Search

We devise and analyze |$C^0$| -conforming hybrid high-order (HHO) methods to approximate biharmonic problems with either clamped or simply supported boundary conditions. |$C^0$| -conforming HHO methods hinge on cell unknowns that are |$C^0$| -conforming polynomials of order |$(k+2)$| approximating the solution in the mesh cells and on face unknowns, which are polynomials of order |$k\ge 0$| approximating the normal derivative of the solution on the mesh skeleton. Such methods deliver |$O(h^{k+1})$| |$H^2$| -error estimates for smooth solutions. An important novelty in the error analysis is to lower the minimal regularity requirement on the exact solution. The technique to achieve this has a broader applicability than just |$C^0$| -conforming HHO methods, and to illustrate this point, we outline the error analysis for the well-known |$C^0$| -conforming interior penalty discontinuous Galerkin methods as well. The present technique does not require a |$C^1$| -smoother to evaluate the right-hand side in case of rough loads; loads in |$W^{-1,q}$|⁠ , |$q>\frac {2d}{d+2}$|⁠ , are covered, but not in |$H^{-2}$|⁠. Finally, numerical results including comparisons to various existing methods showcase the efficiency of the proposed |$C^0$| -conforming HHO methods. [ABSTRACT FROM AUTHOR]

Low-dose CT images contain severe mottle noise and streak artifacts, which seriously affect the physician’s diagnosis of the disease. Hence, in this paper, we propose a novel anisotropic fourth-order diffusion model for low-dose CT image processing. The proposed diffusion model uses both image gradient magnitude and weighted residual local energy to determine the diffusion coefficient. Gradient magnitude is used to detect the image edges, while the weighted residual local energy preserves textures and details in the image. In addition, the fidelity term is introduced into the diffusion model to avoid excessive smoothing and weaken the blocky effects. Experimental results show that when compared with the anisotropic fourth-order diffusion model, the proposed algorithm protects the texture details and suppresses the blocky effects. In comparison with other state-of-the-art algorithms, the proposed model effectively suppresses mottle noise and streak artifacts while simultaneously improving the low-dose CT image quality.

We devise and analyze two hybrid high-order (HHO) methods for the numerical approximation of the biharmonic problem. The methods support polyhedral meshes, rely on the primal formulation of the problem, and deliver O(hk+1) H²-error estimates when using polynomials of order k ≥ 0 to approximate the normal derivative on the mesh (inter)faces. Both HHO methods hinge on a stabilization in the spirit of Lehrenfeld and Sch\"oberl for second-order PDEs. The cell unknowns are polynomials of order (k+2) that can be eliminated locally by means of static condensation. The face unknowns approximating the trace of the solution on the mesh (inter)faces are polynomials of order (k + 1) in the first HHO method, which is valid in dimension two and uses an original stabilization involving the canonical hybrid finite element, and they are of order (k + 2) for the second HHO method, which is valid in arbitrary dimension and uses only L²-orthogonal projections in the stabilization. A comparative discussion with the weak Galerkin methods from the literature is provided, highlighting the close connections and the improvements proposed herein. Additionally, we show how the two HHO methods can be combined with a Nitsche-like boundary-penalty technique to weakly enforce the boundary conditions. An originality in the devised Nitsche's technique is to avoid any penalty parameter that must be large enough. Finally, numerical results showcase the efficiency of the proposed methods and indicate that the HHO methods can generally outperform discontinuous Galerkin methods and even be competitive with C0-interior penalty methods on triangular meshes. [ABSTRACT FROM AUTHOR]

This paper aims to propose a novel contextual adaptive fourth-order smoothing method for noise removal. The classical Laplacian operator is first modified into a multi-scale and robust form, which combines two distinct discontinuity measures simultaneously, i.e., inhomogeneity and local spatial gradient. Then the contextual adaptive fourth-order smoothing method is proposed based on the new Laplacian operator. Moreover, a gain control function is employed for the gain control of smoothing in terms of contextual discontinuities to better preserve the important features. Experimental results support that the proposed method achieves the best performance among the comparative methods with respect to some objective evaluation metrics and visual effects. [ABSTRACT FROM AUTHOR]

It is a difficult issue in image restoration to eliminate noise while avoiding the staircase effect and preserving edges. The anisotropic diffusion model proposed by Perona and Malik (PM) and the total variation model presented by Rudin, Osher, and Fatemi (ROF) are widely used to restore an image. However, the well-known defect of the two classic models is that they tend to cause the staircase effect. We propose a well-balanced anisotropic diffusion (WBAD) model by considering an adaptive balance parameter. The balance can be made in a selective way, meaning that it will alternate between the PM diffusion and ROF diffusion in accordance with the image features. The proposed WBAD model can preserve edges well while reducing noise, but it also causes less staircasing effect in the less smooth regions because it acts like the PM diffusion in these regions. Considering that the fourth-order PDEs can reduce the staircasing effect, we introduce a hybrid image restoration model based on an adaptive weight parameter to take advantage of the WBAD model and the fourth-order model. The experimental results illustrate that our algorithm can effectively remove noise and preserve edges. The higher values of peak signal-to-noise ratio and MSSIM highlight the better performance of our hybrid image restoration model. [ABSTRACT FROM AUTHOR]

In this article, we study the following problem with Navier boundary conditions $$displaylines{ Delta _{p(x)}^2u=lambda | u| ^{p(x)-2}u+f(x,u)quad hbox{in }Omega , cr u=Delta u=0quad hbox{on }partial Omega . }$$ Where $Omega $ is a bounded domain in $mathbb{R}^{N}$ with smooth boundary $partial Omega $, $Ngeq 1$, $Delta _{p(x)}^2u:=Delta (|Delta u| ^{p(x)-2}Delta u) $, is the $p(x)$-biharmonic operator, $lambda leq 0$, $p$ is a continuous function on $overline{Omega } $ with $inf_{xin overline{Omega }} p(x)>1$ and $f:Omega imes mathbb{R}o mathbb{R}$ is a Caratheodory function. Using the Mountain Pass Theorem, we establish the existence of at least one solution of this problem. Especially, the existence of infinite many solutions is obtained.

