483 research outputs found
Discrete Approximations of a Controlled Sweeping Process
The paper is devoted to the study of a new class of optimal control problems
governed by the classical Moreau sweeping process with the new feature that the polyhe-
dral moving set is not fixed while controlled by time-dependent functions. The dynamics of
such problems is described by dissipative non-Lipschitzian differential inclusions with state
constraints of equality and inequality types. It makes challenging and difficult their anal-
ysis and optimization. In this paper we establish some existence results for the sweeping
process under consideration and develop the method of discrete approximations that allows
us to strongly approximate, in the W^{1,2} topology, optimal solutions of the continuous-type
sweeping process by their discrete counterparts
Passing to the limit in maximal slope curves: from a regularized Perona-Malik equation to the total variation flow
We prove that solutions of a mildly regularized Perona-Malik equation
converge, in a slow time scale, to solutions of the total variation flow. The
convergence result is global-in-time, and holds true in any space dimension.
The proof is based on the general principle that "the limit of gradient-flows
is the gradient-flow of the limit". To this end, we exploit a general result
relating the Gamma-limit of a sequence of functionals to the limit of the
corresponding maximal slope curves.Comment: 19 page
Diffusion-Steered Super-Resolution Image Reconstruction
For decades, super-resolution has been a widely applied technique to improve the spatial resolution of an image without hardware modification. Despite the advantages, super-resolution suffers from ill-posedness, a problem that makes the technique susceptible to multiple solutions. Therefore, scholars have proposed regularization approaches as attempts to address the challenge. The present work introduces a parameterized diffusion-steered regularization framework that integrates total variation (TV) and Perona-Malik (PM) smoothing functionals into the classical super-resolution model. The goal is to establish an automatic interplay between TV and PM regularizers such that only their critical useful properties are extracted to well pose the super-resolution problem, and hence, to generate reliable and appreciable results. Extensive analysis of the proposed resolution-enhancement model shows that it can respond well on different image regions. Experimental results provide further evidence that the proposed model outperforms
The Residual Method for Regularizing Ill-Posed Problems
Although the \emph{residual method}, or \emph{constrained regularization}, is
frequently used in applications, a detailed study of its properties is still
missing. This sharply contrasts the progress of the theory of Tikhonov
regularization, where a series of new results for regularization in Banach
spaces has been published in the recent years. The present paper intends to
bridge the gap between the existing theories as far as possible. We develop a
stability and convergence theory for the residual method in general topological
spaces. In addition, we prove convergence rates in terms of (generalized)
Bregman distances, which can also be applied to non-convex regularization
functionals. We provide three examples that show the applicability of our
theory. The first example is the regularized solution of linear operator
equations on -spaces, where we show that the results of Tikhonov
regularization generalize unchanged to the residual method. As a second
example, we consider the problem of density estimation from a finite number of
sampling points, using the Wasserstein distance as a fidelity term and an
entropy measure as regularization term. It is shown that the densities obtained
in this way depend continuously on the location of the sampled points and that
the underlying density can be recovered as the number of sampling points tends
to infinity. Finally, we apply our theory to compressed sensing. Here, we show
the well-posedness of the method and derive convergence rates both for convex
and non-convex regularization under rather weak conditions.Comment: 29 pages, one figur
Comparison of numerical methods for the reconstruction of elastic obstacles from the far-field data of scattered acoustic waves
We consider the inverse problem for an elastic body emerged in a fluid due to an acoustic wave. The shape of this obstacle is to be reconstructed from the far-field pattern of the scattered wave. For the numerical solution in the two-dimensional case, we compare a simple Newton type iteration method with the Kirsch-Kress algorithm. Our computational tests reveal that the Kirsch-Kress method converges faster for obstacles with very smooth boundaries. The simple Newton method, however, is more stable in the case of not so smooth domains and more robust with respect to measurement errors
Optimal control problems in transport dynamics
In the present paper we deal with an optimal control problem related to a model in population dynamics; more precisely, the goal is to modify the behavior of a given density of individuals via another population of agents interacting with the first. The cost functional to be minimized to determine the dynamics of the second population takes into account the desired target or configuration to be reached as well as the quantity of control agents. Several applications may fall into this framework, as for instance driving a mass of pedestrian in (or out of) a certain location; influencing the stock market by acting on a small quantity of key investors; controlling a swarm of unmanned aerial vehicles by means of few piloted drones
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