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A modified Chorin–Teman (Euler non-incremental) projection method and a modified Euler incremental projection method for non inf-sup stable mixed finite elements are analyzed. The analysis of the classical Euler non-incremental and Euler incremental methods are obtained as a particular case. We first prove that the modified Euler non-incremental scheme has an inherent stabilization that allows the use of non inf-sup stable mixed finite elements without any kind of extra added stabilization. We show that it is also true in the case of the classical Chorin–Temam method. For the second scheme, we study a stabilization that allows the use of equal-order pairs of finite elements. The relation of the methods with the so-called pressure stabilized Petrov Galerkin method (PSPG) is established. The influence of the chosen initial approximations in the computed approximations to the pressure is analyzed. Numerical tests confirm the theoretical results, Research sup-ported by Spanish MINECO under grants MTM2013-42538-P (MINECO, ES) and MTM2016-78995-P (AEI/FEDER UE)

The Fundamental Clustering Problems Suite (FCPS) offers a variety of clustering challenges that any algorithm should be able to handle given real-world data. The FCPS consists of datasets with known a priori classifications that are to be reproduced by the algorithm. The datasets are intentionally created to be visualized in two or three dimensions under the hypothesis that objects can be grouped unambiguously by the human eye. Each dataset represents a certain problem that can be solved by known clustering algorithms with varying success. In the R package “Fundamental Clustering Problems Suite” on CRAN, user-defined sample sizes can be drawn for the FCPS. Additionally, the distances of two high-dimensional datasets called Leukemia and Tetragonula are provided here. This collection is useful for investigating the shortcomings of clustering algorithms and the limitations of dimensionality reduction methods in the case of three-dimensional or higher datasets. This article is a simultaneous co-submission with Swarm Intelligence for Self-Organized Clustering [1].

We present a displacement-based Galerkin meshfree method for the analysis of nearly-incompressible linear elastic solids, where low-order simplicial tessellations (i.e., 3-node triangular or 4-node tetrahedral meshes) are used as a background structure for numerical integration of the weak form integrals and to get the nodal information for the computation of the meshfree basis functions. In this approach, a volume-averaged nodal projection operator is constructed to project the dilatational strain into an approximation space of equal- or lower-order than the approximation space for the displacement field resulting in a locking-free method. The stability of the method is provided via bubble-like basis functions. Because the notion of an ‘element’ or ‘cell’ is not present in the computation of the meshfree basis functions such low-order tessellations can be used regardless of the order of the approximation spaces desired. First- and second-order meshfree basis functions are chosen as a particular case in the proposed method. Numerical examples are provided in two and three dimensions to demonstrate the robustness of the method, its ability to avoid volumetric locking in the nearly-incompressible regime, and its improved performance when compared to the MINI finite element scheme on the simplicial mesh.

Open source computational fluid dynamics (CFD) codes provide suitable environments for implementation, testing and rapid dissemination of algorithms typically used for large-eddy simulations (LES) and direct numerical simulations (DNS). In particular, it is important to test low-dissipative algorithms in unstructured codes of industrial relevance. In the present paper, the implementation of incompressible, explicit Runge-Kutta (RK) based projection methods into the OpenFOAM (R) library is discussed. We search for low-dissipative alternatives to the second order time integration methods which are commonly used together with the standard pressure correction approach PISO (Pressure Implicit with Splitting of Operators) in many commercial and open source codes including OpenFOAM (R). The practical implementation of the projection methods in OpenFOAM (R) is provided together with theory. The method is tested with the classical fourth order RK-method and the accelerated third order RK-method. Four numerical experiments are carried out in order to cross-validate the solvers and in order to investigate the drawbacks/ benefits of the solution approaches. The test problems are: (1) 2d lid driven cavity flow at Re = 2500, (2) DNS of 3d turbulent channel flow at Re-tau = 180, (3) LES of a 3d mixing layer, and (4) the 2d inviscid Taylor-Green vortex. The RK-methods are benchmarked against the standard OpenFOAM (R) LES/DNS solver based on the PISO pressure correction method. The results indicate that the RK projection methods provide low-dissipative alternatives to the PISO method. The turbulent test cases show also that the RK-methods have a good computational efficiency. (C) 2014 Elsevier Ltd. All rights reserved. (Less)

