Viscosity Approximation Research Articles

Benchmarking the vertically integrated ice-sheet model IMAU-ICE (version 2.0)

Abstract. Ice-dynamical processes constitute a large uncertainty in future projections of sea-level rise caused by anthropogenic climate change. Improving our understanding of these processes requires ice-sheet models that perform well at simulating both past and future ice-sheet evolution. Here, we present version 2.0 of the ice-sheet model IMAU-ICE, which uses the depth-integrated viscosity approximation (DIVA) to solve the stress balance. We evaluate its performance in a range of benchmark experiments, including simple analytical solutions and both schematic and realistic model intercomparison exercises. IMAU-ICE has adopted recent developments in the numerical treatment of englacial stress and sub-shelf melt near the grounding line, which result in good performance in experiments concerning grounding-line migration (MISMIP, MISMIP+) and buttressing (ABUMIP). This makes it a model that is robust, versatile, and user-friendly, which will provide a firm basis for (palaeo-)glaciological research in the coming years.

On Mann implicit composite subgradient extragradient methods for general systems of variational inequalities with hierarchical variational inequality constraints

In a real Hilbert space, let the VIP, GSVI, HVI, and CFPP denote a variational inequality problem, a general system of variational inequalities, a hierarchical variational inequality, and a common fixed-point problem of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping, respectively. We design two Mann implicit composite subgradient extragradient algorithms with line-search process for finding a common solution of the CFPP, GSVI, and VIP. The suggested algorithms are based on the Mann implicit iteration method, subgradient extragradient method with line-search process, and viscosity approximation method. Under mild assumptions, we prove the strong convergence of the suggested algorithms to a common solution of the CFPP, GSVI, and VIP, which solves a certain HVI defined on their common solutions set.

The Modified Viscosity Approximation Method with Inertial Technique and Forward–Backward Algorithm for Convex Optimization Model

In this paper, we propose a new accelerated algorithm for finding a common fixed point of nonexpansive operators, and then, a strong convergence result of the proposed method is discussed and analyzed in real Hilbert spaces. As an application, we create a new accelerated viscosity forward–backward method (AVFBM) for solving nonsmooth optimization problems of the sum of two objective functions in real Hilbert spaces, and the strong convergence of AVFBM to a minimizer of the sum of two convex functions is established. We also present the application and simulated results of AVFBM for image restoration and data classification problems.

On Strengthened Inertial-Type Subgradient Extragradient Rule with Adaptive Step Sizes for Variational Inequalities and Fixed Points of Asymptotically Nonexpansive Mappings

In a real Hilbert space, let the VIP denote a pseudomonotone variational inequality problem with Lipschitz continuity operator, and let the CFPP indicate a common fixed-point problem of finitely many nonexpansive mappings and an asymptotically nonexpansive mapping. On the basis of the Mann iteration method, the viscosity approximation method and the hybrid steepest-descent method, we propose and analyze two strengthened inertial-type subgradient extragradient rules with adaptive step sizes for solving the VIP and CFPP. With the help of suitable restrictions, we show the strong convergence of the suggested rules to a common solution of the VIP and CFPP, which is the unique solution of a hierarchical variational inequality (HVI).

Periodic entropy solution to a conservation law with nonlocal source arising in radiative gas

We establish the global well-posedness and large-time asymptotic behaviour of spatially periodic entropy solutions to a scalar conservation law with nonlocal source arising in radiative gas. The global existence is established by the L1-contraction and the comparison principle for the vanishing viscosity approximation. Moreover, we show that if the initial data is periodic, the source term induces the solution to decay in L2-norm at an exponential rate to the mean value of initial data over one space period.

Generalized viscosity approximation method for minimization and fixed point problems of quasi-pseudocontractive mappings in Hadamard spaces

In this paper, we introduce a generalized viscosity method which includes a sequence of contractive mappings for approximating solutions of fixed point problem of finitely many resolvents of a lower semi-continuous function together with a fixed point problem of countable infinite family of [Formula: see text]-Lipschitzian and quasi-pseudocontractive mappings in Hadamard spaces. We establish a strong convergence result without imposing a compactness condition of the proposed algorithm to an element in the intersection of the set of solutions of fixed point problem of finitely many resolvents of a lower semi-continuous function and fixed point problem of countable infinite family of quasi-pseudocontractive mappings. We display a numerical example to show the applicability of our main result. Our results complement and extend many recent results in literature.

