Backward step control for Hilbert space problems
Backward step control for Hilbert space problems
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DOI：
10.1007/s11075-018-0539-6
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发表时间：
2016-08
影响因子：
2.1
通讯作者：
A. Potschka
中科院分区：
文献类型：
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作者：
A. Potschka
我们分析后向步骤控制全球化，以找到从 Banach 空间映射到 Hilbert 空间的 Gâteaux 可微函数的零点。结果包括全局收敛到一个独特的解决方案，其特征是通过广义牛顿流传播初始猜测，并保证离散非线性残差范数下降的界限和（也是数字上）易于控制的渐近线性残差收敛速度。收敛理论可用于构造有效的数值方法，我们在 Krylov-Newton 方法和离散化逼近多级框架的情况下进行了演示。两种方法都通过 Krylov 子空间或通过自适应离散化来优化渐近线性残差收敛速度，从而产生实用且有效的停止标准和细化策略，以平衡非线性残差与线性系统的相对残差。我们将这些方法应用于非线性椭圆边值问题，并给出了 Carrier 方程和最小曲面方程的数值结果。
We analyze backward step control globalization for finding zeros of Gâteaux-differentiable functions that map from a Banach space to a Hilbert space. The results include global convergence to a distinctive solution characterized by propagating the initial guess by a generalized Newton flow with guaranteed bounds on the discrete nonlinear residual norm decrease and an (also numerically) easily controllable asymptotic linear residual convergence rate. The convergence theory can be exploited to construct efficient numerical methods, which we demonstrate for the case of a Krylov–Newton method and an approximation-by-discretization multilevel framework. Both approaches optimize the asymptotic linear residual convergence rate, either over the Krylov subspace or through adaptive discretization, which in turn yields practical and efficient stopping criteria and refinement strategies that balance the nonlinear residuals with the relative residuals of the linear systems. We apply these methods to the class of nonlinear elliptic boundary value problems and present numerical results for the Carrier equation and the minimum surface equation.