Stability of the minimum residual method for solving linear operator equations

In this paper, we propose a minimum residual method for solving linear operator equations with linear boundary conditions, constructed using orthonormal bases in a separable Hilbert space. The convergence of the method is proven rigorously. Subsequently, by demonstrating that the condition number of the coefficient matrix in the normal equation remains bounded, we establish the stability of the minimum residual method. Finally, numerical examples are provided to validate the stability of the proposed method.