Adjoint-based exact Hessian-vector multiplication using symplectic Runge–Kutta methods
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We consider a function of the numerical solution of an initial value problem, its Hessian matrix with respect to the initial data, and the computation of a Hessian-vector multiplication. A simple way of approximating the Hessian-vector multiplication is to integrate the so-called second-order adjoint system numerically. However, the error in the approximation could be significant unless the numerical integration is sufficiently accurate. This paper presents a novel algorithm that computes the intended Hessian-vector multiplication exactly. 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 important role.
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