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In this paper, we consider the algorithm proposed in recent years by Censor, Gibali and Reich, which solves split variational inequality problem, and Korpelevich’s extragradient method, which solves variational inequality problems. As our main result, we propose an iterative method for finding an element to solve a class of split variational inequality problems under weaker conditions and get a weak convergence theorem. As applications, we obtain some new weak convergence theorems by using our weak convergence result to solve related problems in nonlinear analysis and optimization.

This study explores for the first time the possibility of using an evolutionary algorithm (EA) for the determination of atomistic rates for use in kinetic Monte Carlo (KMC) simulations of anisotropic etching of silicon for the manufacture of microelectromechanical systems (MEMS). Traditionally, KMC rates are determined based on (i) computationally expensive density functional theory (DFT) calculations or, when possible, (ii) a combination of physical insight and a labor-intensive, manual procedure where, e.g., experimental and simulated surface morphologies are visually matched for a collection of surface orientations. Compared to these approaches, the evolutionary KMC method proposed in this study provides a more flexible, autonomous procedure to describe correctly a wide variety of etching conditions. We focus on the use of a functional representation of the atomistic rates, referred to as the removal probability function (RPF). This simplifies the EA search space to just a few parameters and reduces the number of required experimental data points to just a few etch rates. By proposing two alternative RPFs with four and six parameters, respectively, we show that the ability to explain the orientation dependence of the etch rate for a wide variety of etchants increases with the number of parameters and conclude that the six-parameter RPF provides sufficiently good simulations for a wide range of etching conditions, including KOH, KOH+IPA, TMAH and TMAH+Triton at different concentrations and temperatures. By uncovering the relationships between the parameters and the concentration of the etchant, it is possible to extend the simulations to nonmeasured etching conditions. Although the use of an RPF effectively restricts the search to a subspace of the atomistic rates, the present results suggest that physically meaningful KMC rates can probably be determined in the near future by direct comparison of macroscopic experiments and simulations through the use of evolutionary strategies.

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