A new non-divergence diffusion equation with variable exponent for multiplicative noise removal

A new kind of model based on double degenerate parabolic equation in nondivergence form is proposed for multiplicative noise removal. We first utilize the regularization method to illustrate the existence of the weak solution for the problem and the non-expansion of support of the solution is also provided. In the numerical aspect, experiments and comparisons with other denoising models are presented in order to state the denoising capability of our model.     

T-stability of the semi-implicit Euler method for delay differential equations with multiplicative noise

The paper deals with the T-stability of the semi-implicit Euler method for delay differential equations with multiplicative noise. A difference equation is obtained by applying the numerical method to a linear test equation, in which the Wiener increment is approximated by a discrete random variable with two-point distribution. The conditions under which the method is T-stable are considered and the numerical experiments are given.     

A hybrid convex variational model for image restoration

We propose a new hybrid model for variational image restoration using an alternative diffusion switching non-quadratic function with a parameter. The parameter is chosen adaptively so as to minimize the smoothing near the edges and allow the diffusion to smooth away from the edges. This model belongs to a class of edge-preserving regularization methods proposed in the past, the ?-function formulation. This involves a minimizer to the associated energy functional. We study the existence and uniqueness of the energy functional of the model. Using real and synthetic images we show that the model is effective in image restoration.     

Random attractors of reaction-diffusion equations with multiplicative noise in L

In this paper, we study the random dynamical system (RDS) generated by the reaction-diffusion equation with multiplicative noise and prove the existence of a random attractor for such RDS in L^p(D) for any p?2.     

A non-uniform discretization of stochastic heat equations with multiplicative noise on the unit sphere

We investigate a discretization of a class of stochastic heat equations on the unit sphere with multiplicative noise. A spectral method is used for the spatial discretization and the truncation of the Wiener process, while an implicit Euler scheme with non-uniform steps is used for the temporal discretization. Some numerical experiments inspired by Earth’s surface temperature data analysis GISTEMP provided by NASA are given.     

Existence of solutions for a perturbed Dirichlet problem without growth conditions

We present some results on the existence and multiplicity of solutions for boundary value problems involving equations of the type −Δu=f(x,u)+λg(x,u), where Δ is the Laplacian operator, λ is a real parameter and , are two Carathéodory functions having no growth conditions with respect to the second variable. The approach is variational and mainly based on a critical point theorem by B. Ricceri.     

Continuous level anisotropic diffusion for noise removal

This paper presents a new approach to anisotropic diffusion and noise removal. Several functionals are introduced to a variational model. The diffusion behavior is governed by a nonlinear partial differential equation. A dynamic threshold function plays an important role in the continuous level anisotropic diffusion and a related optimization problem is presented. The noise can be removed while the edge well preserved. Multi-level noise or multi-level edge can be handled automatically. Finally, the accuracy and efficiency of the proposed method are verified by several numerical experiments.     

A new algorithm for solving certain variational problems

A technique for finding the solution of discrete, multistate dynamic programming problems is applied to solve certain variational problems. The algorithm is a method of successive approximations using a general two-stage solution. The advantage of the method is that it provides a means of reducing Bellman's curse of dimensionality. An example on the Plateau problem or the minimal surface area problem is considered, and the algorithm is found to be computationally efficient.This research was supported in part by NRC—Canada, Grant No. A-4051.The authors wish to thank the referees for helpful comments and also for bringing to their attention the method of local variations.     

Strong convergence of a stochastic Rosenbrock-type scheme for the finite element discretization of semilinear SPDEs driven by multiplicative and additive noise

This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part, usually called stochastic dominated transport equations. Most standard numerical schemes lose their good stability properties on such equations, including the current linear implicit Euler method. We discretize the SPDE in space by the finite element method and propose a novel scheme called stochastic Rosenbrock-type scheme for temporal discretization. Our scheme is based on the local linearization of the semi-discrete problem obtained after space discretization and is more appropriate for such equations. We provide a strong convergence of the new fully discrete scheme toward the exact solution for multiplicative and additive noise and obtain optimal rates of convergence. Numerical experiments to sustain our theoretical results are provided.     

A new uncertain insurance model with variational lower limit

Uncertainty theory provides a new tool to deal with uncertainty. The paper employs it to propose a new uncertain insurance model with variational lower limit, and gives a ruin index and uncertainty distribution for the uncertain insurance risk process that claim process is a renewal reward process. The model extends and improves uncertain insurance model presented by Liu. Finally, it also provides examples to illustrate the effectiveness of the model.     

