Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA note on the minimization of a Tikhonov functional with $ell^1$-penalty
405
0
0.0
(
0
)
Added by
Simon Hubmer
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
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 transform the Tikhonov functional into one with an $ell_2$ penalty term but a nonlinear operator. The transformed problem can then be analyzed and minimized using standard methods. However, by the nature of this transform, the resulting functional is only once continuously differentiable, which prohibits the use of second order methods. Hence, in this paper, we propose a different transformation, which leads to a twice differentiable functional that can now be minimized using efficient second order methods like Newtons method. We provide a convergence analysis of our proposed scheme, as well as a number of numerical results showing the usefulness of our proposed approach.
rate research
Read More
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 derivative from scattered data with random noise. Since the noise level is not assumed small, we will use the amount of data for reducing the random error, and use a relatively small number of knots for interpolation. An indicator function for our algorithm is constructed. It indicates where the numerical results are good or may not be good. The corresponding error estimates are obtained. We show how to choose the number of interpolation knots in the reconstruction process for balancing the random errors and interpolation errors. Numerical examples show the effectiveness and rapidity of our method. It should be remarked that the algorithm in this paper can be used for on-line data.
In this corrigendum, we offer a correction to [J. Korean. Math. Soc., 54 (2017), pp. 461--477]. We construct a counterexample for the strengthened Cauchy--Schwarz inequality used in the original paper. In addition, we provide a new proof for Lemma 5 of the original paper, an estimate for the extremal eigenvalues of the standard unpreconditioned FETI-DP dual operator.
Subspace recycling iterative methods and other subspace augmentation schemes are a successful extension to Krylov subspace methods in which a Krylov subspace is augmented with a fixed subspace spanned by vectors deemed to be helpful in accelerating convergence or conveying knowledge of the solution. Recently, a survey was published, in which a framework describing the vast majority of such methods was proposed [Soodhalter et al, GAMM-Mitt. 2020]. In many of these methods, the Krylov subspace is one generated by the system matrix composed with a projector that depends on the augmentation space. However, it is not a requirement that a projected Krylov subspace be used. There are augmentation methods built on using Krylov subspaces generated by the original system matrix, and these methods also fit into the general framework. In this note, we observe that one gains implementation benefits by considering such augmentation methods with unprojected Krylov subspaces in the general framework. We demonstrate this by applying the idea to the R$^3$GMRES method proposed in [Dong et al. ETNA 2014] to obtain a simplified implementation and to connect that algorithm to early augmentation schemes based on flexible preconditioning [Saad. SIMAX 1997].
We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Our main interest is to obtain preasymptotic estimates for the corresponding sampling numbers. We obtain results in terms of the decay of related singular numbers of the compact embedding into $L_2(D,varrho_D)$ multiplied with the supremum of the Christoffel function of the subspace spanned by the first $m$ singular functions. Here the measure $varrho_D$ is at our disposal. As an application we obtain near optimal upper bounds for the sampling numbers for periodic Sobolev type spaces with general smoothness weight. Those can be bounded in terms of the corresponding benchmark approximation number in the uniform norm, which allows for preasymptotic bounds. By applying a recently introduced sub-sampling technique related to Weavers conjecture we mostly lose a $sqrt{log n}$ and sometimes even less. Finally we point out a relation to the corresponding Kolmogorov numbers.
A main drawback of classical Tikhonov regularization is that often the parameters required to apply theoretical results, e.g., the smoothness of the sought-after solution and the noise level, are unknown in practice. In this paper we investigate in new detail the residuals in Tikhonov regularization viewed as functions of the regularization parameter. We show that the residual carries, with some restrictions, the information on both the unknown solution and the noise level. By calculating approximate solutions for a large range of regularization parameters, we can extract both parameters from the residual given only one set of noisy data and the forward operator. The smoothness in the residual allows to revisit parameter choice rules and relate a-priori, a-posteriori, and heuristic rules in a novel way that blurs the lines between the classical division of the parameter choice rules. All results are accompanied by numerical experiments.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
