An Adaptive Multi - Scale Conjugate Gradient Method for Distributed Parameter Estimation of 2 - D Wave Equation ∗

An adaptive multi-scale conjugate gradient method for distributed parameter estimations (or inverse problems) of wave equation is presented. The identification of the coefficients of wave equations in two dimensions is considered. First, the conjugate gradient method for optimization is adopted to solve the inverse problems. Second, the idea of multi-scale inversion and the necessary conditions that the optimal solution should be the fixed point of multi-scale inversion method is considered. An adaptive multi-scale inversion method for the inverse problem is developed in conjunction with the conjugate gradient method. Finally, some numerical results are shown to indicate the robustness and effectiveness of our method.