Abstract
This paper presents an efficient way to compute second-order gradients by using the adjoint method for PDE-constrained optimization. The gradient thus obtained will then be used in an optimization algorithm. We propose a conjugate gradient combined with the trust-region method, which may have a quadratic convergence rate of Newton's method. Furthermore, we compare the proposed algorithm to a quasi-Newton method (BFGS). We apply the method for production optimization of oil reservoirs. Two numerical cases are presented, showing that our proposed method requires fewer function and gradient evaluations.