Explicit Iterative Algorithms Research Articles

Variational Principles in Quaternionic Analysis with Applications to the Stationary MHD Equations

In this paper we aim to combine tools from variational calculus with modern techniques from quaternionic analysis that involve Dirac type operators and related hypercomplex integral operators. The aim is to develop new methods for showing geometry independent explicit global existence and uniqueness criteria as well as new computational methods with special focus to the stationary incompressible viscous magnetohydrodynamic equations. We first show how to specifically apply variational calculus in the quaternionic setting. To this end we explain how the mountain pass theorem can be successfully applied to guarantee the existence of (weak) solutions. To achieve this, the quaternionic integral operator calculus serves as a key ingredient allowing us to apply Schauder’s fixed point theorem. The advantage of the approach using Schauder’s fixed point theorem is that it is also applicable to large data since it does not require any kind of contraction property. These considerations will allow us to provide explicit iterative algorithms for its numerical solution. Finally to obtain more precise a-priori estimates one can use in the situations dealing with small data the Banach fixed point theorem which then also grants the uniqueness.

A novel class of forward-backward explicit iterative algorithms using inertial techniques to solve variational inequality problems with quasi-monotone operators

<abstract><p>The theory of variational inequalities is an important tool in physics, engineering, finance, and optimization theory. The projection algorithm and its variants are useful tools for determining the approximate solution to the variational inequality problem. This paper introduces three distinct extragradient algorithms for dealing with variational inequality problems involving quasi-monotone and semistrictly quasi-monotone operators in infinite-dimensional real Hilbert spaces. This problem is a general mathematical model that incorporates a set of applied mathematical models as an example, such as equilibrium models, optimization problems, fixed point problems, saddle point problems, and Nash equilibrium point problems. The proposed algorithms employ both fixed and variable stepsize rules that are iteratively transformed based on previous iterations. These algorithms are based on the fact that no prior knowledge of the Lipschitz constant or any line-search framework is required. To demonstrate the convergence of the proposed algorithms, some simple conditions are used. Numerous experiments have been conducted to highlight the numerical capabilities of algorithms.</p></abstract>

Generalized system of relaxed cocoercive variational inequalities governed by nonexpansive mappings

The objective of this paper is to study the generalized systems of relaxed cocoercive variational inequalities associated with six nonlinear operators. We proposed three steps explicit iterative methods involving projection operators and give the equivalent fixed point problem of the generalized systems of relaxed cocoercive variational inequalities. This fixed point problem suggests to us the three steps explicit iterative algorithms to obtain an approximate solution of the system under consideration. Strong convergence results are obtained.

K-Steps Explicit Iterative Algorithms to Solve Generalized System of Nonlinear Mixed Variational Inequalities

The objective of this paper is to consider new generalized systems of nonlinear mixed variational inequalities including 3k-distinct nonlinear relaxed cocoercive operators. We proposed k-steps explicit iterative methods including projection operators for this considered system and present the equivalent fixed point problem of this new generalized systems of variational inequalities. This equivalent fixed point problem suggest to us k-steps explicit iterative algorithms to obtain an approximate solution of the considered system. Convergence results of k-step explicit iterative algorithms are obtained.

Modified Popov's explicit iterative algorithms for solving pseudomonotone equilibrium problems

This paper proposes two algorithms that are based on a subgradient and an inertial scheme with the explicit iterative method for solving pseudomonotone equilibrium problems. The weak convergence of both algorithms is well-established under standard assumptions on the cost bifunction. The advantage of these algorithms is that they did not require any line search procedure or any knowledge about bifunction Lipschitz-type constants for step-size evaluation. A practical explanation of this is that they use a sequence of step-size that are revised at each iteration based on some previous iteration. For a numerical experiment, we consider a well-known Nash-Cournot equilibrium model of electricity markets and also other test problems to assist the well-established convergence results and be able to see that our proposed algorithms have a competitive advantage over the time of execution and the number of iterations.

