Retrieving the variable coefficient for a nonlinear convection–diffusion problem with spectral conjugate gradient method

Abstract

In the framework of variational adjoint system, the spectral conjugate gradient (SCG) method, coupled with adjoint method, is adopted in this study as the functional minimization tool to retrieve the variable coefficient for the nonlinear convection–diffusion equation. The main steps focus on the determination of the search step size and the search direction. The former is deduced and then achieved numerically by solving the given sensitivity equation. The latter is formulated by the spectral gradient. The effectiveness of this solution strategy is evaluated by comparing it with L-BFGS iterative descent algorithm using different data types (dense and sparse, noise-free and noisy data). The effects of different forms of initial guess distribution of variable coefficient, as well as various levels of noise, on the retrieval results are also covered. The results indicate that (1) the performance of the SCG is superior to that of the L-BFGS in both the anti-noise ability within a suitable noise range and the enhancement of numerical accuracy. This is mainly due to the fact: first, the SCG adopts optimal step length calculated via the sensitivity problem at each iteration, and second, the SCG itself belongs to the iterative regularisation class; (2) it is demonstrated in our numerical test examples that the effects on the retrieval results caused by the changes of different initial guesses for the variable coefficient are much less in the SCG than in the L-BFGS.