Comparison of advanced large-scale minimization algorithms on solution of inverse problems

We compare the performance of several robust large-scale minimization algorithms applied for the minimization of the cost functional in the solution of inverse problems related to parameter estimation applied to the parabolized Navier-Stokes equations. The methods compared consist of Quasi-Newton (BFGS), a limited memory Quasi-Newton (L-BFGS) [1], Hessian Free Newton method [2] and a new hybrid algorithm proposed by Morales and Nocedal [3]. Introduction The following specific issues characterize the inverse CFD problems posed in the variational statement: • High CPU time required for the single cost functional computation • The computation of the gradient usually is performed using the adjoint model, which requires the same computational effort as the direct model. • The instability (due to Ill-posedness) prohibits using Newton type algorithms without explicit regularization due to the Hessian being indefinite. The conjugate gradient method is widely used for inverse problems [6] because it provides regularization implicitly by neglecting non-dominant Hessian eigenvectors. The large CPU time required for the single cost functional computation justifies the high importance attached to the choice of most efficient optimization methods. From this perspective we will compare conjugate gradient method along with several quasi-Newton and truncated Newton large-scale unconstrained minimization methods for identification of entrance boundary parameters from measurements taken in a downstream flow-field sections. Test problem We consider the identification of unknown parameters (f∞(Y)=(ρ(Y), U (Y), V (Y),T (Y))) on the entrance boundary (Fig. 1) from measurements in a flow-field section ) , ( exp m m Y X f as the test for the inverse computational fluid dynamics (CFD) problem. The algorithm consists of the flow-field calculation (direct model), the discrepancy gradient computation using both forward and adjoint models and an optimization method. The problem has all the features of ill-posed Inverse CFD problems but can be solved relatively fast when using the approximation of parabolized Navier-Stokes equations. Direct Problem The two-dimensional parabolized Navier-Stokes equations are used here in a form similar to that carried out in Refs. [5,6]. The flow (Fig. 1) is laminar and supersonic along the X coordinate. These equations describe an under-expanded jet in the supersonic flow. 0 ) ( ) ( = + Y V X U ∂ ρ ∂ ∂ ρ ∂