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ABSTRACT A mediation analysis approach is proposed for multiple exposures, multiple mediators, and a continuous scalar outcome under the linear structural equation modeling framework. It assumes that there exist orthogonal components that demonstrate parallel mediation mechanisms on the outcome, and thus is named principal component mediation analysis (PCMA). Likelihood‐based estimators are introduced for simultaneous estimation of the component projections and effect parameters. The asymptotic distribution of the estimators is derived for low‐dimensional data. A bootstrap procedure is introduced for inference. Simulation studies illustrate the superior performance of the proposed approach. Applied to a proteomics‐imaging dataset from the Alzheimer's disease neuroimaging initiative (ADNI), the proposed framework identifies protein deposition – brain atrophy – memory deficit mechanisms consistent with existing knowledge and suggests potential AD pathology by integrating data collected from different modalities. [ABSTRACT FROM AUTHOR]

The one parameter conjugate gradient method by Hager and Zhang (Pac J Optim, 2(1):35–58, 2006) represents a family of descent iterative methods for solving large‐scale minimization problems. The nonnegative parameter of the scheme determines the weight of conjugacy and descent, and by extension, the numerical performance of the method. The scheme, however, does not converge globally for general nonlinear functions, and when the parameter approaches 0, the scheme reduces to the conjugate gradient method by Hestenes and Stiefel (J Res Nat Bur Stand, 49:409–436, 1952), which in practical sense does not perform well due to the jamming phenomenon. By carrying out eigenvalue analysis of an adaptive two parameter Hager–Zhang type method, a new scheme is presented for system of monotone nonlinear equations with its application in compressed sensing. The proposed scheme was inspired by nice attributes of the Hager–Zhang method and the various schemes designed with double parameters. The scheme is also applicable to nonsmooth nonlinear problems. Using fundamental assumptions, analysis of the global convergence of the scheme is conducted and preliminary report of numerical experiments carried out with the scheme and some recent methods indicate that the scheme is promising. [ABSTRACT FROM AUTHOR]

First-order optimization methods have long been established as effective methods for solving large-scale unconstrained optimization problems. Over time, these methods have been extended to address the task of finding zeros of single-valued monotone maps within the framework of finite-dimensional Hilbert spaces. Notably, Abubakar et al. (Comput Appl Math 39(2):129, 2020) recently introduced a projection method that incorporates a convex combination of two distinct positive spectral coefficients to find the zeros of a single-valued monotone map. Motivated by their work, this paper generalises the Abubakar et al. technique to a more general setting of infinite-dimensional Hilbert spaces. Under mild assumptions, we establish weak convergence of the generated sequence of iterates. Furthermore, the extended method does not involve computing the resolvent of the monotone operator. [ABSTRACT FROM AUTHOR]

Solving large-scale systems of nonlinear equations (SoNE) is a central task in mathematics that traverses different areas of applications. There are several derivative-free methods for finding SoNE solutions. However, most of the methods contributed to find SoNE solutions involve a monotone cost function. Methods dealing with pseudomonotone cost function remain rare. In this paper, we introduce two classes of derivative-free spectral three-term methods to solve large-scale continuous pseudomonotone SoNE. We combine the projection method of Solodov and Svaiter with the structure of the recently developed spectral three-term conjugate gradient method for unconstrained optimization by Amini and Faramarzi. We prove that the proposed methods possess sufficient descent property, trust region property, and global convergence without relying on Lipschitz continuity. Numerical experiments show that the proposed methods are efficient and competitive with existing methods. Finally, the proposed methods have been successfully applied to recover a sparse signal from incomplete and contaminated sampling measurements, yielding promising results. [ABSTRACT FROM AUTHOR]

In this paper, we propose an extragradient type projection algorithm for solving quasimonotone variational inequalities in $ \mathbb {R}^n $ R n . Under the assumption that f is a quasimonotone and continuous mapping and the solution set of the dual variational inequality problem is nonempty, the globally convergence of the algorithm is established. Numerical experiments are also reported. [ABSTRACT FROM AUTHOR]

