This paper introduces a novel Transformed Primal-Dual with variable-metric/preconditioner (TPDv) algorithm, designed to efficiently solve affine constrained optimization problems common in nonlinear partial differential equations (PDEs). Diverging from traditional methods, TPDv iteratively updates time-evolving preconditioning operators, enhancing adaptability. The algorithm is derived and analyzed, demonstrating global linear convergence rates under mild assumptions. Numerical experiments on challenging nonlinear PDEs, including the Darcy-Forchheimer model and a nonlinear electromagnetic problem, showcase the algorithm's superiority over existing methods in terms of iteration numbers and computational efficiency. The paper concludes with a comprehensive convergence analysis.

翻译：本文提出了一种新型的转化原始-对偶变度量/预条件子（TPDv）算法，旨在高效求解非线性偏微分方程（PDEs）中常见的仿射约束优化问题。与传统方法不同，TPDv算法通过迭代更新时变预条件算子来增强适应性。本文推导并分析了该算法，证明了在温和假设下具有全局线性收敛速率。针对具有挑战性的非线性偏微分方程（包括Darcy-Forchheimer模型和一类非线性电磁问题）进行的数值实验表明，该算法在迭代次数和计算效率方面均优于现有方法。最后，本文给出了完整的收敛性分析。