变分资料同化中伴随敏感度计算优化技术

Optimization Techniques for Adjoint Sensitivity Computation in Variational Data Assimilation

摘要: 针对变分资料同化中目标泛函梯度计算复杂和实现困难等问题，研究了离散伴随敏感度计算优化技术。首先通过与梯度直接计算方法的比较，说明了伴随方程在变分资料同化中的重要作用，通过理论推导和分析展示了当分析变量维数特别巨大时伴随敏感度计算方法在计算效率方面的显著优势；其次考虑到在伴随敏感度计算中Jacobian矩阵是重要影响因素，结合大气预报方程组离散格式分析了伴随方程中Jacobian矩阵的稀疏性及非零元素分布特征；再次，针对变分资料同化中Jacobian矩阵的稀疏性给出了一种压缩存储策略；最后利用有向无环图对函数计算、切线性和伴随计算过程进行了表示，通过示例说明了基于计算图顶点消除的伴随代码优化技术，采用顶点消除方法对四维变分资料同化极小化过程中调用的热点伴随程序进行优化改进后，将大大减小计算量。

Abstract: In order to solve the problems of complex and difficult implementation for calculating gradients of cost function in variational data assimilation, optimization techniques for the discrete adjoint sensitivity calculation are studied. Firstly, by comparing with the direct method of gradient com-putation, the important role of adjoint equation in variational data assimilation is illustrated. Through theoretical derivation and detailed analysis, it is shown that the adjoint sensitivity calculation method has significant advantages in terms of computational efficiency when the dimension of the analysis vector is very large. Secondly, because the Jacobian matrix is an important factor in the adjoint sensitivity calculation, the sparse and non-zero element distribution characteristics of it are analyzed in combination with the numerical discrete schemes in atmospheric prediction models. Thirdly, a compression strategy is given to store Jacobian matrix in variational data assimilation using its sparsity. Finally, different directed acyclic graphs are used to represent the function calculation, its tangent linear and adjoint calculation processes. A fast algorithm is proposed based on graph vertex elimination to accelerate adjoint sensitivity computation demonstrated by an example. By using the graph vertex elimination method, the hotspot programs called during the minimization of the four-dimensional variational data assimilation can be optimized and improved, which will greatly reduce the cost of computation.