Observational studies are common in epidemiology, medicine, economics, and other research fields. Causal inference has been regarded as one of the most effective tools for conducting statistical inference with observational studies without random treatment assignments. The validity of most existing causal inference procedures depends on the underlying outcome regression and/or propensity score modeling assumptions. In this paper, we propose two new Bayesian multiply robust estimation approaches for causal inference based on the loss-likelihood bootstrap. The proposed methods enjoy the multiple robustness property such that the estimators of treatment effect are consistent if at least one of the outcome regression models or propensity score models is correctly specified. Convenient statistical inference can be conducted using the Bayesian posterior credible intervals. A simulation study shows the benefits of our proposed methods. We further apply our proposed methods by using 2009–2010 National Health and Nutrition Examination Survey (NHANES) data to examine the effect of exposure to perfluoroalkyl acids (PFAs) on kidney function. [ABSTRACT FROM AUTHOR]

Numerical solving the Schrödinger equation with incommensurate potentials presents a great challenge since its solutions could be space-filling quasiperiodic structures without translational symmetry nor decay. In this paper, we propose two high-accuracy numerical methods to solve the time-dependent quasiperiodic Schrödinger equation. Concretely, we discretize the spatial variables by the quasiperiodic spectral method and the projection method, and the time variable by the second-order operator splitting method. The corresponding convergence analysis is also presented and shows that the proposed methods both have spectral convergence rates in space and second order accuracy in time, respectively. Meanwhile, we analyse the computational complexity of these numerical algorithms. One- and two-dimensional numerical results verify these convergence conclusions, and demonstrate that the projection method is more efficient. [ABSTRACT FROM AUTHOR]

In this article, two Hager–Zhang (HZ) type projection algorithms are presented for large-dimension nonlinear monotone problems and sparse signal recovery in compressed sensing. This goal is attained by conducting singular value analysis of a nonsingular HZ-type search direction matrix as well as applying the idea by Piazza and Politi [J Comput Appl Math. 2002;143(1):141–144] and minimizing the Frobenius norm of an orthornormal matrix. The paper attempts to fill the gap in the work of Hager and Zhang [Pac J Optim. 2006;2(1):35–58], Waziri et al. [Appl Math Comput. 2019;361:645–660], Sabi'u et al. [Appl Numer Math. 2020;153:217–233] and Babaie-Kafaki [4OR-Q J Oper Res. 2014;12:285-292], where the sufficient descent or global convergence condition is not satisfied when the HZ parameter is in the interval $ (0,\frac {1}{4}) $ (0 , 1 4). The proposed schemes are also suitable for solving non-smooth nonlinear problems. Also, by employing some mild conditions, global convergence of the schemes are established, while numerical comparison with four effective HZ-type methods show that the new methods are efficient. Furthermore, to illustrate their practical application, both methods are applied to solve the $ \ell _1 $ ℓ 1 -norm regularization problems to recover a sparse signal in compressive sensing. The experiments conducted in that regard show that the methods are promising and perform better than two other methods in the literature. [ABSTRACT FROM AUTHOR]

We study a class of general M-estimators in the semi-supervised setting, wherein the data are typically a combination of a relatively small labeled dataset and large amounts of unlabeled data. A new estimator, which efficiently uses the useful information contained in the unlabeled data, is proposed via a projection technique. We prove consistency and asymptotic normality, and provide an inference procedure based on K -fold cross-validation. The optimal weights are derived to balance the contributions of the labeled and unlabeled data. It is shown that the proposed method, by taking advantage of the unlabeled data, produces asymptotically more efficient estimation of the target parameters than the supervised counterpart. Supportive numerical evidence is shown in simulation studies. Applications are illustrated in analysis of the homeless data in Los Angeles. for this article are available online. [ABSTRACT FROM AUTHOR]

We consider a class of convex optimization problems in a Hilbert space that can be solved by performing a single pro jection, i.e., by pro jecting an infeasible point onto the feasible set. Our results improve those established for the linear programming setting in Nurminski (2015) by considering problems that (i) may have multiple solutions, (ii) do not satisfy strict complementarity conditions, and (iii) possess nonlinear convex constraints. As a by-product of our analysis, we provide a quantitative estimate on the required distance between the infeasible point and the feasible set in order for its projection to be a solution of the problem. Our analysis relies on a "sharpness" property of the constraint set, a new property we introduce here. [ABSTRACT FROM AUTHOR]