A novel variational model for image segmentation

In this paper, we propose a new variational model for image segmentation. Our model is inspired by the complex Ginzburg-Landau model and the semi-norm defined by us. This new model can detect both the convex and concave parts of images. Moreover, it can also detect non-closed edges as well as quadruple junctions. Compared with other methods, the initialization is completely automatic and the segmented images obtained by using our new model could keep fine structures and edges of the original images very effectively. Finally, numerical results show the effectiveness of our model.     

Existence of nonzero solutions for a class of generalized variational inequalities

The existence of nonzero solutions for a class of generalized variational inequalities is studied by topological degree theory for multi-valued mappings in finite dimensional spaces and reflexive Banach space. One of the mappings concerned here is nonlinear with coercive or monotone and other is set-contractive or upper semi-continuous. Under some suitable assumptions, some existence theorems of nonzero solutions for this generalized variational inequalities are obtained. This work was supported by the Young Talent Foundation of Zhejiang Gongshang University and the Foundation of Department of Education of Zhejiang Province No. 20070628.     

On the existence and uniqueness of solutions to an asymptotic equation of a variational wave equation

We prove the global existence and uniqueness of admissible weak solutions to an asymptotic equation of a nonlinear hyperbolic variational wave equation with nonnegative L ²(ℝ) initial data. The work of Ping Zhang is supported by the Chinese postdoctor’s foundation, and that of Yuxi Zheng is supported in part by NSF DMS-9703711 and the Alfred P. Sloan Research Fellows award.     

Conservative solutions to a system of asymptotic variational wave equations

This paper is concerned with an asymptotic variational wave system which models weakly nonlinear waves for a system of variational wave equations arising in the theory of nematic liquid crystals and a few other physical contexts. By constructing a global semigroup, we establish the well-posedness of the initial–boundary value problem within the class of energy-conservative solutions for initial data of finite energy.     

Variational solutions for partial differential equations driven by a fractional noise

In this article we develop an existence and uniqueness theory of variational solutions for a class of nonautonomous stochastic partial differential equations of parabolic type defined on a bounded open subset D⊂R^d and driven by an infinite-dimensional multiplicative fractional noise. We introduce two notions of such solutions for them and prove their existence and their indistinguishability by assuming that the noise is derived from an L²(D)-valued fractional Wiener process W^H with Hurst parameter , whose covariance operator satisfies appropriate integrability conditions, and where γ∈(0,1] denotes the Hölder exponent of the derivative of the nonlinearity in the stochastic term of the equations. We also prove the uniqueness of solutions when the stochastic term is an affine function of the unknown random field. Our existence and uniqueness proofs rest upon the construction and the convergence of a suitable sequence of Faedo-Galerkin approximations, while our proof of indistinguishability is based on certain density arguments as well as on new continuity properties of the stochastic integral we define with respect to W^H.     

Attractors for stochastic lattice dynamical systems with a multiplicative noise

In this paper, we consider a stochastic lattice differential equation with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and multiplicative white noise at each node. We prove the existence of a compact global random attractor which, pulled back, attracts tempered random bounded sets.      

Adaptive total variation regularization based scheme for Poisson noise removal

To better preserve the edge features, this paper investigates an adaptive total variation regularization based variational model for removing Poisson noise. This edge‐preserving scheme comprises a spatially adaptive diffusivity coefficient, which adjusts the diffusion strength automatically. Compared with the classical total variation based one, numerical simulations distinctly indicate the superiority of our proposed strategy in maintaining the small details while denoising Poissonian image. Copyright © 2012 John Wiley & Sons, Ltd.     

Random attractor for a two-dimensional incompressible non-Newtonian fluid with multiplicative noise

This article proves that the random dynamical system generated by a twodimensional incompressible non-Newtonian fluid with multiplicative noise has a global random attractor, which is a random compact set absorbing any bounded nonrandom subset of the phase space.     

On a variational problem for radial solutions to extremal elliptic equations

On the ball |x| ≤ 1 of R ^m , m ≥ 2, a radial variational problem, related to a priori estimates for solutions to extremal elliptic equations with fixed ellipticity constant α is investigated. Such a problem has been studied and solved [see Manselli Ann. Mat. Pura Appl. (IV), t. LXXXIX:31–54, 1971] in L ^p spaces, with p ≤ m. In this paper, we assume p > m and we prove the existence of a positive number α ₀ = α ₀(p,m) such that if there exists a smooth function maximizing the problem, whose representation is explicitly determined as in Manselli [Ann. Mat. Pura Appl. (IV), t. LXXXIX:31–54, 1971] This fact is no longer true if 0 < α < α ₀.