Explicit Iterative Algorithms for Continuous Coupled Lyapunov Matrix Equations

In this article, two iterative algorithms are proposed to solve the coupled Lyapunov matrix equations associated with continuous-time Markovian jump linear systems. First, an explicit iterative algorithm is presented to solve the considered matrix equations via introducing a tuning parameter. A necessary and sufficient condition is established for the proposed algorithm to be convergent. Furthermore, the optimal value of the tuning parameter achieving the fastest convergence rate is given explicitly. Second, a modified version of the previous explicit algorithm is established by using a new information update scheme. Similarly, a necessary and sufficient condition is provided for the modified explicit iterative algorithm. Based on this convergence condition, some easily verifiable convergence results are further derived, and the optimal tuning parameter is given explicitly. Finally, two examples are presented to validate the obtained theoretical results.

Numerical analysis of one dimensional motion of magma without mass forces

In this paper, we present numerical treatment for a compaction-driven Darcian flow viscoelastic rock magma model. This problem is a strongly coupled system of one quasi-linear parabolic equation and one integro-differential equation for the density and the porosity of the flow. The numerical discretization uses cell-centered finite difference method, combined with semi-implicit and implicit time stepping. Implicit–explicit schemes, as well as implicit–explicit iterative algorithms have been developed to solve the corresponding discrete problems. Some properties (positivity, boundness, conservation) of the numerical solutions are investigated. Convergence study of the iteration processes is also presented. The efficiency and the accuracy of the proposed methods are illustrated numerically by test examples with near-real data.

Explicit iterative algorithms for solving equilibrium problems

The paper introduces two new extragradient algorithms for solving a pseudomonotone equilibrium problem with a Lipschitz-type condition recently presented by Mastroeni in auxiliary problem principle. The algorithm uses variable stepsizes which are updated at each iteration and based on some previous iterates. The advantage of the algorithms is that they are done without the prior knowledge of Lipschitz-type constants and also without any linesearch procedure. The convergence of the algorithms is established under mild assumptions. In the case where the equilibrium bifunction is strongly pseudomonotone, the R-linear rate of convergence of the new algorithms is formulated. Several of fundamental experiments are provided to illustrate the numerical behavior of the algorithms and also to compare with others.

Convergence theorems for composite viscosity approaches to systems variational inequalities in Banach spaces

In this paper, we study a general system of variational inequalities with a hierarchical variational inequality constraint for an infinite family of nonexpansive mappings. We introduce general implicit and explicit iterative algorithms. We prove the strong convergence of the sequences generated by the proposed iterative algorithms to a solution of the studied problems.

Embedding-Based Interpolation on the Special Orthogonal Group

We study schemes for interpolating functions that take values in the special orthogonal group $SO(n)$. Our focus is on interpolation schemes obtained by embedding $SO(n)$ in a linear space, interpolating in the linear space, and mapping the result onto $SO(n)$ via the closest point projection. The resulting interpolants inherit both the order of accuracy and the regularity of the underlying interpolants on the linear space. This enables one to construct interpolants having $C^1$-continuity in a straightforward way, even when the data do not lie in a single coordinate chart. The values and derivatives of the interpolants admit efficient evaluation via either explicit formulas or iterative algorithms, which we detail for two choices of embeddings: the embedding of $SO(n)$ in the space of $n\times n$ matrices and, when $n=3$, the identification of $SO(3)$ with the set of unit quaternions. Along the way, we point out a connection between these interpolation schemes and geodesic finite elements. We illustrate the utility of these interpolation schemes by numerically computing minimum acceleration curves on $SO(n)$, a task which is handled naturally with $SO(n)$-valued finite elements having $C^1$-continuity.

Implicit and explicit iterative algorithms for hierarchical variational inequality in uniformly smooth Banach spaces

The purpose of this paper is to solve the hierarchical variational inequality with the constraint of a general system of variational inequalities in a uniformly convex and 2-uniformly smooth Banach space. We introduce implicit and explicit iterative algorithms which converge strongly to a unique solution of the hierarchical variational inequality problem. Our results improve and extend the corresponding results announced by some authors.

