We consider a scalar function depending on a numerical solution of an initial value problem, and its second-derivative (Hessian) matrix for the initial value. The need to extract the information of the Hessian or to solve a linear system having the Hessian as a coefficient matrix arises in many research fields such as optimization, Bayesian estimation, and uncertainty quantification. From the perspective of memory efficiency, these tasks often employ a Krylov subspace method that does not need to hold the Hessian matrix explicitly and only requires computing the multiplication of the Hessian and a given vector. One of the ways to obtain an approximation of such Hessian-vector multiplication is to integrate the so-called second-order adjoint system numerically. However, the error in the approximation could be significant even if the numerical integration to the second-order adjoint system is sufficiently accurate. This paper presents a novel algorithm that computes the intended Hessian-vector multiplication exactly and efficiently. For this aim, we give a new concise derivation of the second-order adjoint system and show that the intended multiplication can be computed exactly by applying a particular numerical method to the second-order adjoint system. In the discussion, symplectic partitioned Runge--Kutta methods play an essential role.

中文翻译：

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基于伴随的精确 Hessian 计算

我们考虑一个取决于初始值问题的数值解的标量函数及其初始值的二阶导数（Hessian）矩阵。在优化、贝叶斯估计和不确定性量化等许多研究领域中，都需要提取 Hessian 的信息或求解以 Hessian 作为系数矩阵的线性系统。从内存效率的角度来看，这些任务通常采用 Krylov 子空间方法，不需要显式地保存 Hessian 矩阵，只需要计算 Hessian 和给定向量的乘法。获得这种 Hessian 向量乘法的近似值的方法之一是对所谓的二阶伴随系统进行数值积分。然而，即使二阶伴随系统的数值积分足够准确，近似中的误差也可能很大。本文提出了一种新颖的算法，可以准确有效地计算预期的 Hessian 向量乘法。为此，我们给出了二阶伴随系统的新的简洁推导，并表明可以通过将特定的数值方法应用于二阶伴随系统来精确计算预期的乘法。在讨论中，辛分区的 Runge--Kutta 方法起着至关重要的作用。我们给出了二阶伴随系统的新的简洁推导，并表明通过将特定的数值方法应用于二阶伴随系统，可以精确地计算出预期的乘法。在讨论中，辛分区的 Runge--Kutta 方法起着至关重要的作用。我们给出了二阶伴随系统的新的简洁推导，并表明通过将特定的数值方法应用于二阶伴随系统，可以精确地计算出预期的乘法。在讨论中，辛分区的 Runge--Kutta 方法起着至关重要的作用。