Nonlocal Setting Research Articles

A new nonlocal variational setting for image processing

We introduce a new nonlocal variational scheme for image denoising. This scheme is motivated by, but different from the nonlocal means filter of Buades et al [9] and the nonlocal TV model proposed by Gilboa-Osher by using nonlocal operators. Our approach is based on general geometric considerations. Experiments show that the corresponding TV model yields denoising results that can compare favorably with those obtained by other methods.

Symmetries and their importance for statistical turbulence theory

The special importance of turbulence maybe comprehended by its ubiquity in innumerable natural and technical systems. Examples for natural turbulent flows are the atmospheric flow and the oceanic current which to calculate is a crucial point in climate research. Classical engineering application involving turbulence are the flows around airplanes or cars or turbulence within jet or reciprocating engines. Though supposed for more than hundred years, only with the advent of super computers it became apparent that the Navier-Stokes equations provide an excellent continuummechanical model for turbulent flows. Still, the exclusive and direct application of the Navier-Stokes equations to practical flow problems at very high Reynolds numbers without invoking any additional assumptions is still several decades away. However, in most applications it is not at all necessary to know all the detailed fluctuations of velocity and pressure present in turbulent flows but for the most part statistical measures are sufficient. This was in fact the key idea of O. Reynolds who was the first to suggest a statistical description of turbulence. The Navier-Stokes equations, however, constitute a non-linear and, due to the pressure Poisson equation, a non-local set of equations. As an immediate consequence of this the equations for the mean or expectation values for velocity and pressure lead to an infinite set of statistical equations, or, if truncated at some level of statistics, an un-closed system is generated.

A pseudo-index approach to fractional equations

The aim of this paper is investigating the existence and multiplicity of weak solutions to non-local equations involving a general integro-differential operator of fractional type, when the nonlinearity is subcritical and asymptotically linear at infinity. More precisely, in presence of an odd symmetric non-linear term, we prove multiplicity results by using a pseudo-index theory related to the genus. As a particular case we derive existence and multiplicity results for non-local equations involving the fractional Laplacian operator. Our theorems, obtained exploiting a novel abstract framework, extend to the non-local setting some results, already known in the literature, in the case of the classical Laplace operator.

A Peano theorem for fuzzy differential equations with evolving membership grade

The Peano theorem on the existence without possible uniqueness of solutions has been a perplexing problem in the theory of fuzzy differential equations. The difficulty appears to be due to the standard use of the supremum metric ▪∞ defined by the supremum over the Hausdorff metric between the level sets of the fuzzy sets. Another may have been the classical formulation of fuzzy differential equations in terms of the Hukuhara derivative of the level sets. Here a Peano theorem is established for fuzzy differential equations formulated in a recent paper by the authors by combining Hüllermeier's suggestion of defining fuzzy differential equations at each level set via differential inclusions with Aubin's morphological equations, which allow non-local set evolution. A major difference from previous publications is the use of the endograph metric ▪end, essentially the Hausdorff metric between the endographs in Rn×[0,1] of fuzzy sets, instead of the supremum metric ▪∞. Another is that the membership grades of the fuzzy sets are also allowed to evolve under the fuzzy differential equations. The result applies for a very general class of fuzzy sets without additional assumptions of fuzzy convexity, compact supports or even normality.

Multiplicity of solutions for a class of superlinear non-local fractional equations

In this paper, we are concerned with the problem driven by a non-local integro-differential operator with homogeneous Dirichlet boundary conditions. As a particular case, we study multiple solutions for the following non-local fractional Laplace equations:where is fixed parameter, is an open bounded subset of with smooth boundary () and is the fractional Laplace operator. By a variant version of the Mountain Pass Theorem, a multiplicity result is obtained for the above-mentioned superlinear problem without Ambrosetti–Rabinowitz condition. Consequently, the result may be looked as a complete extension of the previous work of Wang and Tang to the non-local fractional setting.

