Conjugate Gradient Method for Solving Inverse Problems

The Conjugate Gradient (CG) method is an efficient iterative algorithm for solving linear systems of equations, particularly well-suited for large-scale sparse matrix problems. In inverse problem solving, where data often contains noise or missing values, direct solutions may be unstable, while the CG method provides stable approximate solutions. Code implementations typically leverage sparse matrix storage formats like CSR (Compressed Sparse Row) to optimize memory usage and computational efficiency.

The algorithm starts from an initial guess solution and iteratively improves solution accuracy. Each iteration computes the residual of the current solution and adjusts the search direction using conjugate direction properties, ensuring that each optimization direction remains mutually independent to accelerate convergence. A key implementation aspect involves the Fletcher-Reeves formula for direction updates: β_k = (r_{k+1}^T r_{k+1}) / (r_k^T r_k). The CG method theoretically converges in at most n steps (where n is the matrix dimension), but practical implementations often control computational cost by presetting iteration count k due to numerical precision and problem conditioning considerations.

For input matrix A and column vector b, the algorithm initializes the solution vector (typically as a zero vector) and computes the initial residual r_0 = b - A*x_0. The iterative process then updates the solution vector and residual while adjusting search directions until meeting preset iteration limits or convergence criteria. Key computational steps include: 1) α_k calculation using α_k = (r_k^T r_k) / (p_k^T A p_k), 2) solution update x_{k+1} = x_k + α_k p_k, and 3) residual update r_{k+1} = r_k - α_k A p_k. This method not only conserves computational resources but also effectively handles ill-conditioned matrix problems through preconditioning techniques, making it widely applicable in inverse problem solving.