论文标题
直接优化BPX预科器
Direct optimization of BPX preconditioners
论文作者
论文摘要
我们考虑为正定义线性系统的本地最佳预设器的自动构造。为了实现这一目标，我们引入了一个可区分的损失函数，该损失函数不明确包括最小特征值的估计。然而，由此产生的优化问题等于条件号的直接最小化。为了证明我们的方法，我们构建了一个经过修改的BPX预处理的参数家族。也就是说，我们为粗糙有限的元素空间定义了一组经验基础函数，并调整它们以获得更好的状态数。对于所考虑的模型方程（包括Poisson，Helmholtz，对流扩散，Biharmonic等），我们实现了对称阳性确定线性系统的两倍至二十倍。
We consider an automatic construction of locally optimal preconditioners for positive definite linear systems. To achieve this goal, we introduce a differentiable loss function that does not explicitly include the estimation of minimal eigenvalue. Nevertheless, the resulting optimization problem is equivalent to a direct minimization of the condition number. To demonstrate our approach, we construct a parametric family of modified BPX preconditioners. Namely, we define a set of empirical basis functions for coarse finite element spaces and tune them to achieve better condition number. For considered model equations (that includes Poisson, Helmholtz, Convection-diffusion, Biharmonic, and others), we achieve from two to twenty times smaller condition numbers for symmetric positive definite linear systems.