In this study, the Atangana–Baleanu fractional derivative in the Caputo type (as a kind of non‐local and non‐singular derivative) is used to define a new class of stochastic fractional integro‐differential equations. A projection method (more precisely, a Galerkin approach) based on the piecewise Chebyshev cardinal functions is developed to solve these stochastic fractional equations. To construct this method, the operational matrices of fractional and stochastic integrals of these basis functions are obtained and used in the established method. By approximating the solution of the problem with a finite expansion of the expressed basis functions (in which the expansion coefficients are unknown), a system of algebraic equations is obtained. By solving this system, the expansion coefficients and subsequently the solution of the original stochastic fractional problem are obtained. The convergence analysis of the proposed method is investigated, theoretically and numerically. The accuracy of the established procedure is illustrated by solving several numerical examples. [ABSTRACT FROM AUTHOR]

This paper develops a class of algorithms for general linear systems and eigenvalue problems. These algorithms apply fast randomized dimension reduction ("sketching") to accelerate standard subspace projection methods, such as GMRES and Rayleigh-Ritz. This modification makes it possible to incorporate nontraditional bases for the approximation subspace that are easier to construct. When the basis is numerically full rank, the new algorithms have accuracy similar to classic methods but run faster and may use less storage. For model problems, numerical experiments show large advantages over the optimized MATLAB routines, including a 70x speedup over gmres and a 10x speedup over eigs. [ABSTRACT FROM AUTHOR]

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In the current study, a non-hydrostatic two-dimensional vertical (2DV) numerical model with a shock-capturing technique is developed to simulate irregular wave breaking on a barred beach. The complete form of the 2DV Reynolds-averaged Navier-Stokes (RANS) equations is discretized using the finite volume approach. In the spatial discretization phase, a flux limiter function is employed for the velocity advection terms to prevent non-physical oscillations in regions with steep gradients. A novel method has been introduced during the discretization stage, leading to a significant reduction in computational expenses. For temporal discretization, the Leapfrog method, known for its second-order accuracy in time, is employed to address wave-damping issues. Model validation involves a comparison between simulation results and experimental data pertaining to irregular wave breaking, affirming the satisfactory performance of the model. To evaluate the accuracy of the model, the Root Mean Square Error (RMSE) was calculated. The assessment showed that the model is capable of estimating the significant wave height reported in the experiments used with an RMSE between 0.002 and 0.089, and the mean wave periods with an RMSE between 0.061 and 0.252.

BackgroundStudents’ ability to diagnose various blood disorders could be substantially improved by continuously reviewing approaches toward teaching hematology. This study aims to compare the effectiveness of light microscopes and projected images on students’ learning and determine medical students’ perception of these teaching methods.MethodsA randomized trial was conducted using a crossover design. Two groups, each with 30 students, were subjected to teaching methods based on light microscopes and projected images alternatively.ResultsNo differences were found in the two study groups’ baseline characteristics, such as median age, sex, and prior academic performance, as well as in the pre-test scores. Post-test scores were significantly higher among students subjected to the projection method than in the control group (Mean ± SD = 9.8 ± 1.7 vs. 5.1 ± 1.3, p

In this paper, a semi‐discrete rotational velocity‐correction projection method is proposed to solve the Kelvin–Voigt viscoelastic fluid equations. In this method, the rotational velocity‐correction projection method can preserve the divergence free of the velocity u$$ u $$, and the nonlinear equation was linearized. Then, the unconditional stability and optimal convergence order will be provided by the numerical analysis. Finally, our analysis are confirmed by some numerical results, and the algorithm is effective. [ABSTRACT FROM AUTHOR]

This article concerns the linear hypothesis testing problem in the functional response model, which is one of the regression models considered in functional data analysis. In this model, the response is a function represented as a random process, while the predictors are random variables. To test the linear hypothesis, projection tests are constructed and theoretically justified. Namely, a kind of equivalence between the functional null hypothesis and its projected version is established. Different Gaussian processes and numbers of projections are considered in the implementation of new solutions. Moreover, as there is no one test having the best power for all correlation cases, a simple combining test is also proposed. It has satisfactory power in all cases. In simulation studies, the new tests are compared with existing methods in terms of size control and power. A real data example is also provided to illustrate the results. [ABSTRACT FROM AUTHOR]

