Adjoint-based exact Hessian computation

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