Extrapolation Step Research Articles

An alternated inertial method for pseudomonotone variational inequalities in Hilbert spaces

In this paper, we introduce a new relaxed extrgadient algorithm with alternated inertial extrapolation step and self adaptive variable stepsizes for solving variational inequality problems whose cost operator is pseudomonotone operator in Hilbert spaces. We establish the weak convergence of the proposed algorithm and linear convergence under some standard assumptions. Numerical experiments are given to support theoretical results and comparison with recent related methods.

New Inertial Projection Methods for Solving Multivalued Variational Inequality Problems Beyond Monotonicity

In this paper, we present two new inertial projection-type methods for solving multivalued variational inequality problems in finite-dimensional spaces. We establish the convergence of the sequence generated by these methods when the multivalued mapping associated with the problem is only required to be locally bounded without any monotonicity assumption. Furthermore, the inertial techniques that we employ in this paper are quite different from the ones used in most papers. Moreover, based on the weaker assumptions on the inertial factor in our methods, we derive several special cases of our methods. Finally, we present some experimental results to illustrate the profits that we gain by introducing the inertial extrapolation steps.

Self-adaptive inertial subgradient extragradient scheme for pseudomonotone variational inequality problem

Abstract In this article, we propose a self-adaptive inertial subgradient extragradient algorithm for solving variational inequality problems involving pseudomonotone operator. The scheme uses a variable step sizes and inertial extrapolation step. The step size is self-adaptive, which does not require the prior knowledge of the Lipschitz constant of the underlying operator. Furthermore, under mild assumptions, we prove the weak convergence of the sequence generated by the proposed algorithm to a solution of the considered problem. We give numerical experiments to illustrate the inertial-effect and the computational performance of our proposed algorithm in comparison with the existing state of the art algorithms.

A method with inertial extrapolation step for split monotone inclusion problems

ABSTRACT The purpose of this paper is to study the convergence analysis of an iterative algorithm with inertial extrapolation step for finding an approximate solution of split monotone inclusion problem in real Hilbert spaces. Weak convergence of the sequence of iterates generated from the proposed method is obtained under some mild assumptions. Some special cases of the general problem are given and we give some numerical implementations to support the theoretical analysis and give justification for the addition of the extrapolation step in the proposed method.

Equilibria determination of elastic articulated duoskelion beams in 2D via a Riks-type algorithm

The overall behavior of an articulated beam structure constituted by elements arranged according to a specific chirality is studied. The structure as a whole, due to its slenderness and geometry, is called duoskelion beam. The name duoskelion is a neologism which is inspired by the Greek word δύοσκέλιον (two-legged). A discrete model for shearable beams, formulated recently, is exploited to investigate its mechanics. A purposely designed numerical scheme, adapting the Riks rationale, is used to calculate large displacement and deformation equilibria of duoskelion beams. Aimed at computing the current step correction, the Riks arc-length method is modified and made more efficient by applying a specific orthogonality condition, defined via the stiffness matrix, to an adapted extrapolation step. The robustness of the resulting scheme and its capability to follow equilibrium branches allows, in principle, for the exploration of the whole set of local energy minima in the introduced space of configurations, by using suitably modulated perturbative external loads. The developed numerical tool can be used to understand the mechanics of duoskelion beams. It is proved that there exists a stable principal equilibrium branch in which only compression is observed for any compression load. Additional stable equilibrium branches are found in compression such that the clamped–clamped compressed beam assumes a characteristic S shape which, upon reaching a critical load, is significantly amplified. A mechanically relevant stable equilibrium is also found in extension, being observed the S-shaped configuration experimentally found in Misra et al. (2020).

Some properties of the Runge-Kutta-Legendre super-time-stepping explicit methods

In this paper we will show that the Runge-Kutta-Legendre (RKL) super-time-step methods are built up in stages by combining forward Euler steps with linear extrapolation steps. For second order, we will show that linear interpolation is also used. By using these characteristics a simplified algorithm will be presented. The effect of different types of external boundary conditions are shown. For Neumann (zero-flux) and Periodic the methods are shown to be monotone. For Dirichlet it is shown that there are regions of non-monotonicity where solutions have the potential to go negative. These solutions are nonphysical and will lead to erroneous results if they are feed back into system. To remove these limitations two solution strategies are presented based on different non-uniform fixed meshing philosophies. A number of applications are shown with solutions validated against analytic. For a monotone heat front and a diffused heat pulse, the RKL results are shown to be physically correct and computationally cheaper. For a compact heat pulse it will be shown that adverse effects can occur if the number of steps is too large. It will be shown that results are significantly improved by reducing the number of steps and increasing the number of outer cycles. For general applications the universal approach is to try different numbers of steps and then study any sensitivities. © British Crown Copyright AWE/2020

New strong convergence method for the sum of two maximal monotone operators

This paper aims to obtain a strong convergence result for a Douglas–Rachford splitting method with inertial extrapolation step for finding a zero of the sum of two set-valued maximal monotone operators without any further assumption of uniform monotonicity on any of the involved maximal monotone operators. Furthermore, our proposed method is easy to implement and the inertial factor in our proposed method is a natural choice. Our method of proof is of independent interest. Finally, some numerical implementations are given to confirm the theoretical analysis.