The second eigenvalue of the fractional p-Laplacian

AbstractWe consider the eigenvalue problem for the fractional p-Laplacian in an open bounded, possibly disconnected set ${\Omega\subset\mathbb{R}^{n}}$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfunctions, we show that the second eigenvalue ${\lambda_{2}(\Omega)}$ is well-defined, and we characterize it by means of several equivalent variational formulations. In particular, we extend the mountain pass characterization of Cuesta, De Figueiredo and Gossez to the nonlocal and nonlinear setting. Finally, we consider the minimization problem$\inf\{\lambda_{2}(\Omega):|\Omega|=c\}.$We prove that, differently from the local case, an optimal shape does not exist, even among disconnected sets. A minimizing sequence is given by the union of two disjoint balls of volume ${c/2}$ whose mutual distance tends to infinity.

On a new class of fractional partial differential equations

Abstract In this paper, we study a new class of fractional partial differential equations which are obtained by minimizing variational problems in fractional Sobolev spaces. We introduce a notion of fractional gradient which has the potential to extend many classical results in the Sobolev spaces to the nonlocal and fractional setting in a natural way.

Adaptive Sparse Norm and Nonlocal Total Variation Methods for Image Smoothing

In computer vision and graphics, it is challenging to decompose various texture/structure patterns from input images. It is well recognized that how edges are defined and how this prior information guides smoothing are two keys in determining the quality of image smoothing. While many different approaches have been reported in the literature, sparse norm and nonlocal schemes are two promising tools. In this study, by integrating a texture measure as the spatially varying data-fidelity/smooth-penalty weight into the sparse norm and nonlocal total variation models, two new methods are presented for feature/structure-preserving filtering. The first one is a generalized relative total variation (i.e., GRTV) method, which improves the contrast-preserving and edge stiffness-enhancing capabilities of the RTV by extending the range of the penalty function’s norm from 1 to [0, 1]. The other one is a nonlocal version of generalized RTV (i.e., NLGRTV) for which the key idea is to use a modified texture-measure as spatially varying penalty weight and to replace the local candidate pixels with the nonlocal set in the smooth-penalty term. It is shown that NLGRTV substantially improves the performance of decomposition for regions with faint pixel-boundary.

Existence of weak solutions for non-local fractional problems via Morse theory

In this paper, we study the existence of non-trivial solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions. Non-trivial solutions are obtained by computing the critical groups and Morse theory. Our results extend some classical theorems for semilinear elliptic equations to the non-local fractional setting.

A Resonance Problem for Non-Local Elliptic Operators

In this paper we consider a resonance problem driven by a non-local integrodifferential operator \mathcal L_K with homogeneous Dirichlet boundary conditions. This problem has a variational structure and we find a solution for it using the Saddle Point Theorem. We prove this result for a general integrodifferential operator of fractional type and from this, as a particular case, we derive an existence theorem for the following fractional Laplacian equation \left\{ \begin{alignedat}{2} (-\Delta)^s u&=\lambda a(x)u+f(x,u)& \quad&{\text{in }} \Omega\\ u&=0& &{\text{in }} \mathbb{R}^n\setminus \Omega, \end{alignedat} \right. when \lambda is an eigenvalue of the related non-homogenous linear problem with homogeneous Dirichlet boundary data. Here the parameter s\in (0,1) is fixed, \Omega is an open bounded set of \mathbb R^n , n>2s , with Lipschitz boundary, a is a Lipschitz continuous function, while f is a sufficiently smooth function. This existence theorem extends to the non-local setting some results, already known in the literature in the case of the Laplace operator -\Delta .