This article concerns mathematical image denoising methods incorporating fourth-order partial differential equations (PDEs). We introduce and analyze piecewise planarity conditions (PPCs) with which unconstrained fourth-order variational models in continuum converge to a piecewise planar image. It has been observed that fourth-order variational models holding PPCs can restore better images than models without PPCs and second-order models. Numerical schemes are presented in detail and various examples in image denoising are provided to verify the claim.

In this article, we study the following problem with Navier boundary conditions Δp(x)² u = λ|u|p(x)-2 u + f(x, u) in ω, u = Δ u = 0 on ∂Ω. Where Ω is a bounded domain in ℝN with smooth boundary ∂Ω, N ≥ 1, Δp(x)² u ≔ Δ(|Δu|p(x)-2Δu), is the p(x)-biharmonic operator, λ ≤ 0, p is a continuous function on Ω̄ with inf x∈Ω̄p(x) > 1 and f Ω x ℝ → ℝ is a Caratheodory function. Using the Mountain Pass Theorem, we establish the existence of at least one solution of this problem. Especially, the existence of infinite many solutions is obtained. [ABSTRACT FROM AUTHOR]

In recent articles we have introduced the class of Brownian-time processes (BTPs) and the Linearized Kuramoto–Sivashinsky process (LKSP). Probabilistically, BTPs represent a unifying class for some different exciting processes like the iterated Brownian motion (IBM) of Burdzy (a process with fourth-order properties) and the Brownian–snake of Le Gall (a second-order process); they also include many additional new and quite interesting processes. The LKSP is closely connected to the Kuramoto–Sivashinsky PDEs, one of the most celebrated PDEs in modern applied mathematics. We start by surveying the fourth-order PDE connections to BTPs and the LKSP that we uncovered in two recent articles. In the second part of this paper we introduce BTP-SPDEs, these are SPDEs in which the PDE part is that solved by running a BTP. We consider a BTP-SPDE driven by an additive spacetime white noise on the time-space set ℝ+ × ℝd; and we prove the existence of a unique real-valued, Lp(Ω,ℙ) for all p ≥ 1, BTP solution to such BTP-SPDEs for 1 ≤ d ≤ 3. This contrasts sharply with the standard theory of reaction-diffusion type SPDEs driven by spacetime white noise, in which real-valued solutions are confined to one spatial dimension. Like the PDEs case, BTP-SPDEs also provide a valuable insight into other fourth-order SPDEs of applied science. We carry out such a program in forthcoming articles. [ABSTRACT FROM AUTHOR]

In recent articles we have introduced the class of Brownian-time processes (BTPs) and the Linearized Kuramoto–Sivashinsky process (LKSP). Probabilistically, BTPs represent a unifying class for some different exciting processes like the iterated Brownian motion (IBM) of Burdzy (a process with fourth-order properties) and the Brownian–snake of Le Gall (a second-order process); they also include many additional new and quite interesting processes. The LKSP is closely connected to the Kuramoto–Sivashinsky PDEs, one of the most celebrated PDEs in modern applied mathematics. We start by surveying the fourth-order PDE connections to BTPs and the LKSP that we uncovered in two recent articles. In the second part of this paper we introduce BTP-SPDEs, these are SPDEs in which the PDE part is that solved by running a BTP. We consider a BTP-SPDE driven by an additive spacetime white noise on the time-space set ℝ+ × ℝd; and we prove the existence of a unique real-valued, Lp(Ω,ℙ) for all p ≥ 1, BTP solution to such BTP-SPDEs for 1 ≤ d ≤ 3. This contrasts sharply with the standard theory of reaction-diffusion type SPDEs driven by spacetime white noise, in which real-valued solutions are confined to one spatial dimension. Like the PDEs case, BTP-SPDEs also provide a valuable insight into other fourth-order SPDEs of applied science. We carry out such a program in forthcoming articles. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a nonconforming Morley finite element method for the stationary quasi-geostrophic equation in the ocean circulation. Stability and the inf–sup condition for the discrete solution are proved, and the local existence of a unique solution to the discrete nonlinear system is established based on the assumption of the existence of an isolated solution to the linearized problem and Banach fixed point argument. One principal tool employed is to exploit the enrichment operator from nonconforming space to H 0 2 (Ω). Thereby, not only smallness assumption on data is avoided, but also optimal error estimates in H 2 - and H 1 -norms are proved under minimal regularity condition on the exact solution. Then, for the nonlinear discrete system, the Newton method is applied, which is shown to preserve local quadratic convergence. Moreover, a posteriori error estimator for an adaptive algorithm is derived. Finally, several numerical experiments with a benchmark problem are considered to confirm our theoretical findings. • The stream function formulation of the stationary quasi-geostrophic equations in the ocean circulation is considered. • Approximation is based on nonconforming Morley finite element methods. • Stability and the inf–sup condition for the discrete solution, and optimal error estimates are proved. • A posteriori error estimator is proposed, analyzed, and tested for a benchmark problem. [ABSTRACT FROM AUTHOR]