The viscosity approximation method for accretive operators in Banach spaces

The viscosity approximation method for accretive operators in Banach spaces

The joint influence of normal vibrations and electric field on the stability of a two-layer fluid system

The article focuses on the joint influence of normal vibrations and normal electric field on the stability of a system consisting of two flat horizontal fluid layers. Two cases are considered. In the first one, the system is formed by two layers of dielectric fluids (both layers have the same thickness) and, in the second, the system includes a dielectric fluid layer and an ideal electrically conductive fluid layer. The results obtained for these two cases are qualitatively similar. We study two approximations: the low-viscosity approximation and high-frequency vibrations. For the low-viscosity approximation, the so-called electric instability mode and the resonant mode are analyzed. In this case, the short-wavelength and long-wavelength instability modes are considered. In the low viscosity approximation, there occurs a viscous boundary layer, which is analyzed using a fast coordinate. The problem is studied by applying a multiscale method. It is demonstrated that vibrations increase the critical value of the electric field strength for short-wave instability in the presence of an electric mode. In the case of the long-wave instability, vibrations increase the curvature of the neutral curve near the zero value of the wave number. As for the influence of the electric field on resonant modes, it can lead to the splitting of the first resonance into two or three modes, depending on the values of the Weber number. When splitting is absent, the electric field reduces the critical amplitude of the first resonant mode. Otherwise, the effect of the electric field is more complex. In the high-frequency limit, the expressions describing the critical value of the electric field strength are similar to those for the electric mode considered in the low-viscosity approximation for finite vibration frequencies, and no resonant perturbations occur in this case.

Viscosity approximation of a relaxed alternating CQ algorithm for the split equality problem

Viscosity approximation of a relaxed alternating CQ algorithm for the split equality problem

Hybrid viscosity approximation methods with generalized contractions for zeros of monotone operators and fixed point problems

Hybrid viscosity approximation methods with generalized contractions for zeros of monotone operators and fixed point problems

MK viscosity approximation for fixed points and equilibrium problems in Hilbert spaces

MK viscosity approximation for fixed points and equilibrium problems in Hilbert spaces

On Mann-Type Subgradient-like Extragradient Method with Linear-Search Process for Hierarchical Variational Inequalities for Asymptotically Nonexpansive Mappings

We propose two Mann-type subgradient-like extra gradient iterations with the line-search procedure for hierarchical variational inequality (HVI) with the common fixed-point problem (CFPP) constraint of finite family of nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. Our methods include combinations of the Mann iteration method, subgradient extra gradient method with the line-search process, and viscosity approximation method. Under suitable assumptions, we obtain the strong convergence results of sequence of iterates generated by our methods for a solution to HVI with the CFPP constraint.

Quantitative translations for viscosity approximation methods in hyperbolic spaces

In the setting of hyperbolic spaces, we show that the convergence of Browder-type sequences and Halpern iterations respectively entail the convergence of their viscosity version with a Rakotch map. We also show that the convergence of a hybrid viscosity version of the Krasnoselskii-Mann iteration follows from the convergence of the Browder type sequence. Our results follow from proof-theoretic techniques (proof mining). From an analysis of theorems due to T. Suzuki, we extract a transformation of rates for the original Browder type and Halpern iterations into rates for the corresponding viscosity versions. We show that these transformations can be applied to earlier quantitative studies of these iterations. From an analysis of a theorem due to H.-K. Xu, N. Altwaijry and S. Chebbi, we obtain similar results. Finally, in uniformly convex Banach spaces we study a strong notion of accretive operator due to Brezis and Sibony and extract an uniform modulus of uniqueness for the property of being a zero point. In this context, we show that it is possible to obtain Cauchy rates for the Browder type and the Halpern iterations (and hence also for their viscosity versions).

