We propose in this paper a new minimization algorithm based on a slightly modified version of the scalar auxiliary variable (SAV) approach coupled with a relaxation step and an adaptive strategy. It enjoys several distinct advantages over popular gradient based methods: (i) it is unconditionally energy diminishing with a modified energy which is intrinsically related to the original energy, thus no parameter tuning is needed for stability; (ii) it allows the use of large step-sizes which can effectively accelerate the convergence rate. We also present a convergence analysis for some SAV based algorithms, which include our new algorithm without the relaxation step as a special case. We apply our new algorithm to several illustrative and benchmark problems, and compare its performance with several popular gradient based methods. The numerical results indicate that the new algorithm is very robust, and its adaptive version usually converges significantly faster than those popular gradient descent based methods.

翻译：本文提出一种新型最小化算法，该算法基于标量辅助变量（SAV）方法的微调版本，并融合了松弛步长与自适应策略。与主流梯度类方法相比，该算法具有以下显著优势：（i）在修正能量（该能量与原始能量存在本质关联）框架下无条件实现能量递减，因此无需调节稳定性参数；（ii）允许采用大步长，可有效加速收敛速率。本文同时给出了若干SAV类算法的收敛性分析（作为特例，本文未含松弛步长的新算法亦包含其中）。通过将新算法应用于多个说明性算例与基准测试问题，并将其性能与多种主流梯度下降方法进行对比，数值结果表明该算法具有极强稳健性，其自适应版本的收敛速度通常显著优于主流梯度下降类方法。