The largest C-eigenvalue of piezoelectric tensors determines the highest piezoelectric coupling constant, which reflects the coupling between the elastic and dielectric properties of crystal. Here, a projection method based on discrete normalized dynamical system (PDND) is established for computing the largest C-eigenvalue. Theoretical analysis of the convergence for PDND algorithm is given. In numerical experiments, the longitudinal piezoelectric modulus and the unit uniaxial direction that the extreme piezoelectric effect along took place of different piezoelectric materials are given to display the physical meaning of the C-eigenvalues and eigenvectors. Furthermore, the largest C-eigenvalue and all the corresponding eigenvectors can be obtained, which is the advantage of the proposed method. [ABSTRACT FROM AUTHOR]

Quasiperiodic systems are important space-filling ordered structures, without decay and translational invariance. How to solve quasiperiodic systems accurately and efficiently is a great challenge. A useful approach, the projection method (PM) [J. Comput. Phys., 256 (2014), pp. 428--440], has been proposed to compute quasiperiodic systems. Various studies have demonstrated that the PM is an accurate and efficient method to solve quasiperiodic systems. However, there is a lack of theoretical analysis of the PM. In this paper, we present a rigorous convergence analysis of the PM by establishing a mathematical framework of quasiperiodic functions and their high-dimensional periodic functions. We also give a theoretical analysis of the quasiperiodic spectral method (QSM) based on this framework. Results demonstrate that the PM and QSM both have exponential decay, and the QSM (PM) is a generalization of the periodic Fourier spectral (pseudospectral) method. Then, we analyze the computational complexity of the PM and QSM in calculating quasiperiodic systems. The PM can use a fast Fourier transform, while the QSM cannot. Moreover, we investigate the accuracy and efficiency of the PM, QSM, and periodic approximation method in solving the linear time-dependent quasiperiodic Schr\"odinger equation. [ABSTRACT FROM AUTHOR]

We developed the projection method to derive an analog of the quantum master equation for propagators rather than density matrices themselves. As these propagators are superoperators, we call them superoperator master equations. Furthermore, as the projector maps superoperators to superoperators, we call it a hyperprojector. We gave general perturbative expansions for generators of the weak coupling superoperator master equation and the stroboscopic limit superoperator master equation. After that, we considered a particular example of a hyperprojector that is the infinite time average of unitary dynamics. We call it the averaging hyperprojector. We discussed the properties of this hyperprojector and its physical meaning. Then, we illustrated our general second order superoperator master equations arising in the weak coupling limit and the stroboscopic limit, taking the averaging hyperprojector as an example, which we call effective dynamics. We discussed some properties of these superoperator master equations, in particular, the entropy increase. [ABSTRACT FROM AUTHOR]

Oil spills continue to generate various issues and concerns regarding their effect and behavior in the marine environment, owing to the related potential for detrimental environmental, economic and social implications. It is essential to have a solid understanding of the ways in which oil interacts with the water and the coastal ecosystems that are located nearby. This study proposes a simplified model for predicting the plume-like transport behavior of heavy Bunker C fuel oil discharging downward from an acutely-angled broken pipeline located on the water surface. The results show that the spill overall profile is articulated in three major flow areas. The first, is the source field, i.e., a region near the origin of the initial jet, followed by the intermediate or transport field, namely, the region where the jet oil flow transitions into an underwater oil plume flow and starts to move horizontally, and finally, the far-field, where the oil re-surface and spreads onto the shore at a significant distance from the spill site. The behavior of the oil in the intermediate field is investigated using a simplified injection-type oil spill model capable of mimicking the undersea trapping and lateral migration of an oil plume originating from a negatively buoyant jet spill. A rectangular domain with proper boundary conditions is used to implement the model. The Projection approach is used to discretize a modified version of the Navier-Stokes equations in two dimensions. A benchmark fluid flow issue is used to verify the model and the results indicate a reasonable relationship between specific gravity and depth as well as agreement with the aerial data and a vertical temperature profile plot. [ABSTRACT FROM AUTHOR]

In this paper, we study the estimation of the derivative of a regression function in a standard univariate regression model. The estimators are defined either by derivating nonparametric least-squares estimators of the regression function or by estimating the projection of the derivative. We prove two simple risk bounds allowing to compare our estimators. More elaborate bounds under a stability assumption are then provided. Bases and spaces on which we can illustrate our assumptions and first results are both of compact or noncompact type, and we discuss the rates reached by our estimators. They turn out to be optimal in the compact case. Lastly, we propose a model selection procedure and prove the associated risk bound. To consider bases with a noncompact support makes the problem difficult. [ABSTRACT FROM AUTHOR]

