Inverse problems are in many cases solved with optimization techniques. When the underlying model is linear, first-order gradient methods are usually sufficient. With nonlinear models, due to nonconvexity, one must often resort to second-order methods that are computationally more expensive. In this work we aim to approximate a nonlinear model with a linear one and correct the resulting approximation error. We develop a sequential method that iteratively solves a linear inverse problem and updates the approximation error by evaluating it at the new solution. This treatment convexifies the problem and allows us to benefit from established convex optimization methods. We separately consider cases where the approximation is fixed over iterations and where the approximation is adaptive. In the fixed case we show theoretically under what assumptions the sequence converges. In the adaptive case, particularly considering the special case of approximation by first-order Taylor expansion, we show that with certain assumptions the sequence converges to a critical point of the original nonconvex functional. Furthermore, we show that with quadratic objective functions the sequence corresponds to the Gauss-Newton method. Finally, we showcase numerical results superior to the conventional model correction method. We also show, that a fixed approximation can provide competitive results with considerable computational speed-up.

翻译：逆问题通常采用优化技术求解。当底层模型为线性时，一阶梯度方法通常足够；而面对非线性模型，由于非凸性，往往需要借助计算成本更高的二阶方法。本文旨在用线性模型近似非线性模型，并对由此产生的近似误差进行校正。我们开发了一种序贯方法，通过迭代求解线性逆问题，并在新解处评估近似误差以更新该误差。这种处理方式使问题凸化，从而能够利用成熟的凸优化方法。我们分别考虑了近似在迭代过程中固定不变以及自适应调整的两种情况。对于固定近似情况，我们从理论上给出了序列收敛的假设条件；对于自适应近似情况，特别地，通过一阶泰勒展开近似这一特例，我们证明了在特定假设下序列收敛于原始非凸泛函的临界点。进一步，在二次目标函数情形下，该序列等价于高斯-牛顿法。最后，我们展示了优于传统模型校正方法的数值结果，同时表明固定近似可在显著加速计算的同时取得具有竞争力的效果。