Global convergence of Newton’s method for the regularized p-Stokes equations

IF 1.7 3区 数学 Q2 MATHEMATICS, APPLIED Numerical Algorithms Pub Date : 2024-09-17 DOI:10.1007/s11075-024-01941-6
Niko Schmidt
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Abstract

The motion of glaciers can be simulated with the \(\varvec{p}\)-Stokes equations. Up to now, Newton’s method to solve these equations has been analyzed in finite-dimensional settings only. We analyze the problem in infinite dimensions to gain a new viewpoint. We do that by proving global convergence of the infinite-dimensional Newton’s method with Armijo step sizes to the solution of these equations. We only have to add an arbitrarily small diffusion term for this convergence result. We prove that the additional diffusion term only causes minor differences in the solution compared to the original \(\varvec{p}\)-Stokes equations under the assumption of some regularity. Finally, we test our algorithms on two experiments: A reformulation of the experiment ISMIP-HOM \(\varvec{B}\) without sliding and a block with sliding. For the former, the approximation of exact step sizes for the Picard iteration and exact step sizes and Armijo step sizes for Newton’s method are superior in the experiment compared to the Picard iteration. For the latter experiment, Newton’s method with Armijo step sizes needs many iterations until it converges fast to the solution. Thus, Newton’s method with approximately exact step sizes is better than Armijo step sizes in this experiment.

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正则化 p-Stokes 方程牛顿法的全局收敛性
冰川的运动可以用 \(\varvec{p}\)-Stokes 方程来模拟。迄今为止,牛顿方法仅在有限维环境下分析了这些方程的求解。我们分析了无限维问题,从而获得了新的视角。为此,我们证明了采用 Armijo 步长的无限维牛顿法对这些方程求解的全局收敛性。我们只需添加一个任意小的扩散项,就能得到这一收敛结果。我们证明,在某种规则性假设下,额外的扩散项与原始的 \(\varvec{p}\)-Stokes 方程相比,只会导致解的微小差异。最后,我们在两个实验中测试了我们的算法:一个是不带滑动的 ISMIP-HOM \(\varvec{B}\)实验重构,另一个是带滑动的实验块。对于前者,与皮卡尔迭代法相比,牛顿法的近似精确步长和Armijo步长在实验中更胜一筹。在后一项实验中,采用 Armijo 步长的牛顿法需要多次迭代,直到快速收敛到解为止。因此,在本实验中,采用近似精确步长的牛顿法要优于采用 Armijo 步长的牛顿法。
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来源期刊
Numerical Algorithms
Numerical Algorithms 数学-应用数学
CiteScore
4.00
自引率
9.50%
发文量
201
审稿时长
9 months
期刊介绍: The journal Numerical Algorithms is devoted to numerical algorithms. It publishes original and review papers on all the aspects of numerical algorithms: new algorithms, theoretical results, implementation, numerical stability, complexity, parallel computing, subroutines, and applications. Papers on computer algebra related to obtaining numerical results will also be considered. It is intended to publish only high quality papers containing material not published elsewhere.
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