Related papers: On One Property of Tikhonov Regularization Algorit…
Given a proper convex lower semicontinuous function defined on a Hilbert space and whose solution set is supposed nonempty. For attaining a global minimizer when this convex function is continuously differentiable, we approach it by a…
In this paper, we study the Tikhonov regularization scheme in Hilbert scales for the nonlinear statistical inverse problem with a general noise. The regularizing norm in this scheme is stronger than the norm in Hilbert space. We focus on…
Tikhonov regularization is a common technique used when solving poorly behaved optimization problems. Often, and with good reason, this technique is applied by practitioners in an ad hoc fashion. In this note, we systematically illustrate…
In this paper, the classification algorithm arising from Tikhonov regularization is discussed. The main intention is to derive learning rates for the excess misclassification error according to the convex $\eta$-norm loss function…
Nonlocal operators with integral kernels have become a popular tool for designing solution maps between function spaces, due to their efficiency in representing long-range dependence and the attractive feature of being resolution-invariant.…
We study multi-parameter regularization (multiple penalties) for solving linear inverse problems to promote simultaneously distinct features of the sought-for objects. We revisit a balancing principle for choosing regularization parameters…
In this work we consider the problem of finding optimal regularization parameters for general-form Tikhonov regularization using training data. We formulate the general-form Tikhonov solution as a spectral filtered solution using the…
Regularization is a long-standing challenge for ill-posed linear inverse problems, and a prototype is the Fredholm integral equation of the first kind with additive Gaussian measurement noise. We introduce a new RKHS regularization adaptive…
In this paper the problem of recovering a regularized solution of the Fredholm integral equations of the first kind with Hermitian and square-integrable kernels, and with data corrupted by additive noise, is considered. Instead of using a…
In this paper we present a globally convergent algorithm for the computation of a minimizer of the Tikhonov functional with sparsity promoting penalty term for nonlinear forward operators in Banach space. The dual TIGRA method uses a…
We consider the problem of estimating the slope function in a functional regression with a scalar response and a functional covariate. This central problem of functional data analysis is well known to be ill-posed, thus requiring a…
We consider the estimation of quadratic functionals in a Gaussian sequence model where the eigenvalues are supposed to be unknown and accessible through noisy observations only. Imposing smoothness assumptions both on the signal and the…
The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most popular regularization methods for…
We study Tikhonov regularization for possibly nonlinear inverse problems with weighted $\ell^1$-penalization. The forward operator, mapping from a sequence space to an arbitrary Banach space, typically an $L^2$-space, is assumed to satisfy…
In this paper, we investigate the inverse problem on determining the spatial component of the source term in a hyperbolic equation with time-dependent principal part. Based on a newly established Carleman estimate for general hyperbolic…
The total least squares problem with the general Tikhonov regularization can be reformulated as a one-dimensional parametric minimization problem (PM), where each parameterized function evaluation corresponds to solving an n-dimensional…
In this paper we consider the computation of approximate solutions for inverse problems in Hilbert spaces. In order to capture the special feature of solutions, non-smooth convex functions are introduced as penalty terms. By exploiting the…
The joint bidiagonalization process of a matrix pair $\{A,L\}$ can be used to develop iterative regularization algorithms for large scale ill-posed problems in general-form Tikhonov regularization…
In this paper, we consider the minimization of a Tikhonov functional with an $\ell_1$ penalty for solving linear inverse problems with sparsity constraints. One of the many approaches used to solve this problem uses the Nemskii operator to…
Bayesian regularization is a central tool in modern-day statistical and machine learning methods. Many applications involve high-dimensional sparse signal recovery problems. The goal of our paper is to provide a review of the literature on…