Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data

We study Newton type methods for inverse problems described by nonlinear operator equations $F(u)=g$ in Banach spaces where the Newton equations $F'(u_n;u_{n+1}-u_n) = g-F(u_n)$ are regularized variationally using a general data misfit functional and a convex regularization term. This generalizes the well-known iteratively regularized Gauss-Newton method (IRGNM). We prove convergence and convergence rates as the noise level tends to 0 both for an a priori stopping rule and for a Lepski{\u\i}-type a posteriori stopping rule. Our analysis includes previous order optimal convergence rate results for the IRGNM as special cases. The main focus of this paper is on inverse problems with Poisson data where the natural data misfit functional is given by the Kullback-Leibler divergence. Two examples of such problems are discussed in detail: an inverse obstacle scattering problem with amplitude data of the far-field pattern and a phase retrieval problem. The performence of the proposed method for these problems is illustrated in numerical examples

Convergence rates in expectation for Tikhonov-type regularization of Inverse Problems with Poisson data

In this paper we study a Tikhonov-type method for ill-posed nonlinear operator equations \gdag = F( ag) where \gdag is an integrable, non-negative function. We assume that data are drawn from a Poisson process with density t\gdag where $t>0$ may be interpreted as an exposure time. Such problems occur in many photonic imaging applications including positron emission tomography, confocal fluorescence microscopy, astronomic observations, and phase retrieval problems in optics. Our approach uses a Kullback-Leibler-type data fidelity functional and allows for general convex penalty terms. We prove convergence rates of the expectation of the reconstruction error under a variational source condition as $t\to\infty$ both for an a priori and for a Lepski{\u\i}-type parameter choice rule

Variational Data Assimilation via Sparse Regularization

This paper studies the role of sparse regularization in a properly chosen basis for variational data assimilation (VDA) problems. Specifically, it focuses on data assimilation of noisy and down-sampled observations while the state variable of interest exhibits sparsity in the real or transformed domain. We show that in the presence of sparsity, the $\ell_{1}$-norm regularization produces more accurate and stable solutions than the classic data assimilation methods. To motivate further developments of the proposed methodology, assimilation experiments are conducted in the wavelet and spectral domain using the linear advection-diffusion equation