Strong convergence of the forward–backward splitting algorithms via linesearches in Hilbert spaces

Many real-world problems in applied sciences, engineering and economics can be reformulated as the convex minimization problem of two objective functions. In order to solve this problem, the forward–backward splitting algorithm and its modifications have been invented and used to investigate the convergence analysis. In this work, we discuss strong convergence of the sequence generated by the forward–backward algorithm involving the viscosity approximation method. The stepsizes studied in this paper are defined by two different kinds of linesearches which do not require the Lipschitz continuity condition on the gradient. Finally, we provide numerical experiments to show the efficiency of the proposed methods in signal recovery. It is reported that our algorithms have a good convergence behavior in practice without information of Lipschitz constants of the gradient of functions.

A computational approach to optimal control problems subject to mixed control-state constraints

In this paper, we study a computational approach to Hamilton–Jacobi–Bellman (HJB) equation for singular optimal control subject to mixed control-state constraints. A nonlinear diffusion equation is presented for the viscosity approximation to a partial differential equation by rewriting HJB equation. We construct a so called extremal flow for approximating the optimal objective value of the singular optimal control problem by a differential-algebraic equation. Some examples are given to illustrate this computational approach.

Viscosity approximation method for solving variational inequality problem in real Banach spaces

In this paper, we study the implicit and inertial-type viscosity approximation method for approximating a solution to the hierarchical variational inequality problem. Under some mild conditions on the parameters, we prove that the sequence generated by the proposed methods converges strongly to a solution of the above-mentioned problem in $q$-uniformly smooth Banach spaces. The results obtained in this paper generalize and improve many recent results in this direction.

A fast viscosity forward-backward algorithm for convex minimization problems with an application in image recovery

The purpose of this paper is to invent an accelerated algorithm for the convex minimization problem which can be applied to the image restoration problem. Theoretically, we first introduce an algorithm based on viscosity approximation method with the inertial technique for finding a common fixed point of a countable family of nonexpansive operators. Under some suitable assumptions, a strong convergence theorem of the proposed algorithm is established. Subsequently, we utilize our proposed algorithm to solving a convex minimization problem of the sum of two convex functions. As an application, we apply and analyze our algorithm to image restoration problems. Moreover, we compare convergence behavior and efficiency of our algorithm with other well-known methods such as the forward-backward splitting algorithm and the fast iterative shrinkage-thresholding algorithm. By using image quality metrics, numerical experiments show that our algorithm has a higher efficiency than the mentioned algorithms.

Viscosity approximation of the solution to Burgers' equations with shock layers

ABSTRACT Viscous Burgers' equations with a small viscosity are considered and convergence of vanishing viscosity limit problem is investigated. We examine interior layers of a solution to viscous Burgers' equations, , as a viscosity parameter ε tends to zero. The inviscid model, i.e. when , possesses the structure of scalar hyperbolic conservation laws, hence our studies deliver an important idea that arises in the field of shock discontinuities of nonlinear hyperbolic waves. The heart of the paper is to establish asymptotic expansions and utilize inner solutions of sharp transition, which are called a corrector function. With aid of corrector functions and energy estimates, we improve the convergence rate of to as in ( in ) in the regions including shocks under an entropy condition.

Strong convergence theorems for minimization, variational inequality and fixed point problems for quasi-nonexpansive mappings using modified proximal point algorithms in real Hilbert spaces

In this paper, we investigate the problem of finding a common element of the solution set of convex minimization problem, the solution set of variational inequality problem and the solution set of fixed point problem with an infinite family of quasi-nonexpansive mappings in real Hilbert spaces. Based on the well-known proximal point algorithm and viscosity approximation method, we propose and analyze a new iterative algorithm for computing a common element. Under very mild assumptions, we obtain a strong convergence theorem for the sequence generated by the proposed method. Application to convex minimization and variational inequality problems coupled with inclusion problem is provided to support our main results.,Our proposed method is quite general and includes the iterative methods considered in the earlier and recent literature as special cases.

Strong convergence of the viscosity approximation method for the split generalized equilibrium problem

Strong convergence of the viscosity approximation method for the split generalized equilibrium problem