷积”插值的自适应网格矩阵近似方案。该方案适用于本地翻译不变的操作员，即使这些运营商是高级或全等级的运营商。此类操作员以求解部分微分方程（PDE）的Schur补体方法，作为PDE受限优化的Hessians和反相反问题，作为整体运营商，协方差运算符，以及Dirichlet to dirichlet to-neumann映射。构建近似需要计算操作员的脉冲响应，以在样品点的自适应精制网格中以节点为中心。随机的A-tostori误差估计器驱动适应性。构造后，可以使用快速傅立叶变换有效地将近似值应用于向量。可以有效地将近似值转换为层次矩阵（$ h $ -matrix）格式，然后使用可扩展的$ h $ -matrix arithmetic倒置或分解。随着使用较少的样品点，近似值优雅地降低质量，从而使廉价的较低质量近似被用作预处理。这产生了一种自动化方法来构建用于局部翻译不变的Schur补充的预处理。我们直接解决与边界有关的问题，并证明我们的计划消除了边界文物。我们在空间变化的内核上测试了该方案，在泊松操作员的接口Schur补充的非本地组件以及数据Misfit Hessian上以对流为主的对流扩散逆问题。数值结果表明该方案的表现优于现有方法。
We present an adaptive grid matrix-free operator approximation scheme based on a "product-convolution" interpolation of convolution operators. This scheme is appropriate for operators that are locally translation-invariant, even if these operators are high-rank or full-rank. Such operators arise in Schur complement methods for solving partial differential equations (PDEs), as Hessians in PDE-constrained optimization and inverse problems, as integral operators, as covariance operators, and as Dirichlet-to-Neumann maps. Constructing the approximation requires computing the impulse responses of the operator to point sources centered on nodes in an adaptively refined grid of sample points. A randomized a-posteriori error estimator drives the adaptivity. Once constructed, the approximation can be efficiently applied to vectors using the fast Fourier transform. The approximation can be efficiently converted to hierarchical matrix ($H$-matrix) format, then inverted or factorized using scalable $H$-matrix arithmetic. The quality of the approximation degrades gracefully as fewer sample points are used, allowing cheap lower quality approximations to be used as preconditioners. This yields an automated method to construct preconditioners for locally translation-invariant Schur complements. We directly address issues related to boundaries and prove that our scheme eliminates boundary artifacts. We test the scheme on a spatially varying blurring kernel, on the non-local component of an interface Schur complement for the Poisson operator, and on the data misfit Hessian for an advection dominated advection-diffusion inverse problem. Numerical results show that the scheme outperforms existing methods.
DOI：
10.1364/josaa.21.001593
发表时间：
2004-09-01
影响因子：
1.9
作者：
Preza, C;Conchello, JA
通讯作者：
Conchello, JA
用于大规模反问题的数据可扩展增强拉格朗日 KKT 预处理器
影响因子：
3.1
作者：
Alger, Nick;Villa, Umberto;Bui-Thanh, Tan;Ghattas, Omar
通讯作者：
Ghattas, Omar