发表机构
University of California, Irvine（加州大学尔湾分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本教程从反问题出发，以离散化、面向实现的方式系统讲解PDE约束优化中的伴随梯度计算方法，涵盖推导、实现与验证，并通过振动弦示例展示完整流程。
AI 中文摘要
伴随方法是PDE约束优化中计算梯度的标准方法，也是参数估计、模型校准和最优控制的核心。本教程以离散化、面向实现的方式介绍该方法，强调概念理解。从由偏微分方程控制的反问题出发，构建相应的优化问题，并从迭代下降法的角度推导伴随梯度。明确状态、参数、目标函数和约束的作用，伴随方程直接从梯度计算中获得而非预先假设。特别关注显式推导、实现细节以及数学推导与最终计算算法之间的联系。教程还考察了离散化对伴随构造的影响，解释了时间反向伴随递推的起源，并给出了梯度验证的实用程序。一个振动弦反问题作为贯穿示例，展示了从模型构建到基于伴随优化的完整流程。本教程面向高年级本科生、研究生以及寻求逐步理解伴随梯度计算的实践者。
英文摘要
The adjoint method is a standard approach for computing gradients in PDE-constrained optimization and central to parameter estimation, model calibration, and optimal control. This tutorial presents a discrete, implementation-oriented introduction to the method with an emphasis on conceptual understanding. Starting from an inverse problem governed by a partial differential equation, it develops the corresponding optimization problem and derives the adjoint gradient from the perspective of iterative descent methods. The roles of the state, the parameters, the objective, and the constraint are made explicit, and the adjoint equation is obtained directly from the gradient computation rather than postulated. Particular attention is given to explicit derivations, implementation details, and the connection between the mathematical development and the resulting computational algorithm. The tutorial also examines the influence of discretization on the adjoint construction, explains the origin of the backward-in-time adjoint recursion, and presents practical procedures for gradient verification. A vibrating-string inverse problem serves as a running example, illustrating the complete workflow from model formulation to adjoint-based optimization. The presentation is intended for advanced undergraduate students, graduate students, and practitioners seeking a step-by-step understanding of adjoint gradient computation.