MRI denoising using nonlocal neutrosophic set approach of Wiener filtering

In this paper, a new filtering method is presented to remove the Rician noise from magnetic resonance images (MRI) acquired using single coil MRI acquisition system. This filter is based on nonlocal neutrosophic set (NLNS) approach of Wiener filtering. A neutrosophic set (NS), a part of neutrosophy theory, studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. Now, we apply the neutrosophic set into image domain and define some concepts and operators for image denoising. First, the nonlocal mean is applied to the noisy MRI. The resultant image is transformed into NS domain, described using three membership sets: true (T), indeterminacy (I) and false (F). The entropy of the neutrosophic set is defined and employed to measure the indeterminacy. The ω-Wiener filtering operation is used on T and F to decrease the set indeterminacy and to remove the noise. The experiments have been conducted on simulated MR images from Brainweb database and clinical MR images. The results show that the NLNS Wiener filter produces better denoising results in terms of qualitative and quantitative measures compared with other denoising methods, such as classical Wiener filter, the anisotropic diffusion filter, the total variation minimization and the nonlocal means filter. The visual and the diagnostic quality of the denoised image are well preserved.

Lewy–Stampacchia type estimates for variational inequalities driven by (non)local operators

The purpose of this paper is to derive some Lewy–Stampacchia estimates in some cases of interest, such as the ones driven by non-local operators. Since we will perform an abstract approach to the problem, this will provide, as a byproduct, Lewy–Stampacchia estimates in more classical cases as well. In particular, we can recover the known estimates for the standard Laplacian, the p -Laplacian, and the Laplacian in the Heisenberg group. In the non-local framework we prove a Lewy–Stampacchia estimate for a general integrodifferential operator and, as a particular case, for the fractional Laplacian. As far as we know, the abstract framework and the results in the non-local setting are new.

Effect of molecular-orbital rotations on ground-state energies in the parametric two-electron reduced density matrix method

Different sets of molecular orbitals and the rotations connecting them are of great significance in molecular electronic structure. Most electron correlation methods depend on a reference wave function that separates the orbitals into occupied and unoccupied spaces. Energies and properties from these methods depend upon rotations between the spaces. Some electronic structure methods, such as modified coupled electron pair approximations and the recently developed parametric two-electron reduced density matrix (2-RDM) methods [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)], also depend upon rotations between occupied orbitals and rotations between unoccupied orbitals. In this paper, we explore the sensitivity of the ground-state energies from the parametric 2-RDM method to rotations within the occupied space and within the unoccupied space. We discuss the theoretical origin of the rotational dependence and provide computational examples at both equilibrium and non-equilibrium geometries. We also study the effect of these rotations on the size extensivity of the parametric 2-RDM method. Computations show that the orbital rotations have a small effect upon the parametric 2-RDM energies in comparison to the energy differences observed between methodologies such as coupled cluster and parametric 2-RDM. Furthermore, while the 2-RDM method is rigorously size extensive in a local molecular orbital basis set, calculations reveal negligible deviations in nonlocal molecular orbital basis sets such as those from canonical Hartree-Fock calculations.

Cell polarisation model: The 1D case

We study the dynamics of a one-dimensional non-linear and non-local drift-diffusion equation set in the half-line, with the coupling involving the trace value on the boundary. The initial mass M of the density determines the behaviour of the equation: attraction to self-similar profile, to a steady state of finite time, blow-up for supercritical mass. Using the logarithmic Sobolev and the HWI inequalities we obtain a rate of convergence for the sub-critical and critical mass cases. Moreover, we prove a comparison principle on the equation obtained after space integration. This concentration-comparison principle allows proving blow-up of solutions for large initial data without any monotonicity assumption on the initial data.

The Yamabe equation in a non-local setting

Aim of this paper is to study the following elliptic equation driven by a gen- eral non-local integrodifferential operator LK such that LKuCu Cjuj 2 2 uD0 in, uD0 in R n n, wheres2.0;1/, is an open bounded set of R n ,n > 2s, with Lipschitz boundary, is a positive real parameter,2 D2n=.n 2s/ is a fractional critical Sobolev exponent, while LK is the non-local integrodifferential operator

Analysis and Comparison of Different Approximations to Nonlocal Diffusion and Linear Peridynamic Equations

We consider the numerical solution of nonlocal constrained value problems associated with linear nonlocal diffusion and nonlocal peridynamic models. Two classes of discretization methods are presented, including standard finite element methods and quadrature-based finite difference methods. We discuss the applicability of these approaches to nonlocal problems having various singular kernels and study basic numerical analysis issues. We illustrate the similarities and differences of the resulting nonlocal stiffness matrices and discuss whether discrete maximum principles can be established. We pay particular attention to the issue of convergence in both the nonlocal setting and the local limit. While it is known that the nonlocal models converge to corresponding differential equations in the local limit, we elucidate how such limiting behaviors may or may not be preserved in various discrete approximations. Our findings thus offer important insight into applications and simulations of nonlocal models.

