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With the rapid growth of data, how to extract effective information from data is one of the most fundamental problems. In this paper, based on Tikhonov regularization, we propose an effective method for reconstructing the function and its…
This paper is concerned with the regularization of large-scale discrete inverse problems by means of inexact Krylov methods. Specifically, we derive two new inexact Krylov methods that can be efficiently applied to unregularized or…
Tikhonov regularization is a popular approach to obtain a meaningful solution for ill-conditioned linear least squares problems. A relatively simple way of choosing a good regularization parameter is given by Morozov's discrepancy…
This paper provides a new regularization method which is particularly suitable for linear exponentially ill-posed problems. Under logarithmic source conditions (which have a natural interpretation in terms of Sobolev spaces in the…
Regularization and interior point approaches offer valuable perspectives to address constrained nonlinear optimization problems in view of control applications. This paper discusses the interactions between these techniques and proposes an…
In this work we present a novel optimization strategy for image reconstruction tasks under analysis-based image regularization, which promotes sparse and/or low-rank solutions in some learned transform domain. We parameterize such…
Recently, inverse problems have attracted more and more attention in computational mathematics and become increasingly important in engineering applications. After the discretization, many of inverse problems are reduced to linear systems.…
High radiation dose in CT scans increases a lifetime risk of cancer and has become a major clinical concern. Recently, iterative reconstruction algorithms with Total Variation (TV) regularization have been developed to reconstruct CT images…
To deal with the ill-posed nature of the inverse heat conduction problem (IHCP), the regularization parameter alpha can be incorporated into a minimization problem, which is known as Tikhonov regularization method, a popular technique to…
This paper surveys an important class of methods that combine iterative projection methods and variational regularization methods for large-scale inverse problems. Iterative methods such as Krylov subspace methods are invaluable in the…
We investigate geometric regularization strategies for learned latent representations in encoder--decoder reduced-order models. In a fixed experimental setting for the advection--diffusion--reaction (ADR) equation, we model latent dynamics…
We aim at the solution of inverse problems in imaging, by combining a penalized sparse representation of image patches with an unconstrained smooth one. This allows for a straightforward interpretation of the reconstruction. We formulate…
We study the Tikhonov regularization for ill-posed non-linear operator equations in Hilbert scales. Our focus is on the interplay between the smoothness-promoting properties of the penalty and the smoothness inherent in the solution. The…
This paper is concerned with the numerical solution of nonlinear ill-posed operator equations involving convex constraints. We study a Newton-type method which consists in applying linear Tikhonov regularization with convex constraints to…
The iterative refinement method (IRM) has been very successfully applied in many different fields for examples the modern quantum chemical calculation and CT image reconstruction. It is proved that the refinement method can create an exact…
This paper presents an iterative inversion algorithm for computed tomography image reconstruction that performs well in terms of accuracy and speed using limited data. The computational method combines an image domain technique and…
Computing the regularized solution of Bayesian linear inverse problems as well as the corresponding regularization parameter is highly desirable in many applications. This paper proposes a novel iterative method, termed the Projected Newton…
This paper proposes to solve the Total Variation regularized models by finding the residual between the input and the unknown optimal solution. After analyzing a previous method, we developed a new iterative algorithm, named as Residual…
Solving equilibrium problems under constraints is an important problem in optimization and optimal control. In this context an important practical challenge is the efficient incorporation of constraints. We develop a continuous-time method…
Convolutional neural networks are capable of learning powerful representational spaces, which are necessary for tackling complex learning tasks. However, due to the model capacity required to capture such representations, they are often…