Exponential convergence of Sobolev gradient descent for a class of nonlinear eigenproblems
Exponential convergence of Sobolev gradient descent for a class of nonlinear eigenproblems
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DOI：
10.4310/cms.2022.v20.n2.a4
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发表时间：
2019-12
期刊：
影响因子：
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通讯作者：
Ziyun Zhang
中科院分区：
文献类型：
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作者：
Ziyun Zhang
我们建议使用Łojasiewicz不等式作为一般工具来分析Hilbert流形上梯度下降的收敛速度，而不求助于连续的梯度流。利用这一工具，我们证明了具有自适应内积的Soblev梯度下降法对于Gross-Pitaevskii本征问题以指数速度收敛到基态。在一定条件下，该方法可以推广到一类一般的高次优化问题或非线性特征问题。我们通过几个例子来说明这一推广，特别是一个带有超高阶相互作用项的非线性薛定谔本征问题。对这些问题进行了数值实验。
We propose to use the Łojasiewicz inequality as a general tool for analyzing the convergence rate of gradient descent on a Hilbert manifold, without resorting to the continuous gradient flow. Using this tool, we show that a Sobolev gradient descent method with adaptive inner product converges exponentially fast to the ground state for the Gross-Pitaevskii eigenproblem. This method can be extended to a class of general high-degree optimizations or nonlinear eigenproblems under certain conditions. We demonstrate this generalization by several examples, in particular a nonlinear Schrodinger eigenproblem with an extra high-order interaction term. Numerical experiments are presented for these problems.