The analysis of an ill-posed problem using multi-scale resolution and second-order adjoint techniques

A wavelet regularization approach is presented for dealing with an ill-pose d problem of adjoint parameter estimation applied to estimating inflow para meters from down-Row data in an inverse convection case applied to the two- dimensional parabolized Navier-Stokes equations. The wavelet method provide s a decomposition into two subspaces, by identifying both a well-posed as w ell as an ill-posed subspace, the scale of which is determined by finding t he minimal eigenvalues of the Hessian of a cost functional measuring the la ck of fit between model prediction and observed parameters. The control spa ce is transformed into a wavelet space. The Hessian of the cost is obtained either by a discrete differentiation of the gradients of the cost derived from the first-order adjoint or by using the full second-order adjoint. The minimum eigenvalues of the Hessian are obtained either by employing a shif ted iteration method [X. Zou, I.M. Navon, F.X. Le Dimet., Tellus 44A (4) (1 992) 273] or by using the Rayleigh quotient. The numerical results obtained show the usefulness and applicability of this algorithm if the Hessian min imal eigenvalue is greater or equal to the square of the data error dispers ion, in which case the problem can be considered as well-posed (i.e., regul arized). If the regularization fails, i.e., the minimal Hessian eigenvalue is less than the square of the data error dispersion of the problem, the fo llowing wavelet scale should be neglected, followed by another algorithm it eration. The use of wavelets also allowed computational efficiency due to r eduction of the control dimension obtained by neglecting the small-scale wa velet coefficients. (C) 2001 Elsevier Science B.V. All rights reserved.