The wellbore hole simulator as a high-temperature high-pressure thick-walled container, is an experimental device used to simulate the high-temperature high-pressure downhole environment of oilfield. Based on thermodynamics and the large eddy simulation (LES) theory, a physical equation was established. The projection method was applied to solve the temperature field governing equation, and the trapezoidal-rule numerical integration was used to solve the thermal stress governing equation. The discrete scheme for the governing equation was given. The fluid-structure-interaction heat transfer was solved with the virtual density method, and the thermal stress, the pressure stress and their coupling effect of the wellbore hole simulator were numerically analyzed under the principle of stress superimposition. The research results indicate that, the wellbore hole simulator with a minimum wall thickness of 0.18 m could meet the strength requirements for high-temperature high-pressure experiments with 400 ℃ and 220 MPa working parameters. The experiments prove the correctness of the established mathematical model and the numerical solution methods, providing a theoretical basis for the design of thick-walled cylinder containers under high-temperature high-pressure conditions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the multiple-sets split feasibility problem in real Hilbert space and propose a self-adaptive method that uses projections onto quadratic (balls) approximations of the problem's associated sets. Our algorithm has several major advantages over existing methods in the literature. The first is its simple implementation as it uses closed-formula projection onto balls, and the second is that strong convergence is obtained under mild conditions. Several numerical experiments illustrate and compare the performances of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we examine a Cauchy-type problem for one ordinary differential equation with Riemann–Liouville fractional derivatives. For this problem, based on the Lebesgue space of functions that are summable with an arbitrarily fixed degree, we propose a family of pairs of spaces of elements and right-hand sides that provide the well-posed statement of the problem. In these pairs of spaces, we develop a generalized polynomial projection method for solving the problem considered and justify it from the functional-theoretic point of view. Based on the general results obtained, we prove the convergence of the "polynomial" Galerkin method, the collocation method, and the subdomain method for solving the corresponding Cauchy-type problem. [ABSTRACT FROM AUTHOR]

In this research paper, we introduce a novel gradient-free modified three-term conjugate gradient method designed to solve nonlinear equations subject to convex constraints. Our approach incorporates the projection scheme, which enhances the effectiveness of the proposed method. Building upon the modified three-term conjugate gradient method for solving M-tensor systems and ℓ1-norm-based nonsmooth optimization problems, our method can be regarded as an extension of their technique. By making mild assumptions, we establish the theoretical convergence properties of our iterative method. Through extensive numerical experiments, we demonstrate that our proposed approach is not only highly efficient but also outperforms other existing methods in terms of performance and accuracy. [ABSTRACT FROM AUTHOR]

In this paper, a new approach for numerically solving the system of fractional integrodifferential equations is devised. To approximate the issue, we employ Vieta–Fibonacci polynomials as basis functions and derive the projection method for Caputo fractional order for the first time. An efficient transformation reduces the problem to a system of two independent equations. Solving two algebraic equations yields an approximate solution to the problem. The proposed method's efficiency and accuracy are validated. We demonstrate the existence of the solution to the approximate problem and conduct an error analysis. Numerical tests reinforce the interpretations of the theory. [ABSTRACT FROM AUTHOR]

In this paper, we first propose a relaxed regularization projection method involving only a single projection for solving monotone bilevel variational inequality problem in Hilbert spaces and secondly we give an alternated inertial version of the first algorithm. The two proposed algorithms involve self adaptive step-sizes and the algorithms can easily be implemented without the prior knowledge of Lipschitz and strongly monotone constants of operators. Under some mild standard assumptions, we obtain the strong convergence of the two algorithms to the unique solution of the bilevel equilibrium problem. Moreover, some interesting numerical experiments are given to demonstrate the applicability of the results and also to compare with existing algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new outer proximal approach for solving the variational inequality problems in the real Euclidean space, where the feasible set is replaced by its polyhedral outer approximation. First, we prove the quasicontractiveness of the outer proximal operator. Second, we apply this property to present two new algorithms and their convergence under strongly monotone and Lipschitz continuous conditions of the cost mapping. Finally, we give some numerical results for the proposed algorithms and comparison with other known methods. [ABSTRACT FROM AUTHOR]

