Conjugate Gradient Inversion Method for Nonlinear Inversion with MATLAB Implementation

The conjugate gradient method is an efficient nonlinear optimization algorithm widely applied in inversion problems. In the field of nonlinear inversion, this method iteratively seeks the minimum of an objective function, making it particularly suitable for solving large-scale nonlinear inversion problems. The core concept of the conjugate gradient inversion method involves constructing a series of conjugate directions to optimize the search process. Compared to the steepest descent method, it avoids zigzag search paths and converges faster to the optimal solution. The iterative process primarily consists of three key steps: calculating the gradient direction, determining conjugate directions, and updating model parameters. In MATLAB implementation, the conjugate gradient inversion algorithm typically requires: - Designing an appropriate objective function to quantify the disparity between the model and observed data, often implemented using function handles or separate function files - Computing the gradient of the objective function, which can be achieved through finite difference approximations or analytical gradient calculations - Implementing conjugate direction update mechanisms using Polak-Ribière or Fletcher-Reeves formulas - Setting reasonable convergence criteria based on gradient norms or function value changes In practical applications, the conjugate gradient inversion method requires special attention to step size selection strategies (using line search algorithms like Wolfe conditions) and the application of preconditioning techniques, both of which significantly impact the algorithm's convergence speed and stability. For different inversion problems, specific implementation details of the conjugate gradient method may need adjustment through parameter tuning and algorithm customization to achieve optimal performance.