This paper intends to apply the sample-average-approximation (SAA) scheme to solve a system of stochastic equations (SSE), which has many applications in a variety of fields. The SAA is an effective paradigm to address risks and uncertainty in stochastic models from the perspective of Monte Carlo principle. Nonetheless, a numerical conflict arises from the sample size of SAA when one has to make a tradeoff between the accuracy of solutions and the computational cost. To alleviate this issue, we incorporate a gradually reinforced SAA scheme into a differentiable homotopy method and develop a gradually reinforced sample-average-approximation (GRSAA) differentiable homotopy method in this paper. By introducing a series of continuously differentiable functions of the homotopy parameter $t$ ranging between zero and one, we establish a differentiable homotopy system, which is able to gradually increase the sample size of SAA as $t$ descends from one to zero. The set of solutions to the homotopy system contains an everywhere smooth path, which starts from an arbitrary point and ends at a solution to the SAA with any desired accuracy. The GRSAA differentiable homotopy method serves as a bridge to link the gradually reinforced SAA scheme and a differentiable homotopy method and retains the nice property of global convergence the homotopy method possesses while greatly reducing the computational cost for attaining a desired solution to the original SSE. Several numerical experiments further confirm the effectiveness and efficiency of the proposed method.

翻译：本文旨在应用样本均值逼近（SAA）方法求解随机方程组（SSE），该方法在多个领域具有广泛应用。SAA是基于蒙特卡洛原理处理随机模型中风险与不确定性的有效范式。然而，当需要在求解精度与计算成本之间权衡时，SAA的样本量会引发数值矛盾。为缓解这一问题，本文将逐步增强的SAA机制与可微同伦方法相结合，提出了一种逐步增强样本均值逼近（GRSAA）可微同伦方法。通过引入一系列关于同伦参数t（取值范围0到1）的连续可微函数，我们构建了一个可微同伦系统，该系统能在t从1递减至0的过程中逐步增大SAA的样本量。该同伦系统的解集包含一条处处光滑的路径，该路径从任意点起始，最终收敛至满足任意精度的SAA解。GRSAA可微同伦方法搭建了逐步增强SAA机制与可微同伦方法之间的桥梁，既保留了同伦方法全局收敛的优良特性，又大幅降低了求解原随机方程组目标解的计算成本。数值实验进一步验证了所提方法的有效性与高效性。