New inertial method for generalized split variational inclusion problems

<p style='text-indent:20px;'>The purpose of this paper is to introduce a new inertial iterative method for solving split variational inclusion problems in real Hilbert spaces. We prove that the generated sequence converges weakly to the solution of the considered problem under some mild conditions. The major contributions of our results are: (ⅰ) to increase the rate of convergence of the method for solving split variational inclusion problem through the inertial extrapolation step, (ⅱ) to relax the choice of the inertial factor and show the inertial factor can be chosen greater than 1/3 unlike what is previously known before for inertial proximal point method in the literature (ⅲ) to show the numerical efficiency and superiority of our proposed method through some test example.</p>

Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence

The projection methods with vanilla inertial extrapolation step for variational inequalities have been of interest to many authors recently due to the improved convergence speed contributed by the presence of inertial extrapolation step. However, it is discovered that these projection methods with inertial steps lose the Fejér monotonicity of the iterates with respect to the solution, which is being enjoyed by their corresponding non-inertial projection methods for variational inequalities. This lack of Fejér monotonicity makes projection methods with vanilla inertial extrapolation step for variational inequalities not to converge faster than their corresponding non-inertial projection methods at times. Also, it has recently been proved that the projection methods with vanilla inertial extrapolation step may provide convergence rates that are worse than the classical projected gradient methods for strongly convex functions. In this paper, we introduce projection methods with alternated inertial extrapolation step for solving variational inequalities. We show that the sequence of iterates generated by our methods converges weakly to a solution of the variational inequality under some appropriate conditions. The Fejér monotonicity of even subsequence is recovered in these methods and linear rate of convergence is obtained. The numerical implementations of our methods compared with some other inertial projection methods show that our method is more efficient and outperforms some of these inertial projection methods.

New hybrid projection methods for variational inequalities involving pseudomonotone mappings

In this paper, we combine the subgradient extragradient method and the hybrid method with inertial extrapolation step to introduce two strongly convergent methods for solving variational inequalities for which the underlying cost function is a pseudo-monotone operator in real Hilbert spaces. The first proposed method involves step-sizes bounded by the inverse of the Lipschitz constant of the cost function and the second proposed method involves line search approach when the cost function is uniformly continuous. Our strong convergence results are obtained under some relaxed conditions on the inertial factor and the iterative parameters. Our numerical implementations show that our proposed methods are competitive and efficient.

Wall modeled immersed boundary method for high Reynolds number flow over complex terrain

Modeling the effect of complex terrain on high Reynolds number flows is important to improve our understanding of flow dynamics in wind farms and the dispersion of pollen and pollutants in hilly or mountainous terrain as well as the flow in urban areas. Unfortunately, simulating high Reynolds number flows over complex terrain is still a big challenge. Therefore, we present a simplified version of the wall modeled immersed boundary method by Chester et al. (J. Comput. Phys. 2007; 225: 427–448). By preventing the extrapolation and iteration steps in the original method, the proposed approach is much easier to implement and more computationally efficient. Furthermore, the proposed method only requires information that is available to each processor and thus is much more efficient for simulations performed on a large number of cores. These are crucial considerations for algorithms that are deployed on modern supercomputers and will allow much higher grid resolutions to be considered. We validate our method against wind tunnel measurements for turbulent flows over wall-mounted cubes, a two-dimensional ridge, and a three-dimensional hill. We find very good agreement between the simulation results and the measurement data, which shows this method is suitable to model high Reynolds number flows over complex terrain.

Relaxed Inertial Tseng’s Type Method for Solving the Inclusion Problem with Application to Image Restoration

The relaxed inertial Tseng-type method for solving the inclusion problem involving a maximally monotone mapping and a monotone mapping is proposed in this article. The study modifies the Tseng forward-backward forward splitting method by using both the relaxation parameter, as well as the inertial extrapolation step. The proposed method follows from time explicit discretization of a dynamical system. A weak convergence of the iterates generated by the method involving monotone operators is given. Moreover, the iterative scheme uses a variable step size, which does not depend on the Lipschitz constant of the underlying operator given by a simple updating rule. Furthermore, the proposed algorithm is modified and used to derive a scheme for solving a split feasibility problem. The proposed schemes are used in solving the image deblurring problem to illustrate the applicability of the proposed methods in comparison with the existing state-of-the-art methods.

An implicit stabilization strategy for Q-compensated reverse time migration

Reverse time migration with [Formula: see text] compensation ([Formula: see text]-RTM) is an effective approach to enhance the resolution of seismic images because it retrieves the amplitude loss and phase distortion induced by the viscosity of media. According to the crosscorrelation imaging condition, [Formula: see text]-RTM requires compensation for the amplitude loss in the propagation paths of source and receiver wavefields, which can be realized by solving an amplitude-boosted wave equation. However, the amplitude-boosted simulations suffer from numerical instability due to the amplification of high-frequency noise. We have developed a robust stabilization strategy for [Formula: see text]-RTM by incorporating a time-variant filter into the amplitude-boosted wavefield extrapolation step. We modify the Fourier spectrum of the operator that controls the amplitude compensation to be time variant, and we add to the spectrum a stabilization factor. Doing so, we integrate the time-variant filter into the viscoacoustic wave propagator implicitly, and we avoid any explicit filtering operation in [Formula: see text]-RTM. We verify the robustness of this stabilized [Formula: see text]-RTM with two synthetic data examples. We also apply this technique to a field data set to demonstrate the imaging improvements compared to an acoustic RTM and a more traditional [Formula: see text]-RTM method.

