Inexact Newton-type methods based on Lanczos orthonormal method and application for full waveform inversion

The second-order derivative information plays an important role for large-scale full waveform inversion problems. However, exploiting this information requires massive computations and memory requirements. In this study, we develop two inexact Newton methods based on the Lanczos tridiagonalization process to consider the second-order derivative information. Several techniques are developed to improve the computational performance for our proposed methods. We present an effective stopping condition and implement a nonmonotone line search method. A method based on the adjoint-state method is used to efficiently compute Hessian-vector products. In addition, a diagonal preconditioner using the pseudo-Hessian matrix is employed to accelerate solving the Newton equation. Furthermore, we combine these two inexact Newton methods to improve the computational efficiency and the resolution. 2D and 3D experiments are given to demonstrate the convergence and effectiveness of our proposed methods. Numerical results indicate that, compared with the inversion methods based on the first-order derivative, both methods have good computational efficiency. Meanwhile, the method based on MINRES solver performs better than the method with Lanczos_CG due to its ability of utilizing the negative eigenvalue information when solving strongly nonlinear and ill-posed problems.