无记忆约束的广义雅可比矩阵链积的Optimization
Optimization of Generalized Jacobian Chain Products without Memory Constraints
摘要：
雅可比矩阵的有效计算是计算科学和工程中的一个基本挑战。大规模模块化数值模拟程序可以被视为序列的评估，在我们的情况下，可微模块与相应的本地雅可比矩阵。后者通常不可用。假设各个模块的切线和伴随版本作为算法微分的结果给出。经典的（雅可比）矩阵链积公式与矩阵的无矩阵雅可比矩阵和矩阵雅可比产品的切线和伴随的可选评估进行了扩展。我们提出了一个动态规划算法的计算成本最小化的广义雅可比矩阵链产品，而不考虑可用的持久系统内存的限制。换句话说，整个模拟程序的伴随的朴素评估被假设为是可行的选项。不需要检查点。在给定的假设下，我们得到最优解，提高了最好的国家的最先进的方法的因素高达七个随机生成的问题的情况下，不断增长的规模。
英文摘要：
The efficient computation of Jacobians represents a fundamental challenge in computational science and engineering. Large-scale modular numerical simulation programs can be regarded as sequences of evaluations of in our case differentiable modules with corresponding local Jacobians. The latter are typically not available. Tangent and adjoint versions of the individual modules are assumed to be given as results of algorithmic differentiation instead. The classical (Jacobian) matrix chain product formulation is extended with the optional evaluation of matrix-free Jacobian-matrix and matrix-Jacobian products as tangents and adjoints. We propose a dynamic programming algorithm for the minimization of the computational cost of such generalized Jacobian chain products without considering constraints on the available persistent system memory. In other words, the naive evaluation of an adjoint of the entire simulation program is assumed to be a feasible option. No checkpointing is required. Under the given assumptions we obtain optimal solutions which improve the best state of the art methods by factors of up to seven on a set of randomly generated problem instances of growing size.
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