1,721,219 research outputs found
EXISTENCE OF MULTIPLE SOLUTIONS FOR A FOURTH-ORDER PROBLEM WITH VARIABLE EXPONENT
No full textWe provide a new multiplicity result for a weighted p(x)-biharmonic problem on a bounded domain Ω of Rnwith Navier conditions on ∂Ω. Our approach, of variational nature, requires a suitable oscillating behavior of the nonlinearity and the associated weight to be compactly supported in Ω
Theoretical and numerical aspects of a non-stationary preconditioned iterative method for linear discrete ill-posed problems
Full text linkThis work considers some theoretical and computational aspects of the recent paper (Buccini et al., 2021), whose aim was to relax the convergence conditions in a previous work by Donatelli and Hanke, and thereby make the iterative method discussed in the latter work applicable to a larger class of problems. This aim was achieved in the sense that the iterative method presented convergences for a larger class of problems. However, while the analysis presented is correct, it does not establish the superior behavior of the iterative method described. The present note describes a slight modification of the analysis that establishes the superiority of the iterative method. The new analysis allows to discuss the behavior of the algorithm when varying the involved parameters, which is also useful for their empirical estimation
regularization for image reconstruction
No full textThe use of the Laplacian of a properly constructed graph for denoising images has attracted a lot of attention in the last years. Recently, a way to use this instrument for image deblurring has been proposed. Even though the previously proposed method was able to provide extremely accurate reconstructions, it had several limitations, namely it was only applicable when periodic boundary conditions were employed, the regularization parameter had to be hand-tuned, and only convex regularization terms were allowed. In this paper, we propose two automatic methods that do not need the tuning of any parameter and that can be used for different imaging problems. Moreover, thanks to the projection into properly constructed subspaces of fairly small dimension, the proposed algorithms can be used for solving large scale problems
Regularization of inverse problems by an approximate matrix-function technique
Full text linkIn this work, we introduce and investigate a class of matrix-free regularization techniques for discrete linear ill-posed problems based on the approximate computation of a special matrix-function. In order to produce a regularized solution, the proposed strategy employs a regular approximation of the Heavyside step function computed into a small Krylov subspace. This particular feature allows our proposal to be independent from the structure of the underlying matrix. If on the one hand, the use of the Heavyside step function prevents the amplification of the noise by suitably filtering the responsible components of the spectrum of the discretization matrix, on the other hand, it permits the correct reconstruction of the signal inverting the remaining part of the spectrum. Numerical tests on a gallery of standard benchmark problems are included to prove the efficacy of our approach even for problems affected by a high level of noise
Fractional graph Laplacian for image reconstruction
Full text linkImage reconstruction problems, like image deblurring and computer tomography, are usually ill-posed and require regularization. A popular approach to regularization is to substitute the original problem with an optimization problem that minimizes the sum of two terms, an
term and an
term with
. The first penalizes the distance between the measured data and the reconstructed one, the latter imposes sparsity on some features of the computed solution.
In this work, we propose to use the fractional Laplacian of a properly constructed graph in the
term to compute extremely accurate reconstructions of the desired images. A simple model with a fully automatic method, i.e., that does not require the tuning of any parameter, is used to construct the graph and enhanced diffusion on the graph is achieved with the use of a fractional exponent in the Laplacian operator. Since the fractional Laplacian is a global operator, i.e., its matrix representation is completely full, it cannot be formed and stored. We propose to replace it with an approximation in an appropriate Krylov subspace. We show that the algorithm is a regularization method under some reasonable assumptions. Some selected numerical examples in image deblurring and computer tomography show the performance of our proposal
A general framework for ADMM acceleration
Full text linkThe Alternating Direction Multipliers Method (ADMM) is a very popular algorithm for computing the solution of convex constrained minimization problems. Such problems are important from the application point of view, since they occur in many fields of science and engineering. ADMM is a powerful numerical tool, but unfortunately its main drawback is that it can exhibit slow convergence. Several approaches for its acceleration have been proposed in the literature and in this paper we present a new general framework devoted to this aim. In particular, we describe an algorithmic framework that makes possible the application of any acceleration step while still having the guarantee of convergence. This result is achieved thanks to a guard condition that ensures the monotonic decrease of the combined residual. The proposed strategy is applied to image deblurring problems. Several acceleration techniques are compared; to the best of our knowledge, some of them are investigated for the first time in connection with ADMM. Numerical results show that the proposed framework leads to a faster convergence with respect to other acceleration strategies recently introduced for ADMM
Graph approximation and generalized Tikhonov regularization for signal deblurring
No full textGiven a compact linear operator K, the (pseudo) inverse K^† is usually substituted by a family of regularizing operators Rα which depends on K itself. Naturally, in the actual computation we are forced to approximate the true continuous operator K with a discrete operator K^(n) characterized by a finesses discretization parameter n, and obtaining then a discretized family of regularizing operators R_α ^(n). In general, the numerical scheme applied to discretize K does not preserve, asymptotically, the full spectrum of K. In the context of a generalized Tikhonov-type regularization, we show that a graph-based discretization scheme that guarantees, asymptotically, a zero maximum relative spectral error can significantly improve the approximated solutions given by R_α ^(n). This approach is combined with a graph based regularization technique with respect to the penalty term
- …