An inertial extragradient method for solving bilevel equilibrium problems

In this paper, we propose an algorithm with two inertial term extrapolation steps for solving bilevel equilibrium problem in a real Hilbert space. The inertial term extrapolation step is introduced to speed up the rate of convergence of the iteration process. Under some sufficient assumptions on the bifunctions involving pseudomonotone and Lipschitz-type conditions, we obtain the strong convergence of the iterative sequence generated by the proposed algorithm. A numerical experiment is performed to illustrate the numerical behavior of the algorithm and also comparison with some other related algorithms in the literature.

Study of micro–macro acceleration schemes for linear slow-fast stochastic differential equations with additive noise

Computational multi-scale methods capitalize on a large time-scale separation to efficiently simulate slow dynamics over long time intervals. For stochastic systems, one often aims at resolving the statistics of the slowest dynamics. This paper looks at the efficiency of a micro-macro acceleration method that couples short bursts of stochastic path simulation with extrapolation of spatial averages forward in time. To have explicit derivations, we elicit an amenable linear test equation containing multiple time scales. We make derivations and perform numerical experiments in the Gaussian setting, where only the evolution of mean and variance matters. The analysis shows that, for this test model, the stability threshold on the extrapolation step is largely independent of the time-scale separation. In consequence, the micro-macro acceleration method increases the admissible time steps far beyond those for which a direct time discretization becomes unstable.

Inertial proximal strictly contractive Peaceman–Rachford splitting method with an indefinite term for convex optimization

We consider an inertial proximal strictly contractive Peaceman–Rachford splitting method (abbreviated as IPSCPRSM) with an indefinite proximal term for solving convex optimization problems with linear constraints. With the aid of variational inequality, proximal point method and fundamental inequality, we prove global convergence of the proposed method and analyze iteration complexity in the best function value and feasibility residues. The experimental results demonstrate the efficiency of the inertial extrapolation step and the indefinite proximal term even compared with the state-of-the-art methods.

Richardson Extrapolation of Superconvergent Projection-Type Methods for Hammerstein Equations*

For a nonlinear Hammerstein equation with a smooth kernel, a method proposed recently, based on projection onto a space of piecewise polynomials of degree is shown to have convergence of order This paper shows that this method have asymptotic series expansion and the order of convergence can be further improved to by one step of Richardson extrapolation, assuming the calculation to be repeated with each subinterval halved. Some numerical results are given to illustrate this improvement.

Convergence analysis of projection method for variational inequalities

The main contributions of this paper are the proposition and the convergence analysis of a class of inertial projection-type algorithm for solving variational inequality problems in real Hilbert spaces where the underline operator is monotone and uniformly continuous. We carry out a unified analysis of the proposed method under very mild assumptions. In particular, weak convergence of the generated sequence is established and nonasymptotic $O(1/n)$ rate of convergence is established, where $n$ denotes the iteration counter. We also present some experimental results to illustrate the profits gained by introducing the inertial extrapolation steps.

English

We propose a Halpern-type forward-backward splitting with inertial extrapolation step for finding a zero of the sum of accretive operators in Banach spaces. Strong convergence of the sequence of iterates generated by the method proposed is obtained under mild assumptions. We give some numerical results in compressed sensing to validate the theoretical analysis results. Our result is one of the lew available inertial-type methods for zeros of the sum of accretive operators in Banach spaces.

Bregman Proximal Gradient Algorithm With Extrapolation for a Class of Nonconvex Nonsmooth Minimization Problems

In this paper, we consider an accelerated method for solving nonconvex and nonsmooth minimization problems. We propose a Bregman Proximal Gradient algorithm with extrapolation (BPGe). This algorithm extends and accelerates the Bregman Proximal Gradient algorithm (BPG), which circumvents the restrictive global Lipschitz gradient continuity assumption needed in Proximal Gradient algorithms (PG). The BPGe algorithm has a greater generality than the recently introduced Proximal Gradient algorithm with extrapolation (PGe) and, in addition, due to the extrapolation step, BPGe converges faster than the BPG algorithm. Analyzing the convergence, we prove that any limit point of the sequence generated by BPGe is a stationary point of the problem by choosing the parameters properly. Besides, assuming Kurdyka-Łojasiewicz property, we prove that all the sequences generated by BPGe converge to a stationary point. Finally, to illustrate the potential of the new method BPGe, we apply it to two important practical problems that arise in many fundamental applications (and that not satisfy global Lipschitz gradient continuity assumption): Poisson linear inverse problems and quadratic inverse problems. In the tests the accelerated BPGe algorithm shows faster convergence results, giving an interesting new algorithm.