We introduce and analyze two modified subgradient extragradient methods with adaptive step sizes for solving variational inequality problems governed by pseudomonotone and Lipschitz continuous operators in real Hilbert spaces. For the first algorithm, a sufficient condition for weak convergence is established under pseudomonotonicity and Lipschitz continuity assumptions, with a possibly unknown Lipschitz constant. A sufficient condition for weak convergence for the second algorithm is proved under pseudomonotonicity and uniform continuity assumptions. We also establish an R-linear rate of convergence under strong pseudomonotonicity and Lipschitz continuity hypotheses regarding the variational inequality operator. Finally, we present several numerical computational experiments which involve a comparison of our proposed algorithms with other algorithms in some real-life applications. [ABSTRACT FROM AUTHOR]

In this work, a projection method based on modular grad-div stabilization has been proposed for the evolutionary Navier-Stokes equations. The presented method can also be regarded as the improved penalty-projection method. Unlike the penalty-projection method that adds the grad-div term to the momentum equation, the proposed scheme incorporates a minor intrusive step to an existing projection code. By doing so, the proposed scheme not only holds the merits of the penalty-projection method; but also avoids solver failures and improves computational efficiency along with the increase of grad-div parameters. Furthermore, we show that the scheme is unconditionally stable and give out convergence analysis. In the final, numerical tests are done which validate the theoretical results and their efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we focus on a finite element projection method for inductionless magnetohydrodynamics equations. A fully discrete projection method based on Euler semi-implicit scheme is proposed in which continuous elements are used to approximate the Navier–Stokes equations and the divergence-conforming element is used to approximate the current density. The key of the projection method is that it must be compatible with two different spaces for calculating velocity, which leads to solve the pressure by solving a Poisson equation. The results show that the proposed projection scheme meets a discrete energy stability and the discrete current density keeps charge conservation property. In addition, this paper provides a rigorous optimal error analysis of velocity, pressure and current density. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme. [ABSTRACT FROM AUTHOR]

Like the Polak-Ribi`ere-Polyak (PRP) and Hestenes-Stiefel (HS) meth-ods, the classical Liu-Storey (LS) conjugate gradient scheme is widely be-lieved to perform well numerically. This is attributed to the in-built capa-bility of the method to conduct a restart when a bad direction is encoun-tered. However, the scheme’s inability to generate descent search direc-tions, which is vital for global convergence, represents its major shortfall. In this article, we present an LS-type scheme for solving system of mono-tone nonlinear equations with convex constraints. The scheme is based on the approach by Wang et al. (2020) and the projection scheme by Solodov and Svaiter (1998). The new scheme satisfies the important condition for global convergence and is suitable for non-smooth nonlinear problems. Fur-thermore, we demonstrate the method’s application in restoring blurry im-ages in compressed sensing. The scheme’s global convergence is established under mild assumptions and preliminary numerical results show that the proposed method is promising and performs better than two recent meth-ods in the literature.

In this article, three dimensional (3D) lid-driven flow in shallow cavities with a unit square base are studied. The numerical solution of the Navier–Stokes equations modeling incompressible viscous fluid flow in a cavity is obtained via a methodology combining a first order accurate operator-splitting scheme, a $L^2$-projection Stokes solver, a wave-like equation treatment of the advection and finite element space approximations. Numerical results of a lid-driven flow in a cubic cavity show a good agreement with those reported in literature. The critical Reynolds numbers ($\mathit{Re}_{\mathrm{cr}}$) for having flow with increasing of oscillating amplitude (a Hopf bifurcation) in different shallow cavities are obtained and associated oscillating modes are studied.

The this paper, we introduce a pair of Sobolev spaces with special Jacobi–Gegenbauer weights, in which the general boundary-value problem for a class of ordinary integro-differential equations characterized by the positivity of the difference of orders of the inner and outer differential operators is well-posed in the Hadamard sense. Based on this result, we justify the general polynomial projection method for solving the corresponding problem. An application of general results to the proof of the convergence of the polynomial Galerkin method for solving the Cauchy problem in the Sobolev weighted space is given. The convergence rate of the method is characterized in terms of the best polynomial approximations of an exact solution, which automatically responds to the smoothness properties of the coefficients of the equation. [ABSTRACT FROM AUTHOR]

