Now, I'm using L-BFGS method to solve the minimization problem of $\phi = \phi_d + \phi_m$. The problem is ill posed and large scale, so it is necessary to use the approximation of the Hessian matrix to precondition it.
The hessian of $\phi_m$ is easy to compute, but the Hessian of $\phi_d$ is almost impossible to compute, so I want to use $\nabla^{2}\phi_m + \beta I$ as an approximation of $\nabla^{2}\phi$, where $\beta$ is the dominant eigenvalue of $\nabla^{2}\phi_d$. I can only compute the matrix-vector product of $\nabla^{2}\phi_d$ and an arbitrary vector, so is there an method to compute $\beta$?