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We study the convergence of the Augmented Decomposition Algorithm (ADA) proposed in [32] for solving multi-block separable convex minimization problems subject to linear constraints. We show that the global convergence rate of the exact ADA is o(1/ν) under the assumption that there exists a saddle point. We consider the inexact Augmented Decomposition Algorithm (iADA) and establish global and local convergence results under some mild assumptions, by providing a stability result for the maximal monotone operator T associated with the perturbation from both primal and dual perspectives. This result implies the local linear convergence of the inexact ADA for many applications such as the lasso, total variation reconstruction, exchange problem and many other problems from statistics, machine learning and engineering with 1 regularization.

The Reynolds-averaged Navier–Stokes equation for compressible flow over supercritical airfoils under various flow conditions must be rapidly and accurately solved to shorten design cycles for such airfoils. Although deep-learning methods can effectively predict flow fields, the accuracy of these predictions near sensitive regions and their generalizability to large-scale datasets in engineering applications must be enhanced. In this study, a modified vision transformer-based encoder–decoder network is designed for the prediction of transonic flow over supercritical airfoils. In addition, four methods are designed to encode the geometric input with various information points and the performances of these methods are compared. The statistical results show that these methods generate accurate predictions over the complete flow field, with a mean absolute error on the order of 1 × 10−4. To increase accuracy near the shock area, multilevel wavelet transformation and gradient distribution losses are introduced into the loss function. This results in the maximum error that is typically observed near the shock area decreasing by 50%. Furthermore, the models are pretrained through transfer learning on large-scale datasets and fine-tuned on small datasets to improve their generalizability in engineering applications. The results generated by various pretrained models demonstrate that transfer learning yields a comparable accuracy from a reduced training time.