Related papers: A priori parameter choice in Tikhonov regularizati…
We consider the ill-posed operator equation $Ax=y$ with an injective and bounded linear operator $A$ mapping between $\ell^2$ and a Hilbert space $Y$, possessing the unique solution \linebreak $x^\dag=\{x^\dag_k\}_{k=1}^\infty$. For the…
This work is concerned with linear inverse problems where a distributed parameter is known a priori to only take on values from a given discrete set. This property can be promoted in Tikhonov regularization with the aid of a suitable convex…
These lecture notes for a graduate class present the regularization theory for linear and nonlinear ill-posed operator equations in Hilbert spaces. Covered are the general framework of regularization methods and their analysis via spectral…
Most of the literature on the solution of linear ill-posed operator equations, or their discretization, focuses only on the infinite-dimensional setting or only on the solution of the algebraic linear system of equations obtained by…
Despite recent advances in regularisation theory, the issue of parameter selection still remains a challenge for most applications. In a recent work the framework of statistical learning was used to approximate the optimal Tikhonov…
We study Tikhonov regularization for solving ill--posed operator equations where the solutions are functions defined on surfaces. One contribution of this paper is an error analysis of Tikhonov regularization which takes into account…
In this article on variational regularization for ill-posed nonlinear problems, we are once again discussing the consequences of an oversmoothing penalty term. This means in our model that the searched-for solution of the considered…
In this short note, we formulate the convergence rates of the well known Tikhonov regularization scheme for solving the nonlinear ill-posed problems in Banach spaces. For deriving the convergence rates, we employ the novel smoothness…
We consider the nonstationary iterated Tikhonov regularization in Banach spaces which defines the iterates via minimization problems with uniformly convex penalty term. The penalty term is allowed to be non-smooth to include $L^1$ and total…
For solving linear ill-posed problems regularization methods are required when the right hand side is with some noise. In the present paper regularized solutions are obtained by implicit iteration methods in Hilbert scales. % By exploiting…
We consider a modified Tikhonov-type functional for the solution of ill-posed nonlinear inverse problems. Motivated by applications in the field of production engineering, we allow small deviations in the solution, which are modeled through…
We study weighted Tikhonov regularization for large-scale linear discrete ill-posed problems with random noise. Under a polynomial upper-bound assumption on the generalized eigenvalues of the discrete forward operator, we derive stochastic…
In this paper, in the setting of Hilbert spaces, we consider a Tikhonov regularized second-order plus first-order primal-dual dynamical system with asymptotically vanishing damping for a linear equality constrained convex optimization…
In this article, we consider the Tikhonov regularization of an optimal control problem of semilinear partial differential equations with box constraints on the control. We derive a-priori regularization error estimates for the control under…
In this paper, we deal with nonlinear ill-posed problems involving monotone operators and consider Lavrentiev's regularization method. This approach, in contrast to Tikhonov's regularization method, does not make use of the adjoint of the…
Convergence rates results for Tikhonov regularization of nonlinear ill-posed operator equations in abstract function spaces require the handling of both smoothness conditions imposed on the solution and structural conditions expressing the…
Despite a variety of available techniques the issue of the proper regularization parameter choice for inverse problems still remains one of the biggest challenges. The main difficulty lies in constructing a rule, allowing to compute the…
Measuring the error by an l^1-norm, we analyze under sparsity assumptions an l^0-regularization approach, where the penalty in the Tikhonov functional is complemented by a general stabilizing convex functional. In this context, ill-posed…
The new version of a posteriori choice (NVAC) of the regularization parameter in the classical Tikhonov regularization method is considered. Lemmas and theorems on the error and the asymptotic convergence rate of the regularized solution…
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…