Dirichlet’s principle and wellposedness of solutions for a nonlocal [formula omitted]Laplacian system

We prove Dirichlet’s principle for a nonlocal p-Laplacian system which arises in the nonlocal setting of peridynamics when p=2. This nonlinear model includes boundary conditions imposed on a nonzero volume collar surrounding the domain. Our analysis uses nonlocal versions of integration by parts techniques that resemble the classical Green and Gauss identities. The nonlocal energy functional associated with this “elliptic” type system exhibits a general kernel which could be weakly singular. The coercivity of the system is shown by employing a nonlocal Poincaré’s inequality. We use the direct method in calculus of variations to show existence and uniqueness of minimizers for the nonlocal energy, from which we obtain the wellposedness of this steady state diffusion system.

Fuzzy differential equations without fuzzy convexity

Classical fuzzy differential equations defined in terms of the Hukuhara derivative depend critically on the convexity of the level sets and result in expanding level sets. Here Hüllermeier's suggestion of defining fuzzy differential equations at each level set via differential inclusions is combined with ideas of Aubin on morphological equations, which allow nonlocal set evolution, to remove the assumption of fuzzy convexity and thus to allow fuzzy differential equations to be defined for non-convex level sets. This approach uses reachable sets as a more general form of set integration and, in contrast to the Aumann set integral, does not necessarily give rise to convex sets. The results presented in this paper are even more general since they concern fuzzy sets that need to be only closed without additional assumptions of convexity, compactness or even normality. In particular, an existence and uniqueness theorem is established under the assumption that the right-hand sides satisfy a one-sided Lipschitz condition rather than a much stronger Lipschitz condition. Fuzzy delay differential equations are also considered from this new perspective.

Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites

We introduce discrete systems in the form of straight (infinite) and ring-shaped chains, with two symmetrically placed nonlinear sites. The systems can be implemented in nonlinear optics (as waveguiding arrays) and Bose-Einstein condensates (by means of an optical lattice). A full set of exact analytical solutions for symmetric, asymmetric, and antisymmetric localized modes is found, and their stability is investigated in a numerical form. The symmetry-breaking bifurcation, through which the asymmetric modes emerge from the symmetric ones, is found to be of the subcritical type. It is transformed into a supercritical bifurcation if the nonlinearity is localized in relatively broad domains around two central sites, and also in the ring of a small size, i.e., in effectively nonlocal settings. The family of antisymmetric modes does not undergo bifurcations and features both stable and unstable portions. The evolution of unstable localized modes is investigated by means of direct simulations. In particular, unstable asymmetric states, which exist in the case of the subcritical bifurcation, give rise to breathers oscillating between the nonlinear sites, thus restoring an effective dynamical symmetry between them.

Non-local setting and outcome information for violation of Bell's inequality

Bell's theorem is a no-go theorem stating that quantum mechanics cannot be reproduced by a physical theory based on realism, freedom to choose experimental settings and two locality conditions: setting (SI) and outcome (OI) independence. We provide a novel analysis of what it takes to violate Bell's inequality within the framework in which both realism and freedom of choice are assumed, by showing that it is impossible to model a violation without having information in one laboratory about both the setting and the outcome at the distant one. While it is possible that outcome information can be revealed from shared hidden variables, the assumed experimenter's freedom to choose the settings ensures that the setting information must be non-locally transferred even when the SI condition is obeyed. The amount of transmitted information about the setting that is sufficient to violate the CHSH inequality up to its quantum mechanical maximum is 0.736 bits.