Several kind of new numerical schemes for the stationary Navier-Stokes equations based on the virtue of inertial Manifold and Approximate inertial Manifold, which we call them inertial algorithms in this paper, togeth...
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Several kind of new numerical schemes for the stationary Navier-Stokes equations based on the virtue of inertial Manifold and Approximate inertial Manifold, which we call them inertial algorithms in this paper, together with their error estimations are presented. All these algorithms are constructed under an uniform frame, that is to construct some kind of new projections for the Sobolev space in which the true solution is sought. It is shown that the proposed inertial algorithms can greatly improve the convergence rate of the standard Galerkin approximate solution with lower computing effort. And some numerical examples are also given to verify results of this paper.
This study presents two inertial type extragradient algorithms for finding a common solution to the monotone variational inequalities and fixed point problems for rho$$ \rho $$-demicontractive mapping in real Hilbert ...
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This study presents two inertial type extragradient algorithms for finding a common solution to the monotone variational inequalities and fixed point problems for rho$$ \rho $$-demicontractive mapping in real Hilbert spaces. We provide inertial type iterative algorithms with self-adaptive variable step size rules that do not require prior knowledge of the operator value. Our algorithms employ a basic step size rule, which is derived by certain computations at each iteration. Without previous knowledge of the operators Lipschitz constant, two strong convergence theorems were obtained. Finally, we present a number of numerical experiments to evaluate the efficacy and applicability of the proposed algorithms. The conclusions of this study on variational inequality and fixed point problems support and extend previous findings.
We propose a projective splitting type method to solve the problem of finding a zero of the sum of two maximal monotone operators. Our method considers inertial and relaxation steps, and also allows inexact solutions ...
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We propose a projective splitting type method to solve the problem of finding a zero of the sum of two maximal monotone operators. Our method considers inertial and relaxation steps, and also allows inexact solutions of the proximal subproblems within a relative-error criterion. We study the asymptotic convergence of the method, as well as its iteration-complexity. We also discuss how the inexact computations of the proximal subproblems can be carried out when the operators are Lipschitz continuous. In addition, we provide numerical experiments comparing the computational performance of our method with previous (inertial and non-inertial) projective splitting methods.
This paper derives new inexact variants of the Douglas-Rachford splitting method for maximal monotone operators and the alternating direction method of multipliers (ADMM) for convex optimization. The analysis is based...
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This paper derives new inexact variants of the Douglas-Rachford splitting method for maximal monotone operators and the alternating direction method of multipliers (ADMM) for convex optimization. The analysis is based on a new inexact version of the proximal point algorithm that includes both an inertial step and overrelaxation. We apply our new inexact ADMM method to LASSO and logistic regression problems and obtain somewhat better computational performance than earlier inexact ADMM methods.
We propose a relaxed-inertial proximal point algorithm for solving equilibrium problems involving bifunctions which satisfy in the second variable a generalized convexity notion called strong quasiconvexity, introduce...
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We propose a relaxed-inertial proximal point algorithm for solving equilibrium problems involving bifunctions which satisfy in the second variable a generalized convexity notion called strong quasiconvexity, introduced by Polyak (Sov Math Dokl 7:72-75, 1966). The method is suitable for solving mixed variational inequalities and inverse mixed variational inequalities involving strongly quasiconvex functions, as these can be written as special cases of equilibrium problems. Numerical experiments where the performance of the proposed algorithm outperforms one of the standard proximal point methods are provided, too.
In image processing, total variation (TV) regularization models are commonly used to recover the blurred images. One of the most efficient and popular methods to solve the convex TV problem is the alternating directio...
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In image processing, total variation (TV) regularization models are commonly used to recover the blurred images. One of the most efficient and popular methods to solve the convex TV problem is the alternating direction method of multipliers (ADMM) algorithm, recently extended using the inertial proximal point method. Although all the classical studies focus on only a convex formulation, recent articles are paying increasing attention to the nonconvex methodology due to its good numerical performance and properties. In this paper, we propose to extend the classical formulation with a novel nonconvex alternating direction method of multipliers with the inertial technique (IADMM). Under certain assumptions on the parameters, we prove the convergence of the algorithm with the help of the Kurdyka-Lojasiewicz property. We also present numerical simulations on the classical TV image reconstruction problems to illustrate the efficiency of the new algorithm and its behavior compared with the well-established ADMM method.
In this paper, we introduce a three-operator splitting algorithm with deviations for solving the minimization problem composed of the sum of two convex functions minus a convex and smooth function in a real Hilbert sp...
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In this paper, we introduce a three-operator splitting algorithm with deviations for solving the minimization problem composed of the sum of two convex functions minus a convex and smooth function in a real Hilbert space. The main feature of the proposed method is that two per-iteration deviation vectors provide additional degrees of freedom. We propose one-step and two step inertial three-operator splitting algorithms by selecting the deviations along a momentum direction. A numerical experiment for DC regularized sparse recovery problems shows that the proposed algorithms have better performance than the original three-operator splitting algorithm.(C) 2023 IMACS. Published by Elsevier B.V. All rights reserved.
We propose a variation of the forward-backward splitting method for solving structured monotone inclusions. Our method integrates past iterates and two deviation vectors into the update equations. These deviation vect...
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We propose a variation of the forward-backward splitting method for solving structured monotone inclusions. Our method integrates past iterates and two deviation vectors into the update equations. These deviation vectors bring flexibility to the algorithm and can be chosen arbitrarily as long as they together satisfy a norm condition. We present special cases where the deviation vectors, selected as predetermined linear combinations of previous iterates, always meet the norm condition. Notably, we introduce an algorithm employing a scalar parameter to interpolate between the conventional forward-backward splitting scheme and an accelerated O(1/n(2))-convergent forward-backward method that encompasses both the accelerated proximal point method and the Halpern iteration as special cases. The existing methods correspond to the two extremes of the allowed scalar parameter range. By choosing the interpolation scalar near the midpoint of the permissible range, our algorithm significantly outperforms these previously known methods when addressing a basic monotone inclusion problem stemming from minimax optimization.
We discuss here the convergence of the iterates of the "Fast Iterative Shrinkage/Thresholding Algorithm," which is an algorithm proposed by Beck and Teboulle for minimizing the sum of two convex, lower-semic...
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We discuss here the convergence of the iterates of the "Fast Iterative Shrinkage/Thresholding Algorithm," which is an algorithm proposed by Beck and Teboulle for minimizing the sum of two convex, lower-semicontinuous, and proper functions (defined in a Euclidean or Hilbert space), such that one is differentiable with Lipschitz gradient, and the proximity operator of the second is easy to compute. It builds a sequence of iterates for which the objective is controlled, up to a (nearly optimal) constant, by the inverse of the square of the iteration number. However, the convergence of the iterates themselves is not known. We show here that with a small modification, we can ensure the same upper bound for the decay of the energy, as well as the convergence of the iterates to a minimizer.
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