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In this work, we develop a Bayesian framework for solving inverse problems in which the unknown parameter belongs to a space of Radon measures taking values in a separable Hilbert space. The inherent ill-posedness of such problems is…
Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for…
Building up on classical linear formulations, we posit that a broad class of problems in signal synthesis and in signal recovery are reducible to the basic task of finding a point in a closed convex subset of a Hilbert space that satisfies…
In this paper, we propose two algorithms for solving linear inverse problems when the observations are corrupted by noise. A proper data fidelity term (log-likelihood) is introduced to reflect the statistics of the noise (e.g. Gaussian,…
Accurate determination of the regularization parameter in inverse problems still represents an analytical challenge, owing mainly to the considerable difficulty to separate the unknown noise from the signal. We present a new approach for…
The choice of the parameter value for regularized inverse problems is critical to the results and remains a topic of interest. This article explores a criterion for selecting a good parameter value by maximizing the probability of the data,…
In this paper, we consider a modified Levenberg--Marquardt method for solving an ill-posed inverse problem where the forward mapping is not G\^ateaux differentiable. By relaxing the standard assumptions for the classical smooth setting, we…
This paper aims to develop and analyze a numerical scheme for solving the backward problem of semilinear subdiffusion equations. We establish the existence, uniqueness, and conditional stability of the solution to the inverse problem by…
A number of regularization methods for discrete inverse problems consist in considering weighted versions of the usual least square solution. However, these so-called filter methods are generally restricted to monotonic transformations,…
When solving rank-deficient or discrete ill-posed problems by regularization methods, the choice of the regularization parameter is crucial. It is also of interest, the regularization norm used in the selection of the solution. In this…
Regularization is a critical technique for ensuring well-posedness in solving inverse problems with incomplete measurement data. Traditionally, the regularization term is designed based on prior knowledge of the unknown signal's…
In this work, in the context of Linear and Quadratic Programming, we interpret Primal Dual Regularized Interior Point Methods (PDR-IPMs) in the framework of the Proximal Point Method. The resulting Proximal Stabilized IPM (PS-IPM) is…
In recent years, new regularization methods based on (deep) neural networks have shown very promising empirical performance for the numerical solution of ill-posed problems, e.g., in medical imaging and imaging science. Due to the…
This paper discusses the solution of nonlinear integral equations with noisy integral kernels as they appear in nonparametric instrumental regression. We propose a regularized Newton-type iteration and establish convergence and convergence…
There has been a recent interest in imitation learning methods that are guaranteed to produce a stabilizing control law with respect to a known system. Work in this area has generally considered linear systems and controllers, for which…
In the paper, we introduce several accelerate iterative algorithms for solving the multiple-set split common fixed-point problem of quasi-nonexpansive operators in real Hilbert space. Based on primal-dual method, we construct several…
We consider the task of image reconstruction while simultaneously decomposing the reconstructed image into components with different features. A commonly used tool for this is a variational approach with an infimal convolution of…
For linear inverse problems $Y=\mathsf{A}\mu+\xi$, it is classical to recover the unknown signal $\mu$ by iterative regularisation methods $(\widehat \mu^{(m)}, m=0,1,\ldots)$ and halt at a data-dependent iteration $\tau$ using some…
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…
Inverse problems arise in a variety of imaging applications including computed tomography, non-destructive testing, and remote sensing. The characteristic features of inverse problems are the non-uniqueness and instability of their…