| 摘要: | 本計畫的目標是發展可平行化，具擴展性，高效能的，對限制條件最佳化問題衍生的大規模非線性方程組求解的演算法。我們特別專注的是，具有不等式約束條件和不可微分的目標函數問題。這類的問題在計算科學和工程中有廣泛應用，例如流體控制問題，太空任務中的軌道最佳化問題，統計學和資料科學中的正則化最小平方問題。除了因為高維度之外，由於這些特性，使得最佳化問題數值計算是十分具有挑戰性的。半光滑牛頓法是對非光滑系統最常用的方法之一，當系統的非線性不均衡時，常會遇到收斂與否問題。非線性預處理技巧，提供除全域化技術之外的替代方法，不僅可以增強牛頓方法的收斂能性，而且可以加速某些Krylov子空間方法的收斂性。在本研究中，我們將考慮一些非線性迭代方法作為半光滑牛頓算法的預條件子和非線性Krylov子空間方法，如非線性GMRES方法。這些預條件子包括，非線性消去法，該方法已成功地，應用於具有局部非線性強的困難偏微分方程問題，並應用於計算流體力學和流體控制問題等。我們也將考慮了分解部分修正算法，如懲罰法，拉格朗日乘子法，交替方向法和乘子法（ADDM）。 所有考慮的演算法都將利用PETSc上實作，並將在不同的最先進的電腦平台上進行測試，包括叢集電腦，多核系統和CPU / GPU 混合系統，並在這些平台上的平行效能測試。希望這個基於PETSc的科學計算軟體可以對科學界和工業界有幫助。 ;The goal of this proposed project is to develop the parallel, scalable, efficient solution algorithms for solving large-scale nonlinear system of equations arising from constrained optimization problems. We are particularly interested in the problem with inequality constraints and non-differentiable objective functions involved. Such problems represent a broad range of applications in computational sciences and engineerings, such as flow control problems, the dogleg trajectory optimization problems in the space mission, l1- regularized least-squares problems in the statistics and data sciences. Addition to the high dimensionality, such characteristics make the optimization problems more challenging to solve. The semi-smooth Newton method is one of the most popular methods for the non-smooth system also suffer from the convergence issue when the nonlinearity of the system is not well balanced. Nonlinear preconditioning technique provides alternative other than globalization techniques, e.g., linesearch or trust region not only to enhance the robustness of Newton type method but also to accelerate the convergence of some Krylov subspace method In this project, we will study a variety of nonlinear iterative methods as preconditioners for semi-smooth Newton algorithms and nonlinear Krylov subspace method, such as nonlinear Generalized Minimal Residuals MRES (GMRES) method. This includes nonlinear elimination method, which has been successfully applied to some difficult PDE problems with strong local nonlinearity with applications in computational fluid dynamics and flow control problems and others. We also consider the decoupled algorithms, namely the method of penalty method, the method of Lagrange multipliers, and the Alternating direction method and Method of Multipliers (ADDM). All the algorithms considered will be implemented on the top of Portable, Extensible, Toolkits for Scientific computation (PETSc) and will be tested on the different state-of-the-art computer platforms, including a cluster of PCs, the multicore system, and hybrid CPU/GPU system and its parallel performance on these platforms will be studied. Hopefully, this PETSc-based scientific package can be beneficial to scientific community and industry. |