In this work, we analyze some convergent properties of a projection and contraction algorithm for solving a variational inequality problem, where the feasible domain is the solution set of an affine variational inequality problem. We prove that, for solving the problem where the second cost mapping is affine and not necessary for monotone properties, any iterative sequence generated by the algorithm converges to a unique solution provided that the first cost mapping is strongly monotone and Lipschitz continuous. Computational errors of the algorithm are showed. Finally, some preliminary numerical experiences and comparisons are also reported. [ABSTRACT FROM AUTHOR]

Projection methods have been one of the popular methods for solving variational inequality problems in the literature. Several projection methods have been studied for variational inequality problems and most of these methods require at least two evaluations of the cost operator and (or) two projections per iteration. One of the major interests of researchers in variational inequalities is to study projection methods with less computational complexity in terms of reduced number of evaluations of the cost operator and reduced number of projections onto feasible set per iteration especially when the cost operator is non-monotone. In this paper, we propose a simple projection method which requires only one evaluation of the cost operator and one projection onto the feasible set per iteration. We obtain weak convergence results when the cost operator is quasimonotone and Lipschitz continuous with self-adaptive line search strategy. We also give weak convergence analysis when the cost operator is non-monotone and Lipschitz continuous. Some numerical implementations using standard tests are given to illustrate our method and compare with other recently related methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a derivative‐free iterative method for finding the solutions of convex constrained nonlinear equations (CCNE) using the projection strategy. The new approach is free from gradient evaluations at each iteration. Also, the search direction generated by the proposed method satisfies the sufficient descent property, which is independent of the line search. Compared with traditional methods for solving CCNE that assumes Lipschitz continuity and monotonicity to establish the global convergence result, an advantage of our proposed method is that the global convergence result does not require the assumption of Lipschitz continuity. Moreover, the underlying operator is assumed to be pseudomonotone, which is a milder condition than monotonicity. As an applications, we solve the LASSO problem in compressed sensing. Numerical experiments illustrate the performances of our proposed algorithm and provide a comparison with related algorithms. [ABSTRACT FROM AUTHOR]

A novel high-order projection method is presented in this study for the simulation of incompressible turbulent flows, utilising weighted essentially non-oscillatory (WENO) schemes. Spatial discretization involves using WENO schemes for nonlinear convection and standard central differences (CD) for the viscous term. The approach combines various orders of WENO and central difference schemes (WENO3/CD2, WENO5/CD4, and WENO7/CD6) to achieve third, fifth, and seventh-order spatial accuracy for velocity fields, respectively. Validating this method involved a thorough examination using the 2D Taylor Green vortex problem in both space and time, along with simulations of 3D Taylor-Green vortex problems at increasing Reynolds numbers (Re) of 1, 600, 16, 000, and 160, 000. The outcomes encompassed analyses of kinetic energy, enstrophy, energy spectra, and vortex fields, indicating the method's effectiveness in accurately simulating high Reynolds number turbulent flows without additional sub-grid scale models. [ABSTRACT FROM AUTHOR]

In this paper, the COPRAS (Complex Proportional Assessment) method is extended for interval-valued q-rung orthopair fuzzy numbers (IVq-ROFNs) to solve multi-attribute group decision-making (MAGDM) problems. A novel distance measure for IVq-ROFNs is proposed, and its properties are also probed. This distance measure is used in an improved weights determination method for decision-makers. A weighted projection optimization model is developed to evaluate the completely unknown attributes' weights. The projection of assessment values is defined by the positive and negative ideal solutions, which determine the resemblance between two objects by considering their directional angle. An Indian cities' ranking problem for a better solid waste management infrastructure is solved using the proposed approach based on composite indicators, like recycling waste, greenhouse gas emissions, waste generation, landfilling waste, recycling rate, waste-to-energy rate, and composting waste. Numerical comparisons, sensitivity analysis, and other relevant analyses are performed for validation. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new class of variational inequalities involving two operators, which is called the general variational inequality. We have shown that the general variational inequalities are equivalent to the fixed point problem using the projection technique. This equivalent fixed point formulation is used to discuss the existence of solution as well as to investigate several iterative methods for solving general variational inequalities. Some applications of the associated dynamical system coupled with finite difference are explored. Convergence analysis of the proposed methods is considered under suitable conditions. Since general variational inequalities include the variational inequalities, complementarity problems and nonlinear equations as special cases, our results continued to hold for these problems. The techniques and ideas of this paper be starting point for the future research. [ABSTRACT FROM AUTHOR]