An efficient Quasi-Newton method for nonlinear inverse problems via learned singular values
This method offers a more efficient approach for solving nonlinear inverse problems, particularly benefiting time-critical applications in fields like electrical impedance tomography, by reducing the computational burden of derivative calculations.
This paper addresses the computational cost of multi-dimensional function derivatives in nonlinear inverse problems, which typically limits time-sensitive applications. They propose a data-driven Quasi-Newton method that learns a mapping from model outputs to singular values to update the Jacobian, achieving speed-ups without accumulating roundoff errors.
Solving complex optimization problems in engineering and the physical sciences requires repetitive computation of multi-dimensional function derivatives. Commonly, this requires computationally-demanding numerical differentiation such as perturbation techniques, which ultimately limits the use for time-sensitive applications. In particular, in nonlinear inverse problems Gauss-Newton methods are used that require iterative updates to be computed from the Jacobian. Computationally more efficient alternatives are Quasi-Newton methods, where the repeated computation of the Jacobian is replaced by an approximate update. Here we present a highly efficient data-driven Quasi-Newton method applicable to nonlinear inverse problems. We achieve this, by using the singular value decomposition and learning a mapping from model outputs to the singular values to compute the updated Jacobian. This enables a speed-up expected of Quasi-Newton methods without accumulating roundoff errors, enabling time-critical applications and allowing for flexible incorporation of prior knowledge necessary to solve ill-posed problems. We present results for the highly non-linear inverse problem of electrical impedance tomography with experimental data.