An Efficient Numerical Algorithm with Adaptive Regularization for Parameter Estimations

Many engineering applications require numerical solution of partial differential equations (PDEs). This chapter discusses a numerical algorithm for estimating unknown coefficients in a system of two-dimensional parabolic PDES. The new algorithm features efficient calculation of sensitivity coefficients, accurate treatment of measurements at different time steps, and adaptive regularization process. Based on how the sensitivity coefficients are calculated, most of the existing parameter estimation algorithms can be classified into two categories: one is based on the direct difference method, which is computationally inefficient for large-scale problems, and the other is based on the adjoint equation method, in which a separate adjoint equation(s) must be solved. In this study, the first-order perturbation and the multi-time step methods is extended to system of PDEs for efficient calculation of sensitivity coefficient. An efficient and practical adaptive regularization algorithm is also developed in this paper to improve the stability and accuracy of the solution process. Further, numerical results from various test cases are discussed to demonstrate the implementation of the algorithm and improvement in accuracy. (Publisher summary