Discrete variational adjoint assimilations for parameter optimizations via spectral methods

Abstract

In this article, we investigate the parameter optimization problem via the variational adjoint data assimilations in the framework of the spectral approximation of a heat model. Precisely, we consider the optimization for the initial condition of the heat equation by means of the discrete adjoint data assimilations. Spectral methods and the classical Crank–Nicolson scheme are applied to deduce the full discrete system for the continuous model, then the variational adjoint data assimilation approach is used to derive the gradient and the corresponding adjoint system. The main contribution of this work consists of: (i) proposing efficient methods to solve the parameter optimization problems based on the discrete observation data. In particular, an optimal choice of step sizes is presented; (ii) comparing different strategies of calculating the gradient of the objective functions with respect to the initial state corresponding to the complete and incomplete assimilations and (iii) investigating the convergence rate of the optimal parameter with respect to the polynomial degree. Our numerical results show that all proposed methods can properly optimize the initial condition, and the so-called spectral convergence rate is obtained. Moreover, it is found that the computational cost can be significantly reduced if the matrix appearing in the gradient calculation is explicitly constructed in the incomplete assimilations.

Keywords:

• heat equation
• variational adjoint assimilation
• parameter optimization
• spectral methods

Acknowledgements

This work was partially supported by National NSF of China under Grant 10531080, the Excellent Young Teachers Program by the Ministry of Education of China and 973 High Performance Scientific Computation Research Program 2005CB321703.