Explicit and implicit iterative algorithms for strict pseudo-contractions in Banach spaces

Explicit and implicit iterative algorithms for strict pseudo-contractions in Banach spaces

Explicit iterative algorithms for solving coupled discrete‐time Lyapunov matrix equations

In this study, the authors aim to study explicit iterative algorithms for solving coupled discrete-time Lyapunov matrix equations. First, an explicit iterative algorithm based on fixed point theory of dynamic equations is presented via adding a tuning parameter. Second, a necessary and sufficient condition is provided for the convergence of the proposed algorithm. Moreover, the optimal value of the tuning parameter is derived for the fastest convergence of the algorithm. Third, by using the latest updated information, a modified version of the presented explicit iterative algorithm is also established with a necessary and sufficient condition being provided to guarantee the convergence of the modified algorithm. Finally, a numerical example is given to demonstrate the effectiveness of the proposed algorithms.

Strong Convergence of Modified Algorithms Based on the Regularization for the Constrained Convex Minimization Problem

As is known, the regularization method plays an important role in solving constrained convex minimization problems. Based on the idea of regularization, implicit and explicit iterative algorithms are proposed in this paper and the sequences generated by the algorithms can converge strongly to a solution of the constrained convex minimization problem, which also solves a certain variational inequality. As an application, we also apply the algorithm to solve the split feasibility problem.

Hybrid Viscosity Approaches to General Systems of Variational Inequalities with Hierarchical Fixed Point Problem Constraints in Banach Spaces

The purpose of this paper is to introduce and analyze hybrid viscosity methods for a general system of variational inequalities (GSVI) with hierarchical fixed point problem constraint in the setting of real uniformly convex and 2-uniformly smooth Banach spaces. Here, the hybrid viscosity methods are based on Korpelevich’s extragradient method, viscosity approximation method, and hybrid steepest-descent method. We propose and consider hybrid implicit and explicit viscosity iterative algorithms for solving the GSVI with hierarchical fixed point problem constraint not only for a nonexpansive mapping but also for a countable family of nonexpansive mappings inX, respectively. We derive some strong convergence theorems under appropriate conditions. Our results extend, improve, supplement, and develop the recent results announced by many authors.

Algorithms of Common Solutions for Generalized Mixed Equilibria, Variational Inclusions, and Constrained Convex Minimization

We introduce new implicit and explicit iterative algorithms for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of a finite family of generalized mixed equilibrium problems, and the set of solutions of a finite family of variational inclusions in a real Hilbert space. Under suitable control conditions, we prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets.

Solving Generalized Mixed Equilibria, Variational Inequalities, and Constrained Convex Minimization

We propose implicit and explicit iterative algorithms for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of a finite family of generalized mixed equilibrium problems, and the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings in a real Hilbert space. We prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets under very mild conditions.

Hybrid viscosity approach for hierarchical fixed-point problems

In this paper, we propose hybrid implicit and explicit viscosity iterative algorithms for solving general hierarchical fixed-point problems for a countable family of non-expansive mappings in uniformly smooth Banach spaces. These hybrid viscosity algorithms are based on the well-known viscosity approximation method and hybrid steepest-descent method. We obtain some strong convergence theorems under suitable conditions. Our results extend, improve, supplement and develop the recent results in the literature.

A general iterative algorithm for finding a common solution of equilibrium and constrained convex minimization problems

The gradient-projection algorithm (GPA) plays an important role in solving constrained convex minimization problems. Based on Tian’s method, we combine GPA and averaged mapping approach to propose implicit and explicit composite iterative algorithms for finding a common solution of an equilibrium and a constrained convex minimization problem for the first time in this paper. Under suitable conditions, strong convergence theorems are obtained.

Composite Iterative Algorithms for Variational Inequality and Fixed Point Problems in Real Smooth and Uniformly Convex Banach Spaces

We introduce composite implicit and explicit iterative algorithms for solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in a real smooth and uniformly convex Banach space. These composite iterative algorithms are based on Korpelevich's extragradient method and viscosity approximation method. We first consider and analyze a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another composite explicit iterative algorithm in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literatures.