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

A wavelet regularization approach is presented for dealing with an ill-posed problem of adjoint parameter estimation applied to estimating inflow parameters from down-flow data in an inverse convection case applied to the two-dimensional parabolized Navier–Stokes equations. The wavelet method provides a decomposition into two subspaces, by identifying both a well-posed as well as an ill-posed subspace, the scale of which is determined by finding the minimal eigenvalues of the Hessian of a cost functional measuring the lack of fit between model prediction and observed parameters. The control space 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 shifted iteration method [X. Zou, I.M. Navon, F.X. Le Dimet., Tellus 44A (4) (1992) 273] or by using the Rayleigh quotient. The numerical results obtained show the usefulness and applicability of this algorithm if the Hessian minimal eigenvalue is greater or equal to the square of the data error dispersion, in which case the problem can be considered as well-posed (i.e., regularized). If the regularization fails, i.e., the minimal Hessian eigenvalue is less than the square of the data error dispersion of the problem, the following wavelet scale should be neglected, followed by another algorithm iteration. The use of wavelets also allowed computational efficiency due to reduction of the control dimension obtained by neglecting the small-scale wavelet coefficients.