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Consistent treatment of incompletely converged iterative linear solvers in reverse-mode algorithmic differentiation
Computational Optimization and Applications ( IF 2.2 ) Pub Date : 2020-08-03 , DOI: 10.1007/s10589-020-00214-x
Siamak Akbarzadeh , Jan Hückelheim , Jens-Dominik Müller

Algorithmic differentiation (AD) is a widely-used approach to compute derivatives of numerical models. Many numerical models include an iterative process to solve non-linear systems of equations. To improve efficiency and numerical stability, AD is typically not applied to the linear solvers. Instead, the differentiated linear solver call is replaced with hand-produced derivative code that exploits the linearity of the original call. In practice, the iterative linear solvers are often stopped prematurely to recompute the linearisation of the non-linear outer loop. We show that in the reverse-mode of AD, the derivatives obtained with partial convergence become inconsistent with the original and the tangent-linear models, resulting in inaccurate adjoints. We present a correction term that restores consistency between adjoint and tangent-linear gradients if linear systems are only partially converged. We prove the consistency of this correction term and show in numerical experiments that the accuracy of adjoint gradients of an incompressible flow solver applied to an industrial test case is restored when the correction term is used.

中文翻译:

逆模算法微分中不完全收敛的迭代线性求解器的一致处理

算法微分(AD)是一种广泛使用的计算数值模型导数的方法。许多数值模型都包含一个迭代过程来求解非线性方程组。为了提高效率和数值稳定性,通常不将AD应用于线性求解器。取而代之的是,将差分线性求解器调用替换为利用原始调用的线性度的手工生成的派生代码。在实践中,迭代线性求解器通常会过早停止以重新计算非线性外环的线性化。我们表明,在AD的反向模式下,通过部分收敛获得的导数与原始模型和切线线性模型不一致,从而导致不精确的伴随。我们提出一个校正项,如果线性系统仅部分收敛,则它可以恢复伴随和切线线性梯度之间的一致性。我们证明了该校正项的一致性,并在数值实验中表明,使用校正项可以恢复应用于工业测试用例的不可压缩流求解器的伴随梯度的精度。
更新日期:2020-08-03
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