The paper deals with the problems of development of tertiary education during the last decade and the perspective in the second decade of the 21-st century. Therefore, firstly, it is analysed the evolution of the number of students and their distribution within specialization fields in European Union. Also, a special attention is paid to identification of characteristic features of the tertiary education in Romania during the period of integration in European Union. The second part of the paper is dedicated the projection of tertiary education supply in Romania from quantitative and structural point of view. In this context, there are presented the constraints and opportunities for the modelling the evolution of tertiary education generated by demographic dynamics and the convergence with the students distribution on specialization fields registered in European Union as a whole.

Since the appearance of the Barzilai–Borwein (BB) step sizes strategy for unconstrained optimization problems, it received more and more attention of the researchers. It was applied in various fields of the nonlinear optimization problems and recently was also extended to optimization problems with bound constraints. In this paper, we further extend the BB step sizes to more general variational inequality (VI) problems, i.e., we adopt them in projection methods. Under the condition that the underlying mapping of the VI problem is strongly monotone and Lipschitz continuous and the modulus of strong monotonicity and the Lipschitz constant satisfy some further conditions, we establish the global convergence of the projection methods with BB step sizes. A series of numerical examples are presented, which demonstrate that the proposed methods are convergent under mild conditions, and are more efficient than some classical projection-like methods.

In the present paper, delta functions (DFs) are proposed as a new set of basis functions. Their properties and relations to well-known triangular functions (TFs) are described. The simplicity and useful properties of newly proposed sets led us to use them with more accuracy and less computational burden.Furthermore, DFs are applied to propose an efficient method for approximating the solution of integral equations systems. Convergence analysis and the rate of convergence have been considered as well. Some numerical examples are provided to illustrate the computational efficiency and accuracy of the method.

A numerical method for simulating periodic travelling-wave solutions of some nonlinear dispersive wave equations is proposed. The construction of the scheme is based on an efficient computation of the elements that characterize these solutions: the initial profile and the velocity of the wave.

The main goal of this paper is to present recovery type a posteriori error estimators and superconvergence for the nonconforming finite element eigenvalue approximation of self-adjoint elliptic equations by projection methods. Based on the superconvergence results of nonconforming finite element for the eigenfunction we derive superconvergence and recovery type a posteriori error estimates of the eigenvalue. The results are based on some regularity assumption for the elliptic problem and are applicable to the lowest order nonconforming finite element approximations of self-adjoint elliptic eigenvalue problems with quasi-regular partitions. Therefore, the results of this paper can be employed to provide useful a posteriori error estimators in practical computing under unstructured meshes.

We deal with the time-dependent Navier–Stokes equations (NSE) with Dirichlet boundary conditions on the whole domain or, on a part of the domain and open boundary conditions on the other part. It is shown numerically that combining the penalty-projection method with spatial discretization by the Marker And Cell scheme (MAC) yields reasonably good results for solving the above-mentioned problem. The scheme which has been introduced combines the backward difference formula of second-order (BDF2, namely Gear’s scheme) for the temporal approximation, the second-order Richardson extrapolation for the nonlinear term, and the penalty-projection to split the velocity and pressure unknowns. Similarly to the results obtained for other projection methods, we estimate the errors for the velocity and pressure in adequate norms via the energy method.

We consider the problem of how to expand a given subspace for approximating an eigenvalue and eigenvector of a matrix A. Specifically, we consider which vector in the subspace, after multiplied by A, provides optimal expansion of the existing subspace for the eigenvalue problem. We determine the optimal vector, when the quality of subspace for approximation is measured by the angle between the subspace and the eigenvector. We have also derived some characterization of the angle that might lead to more practically useful choice of the expansion vector.

In this paper, we propose a new projection method for solving variational inequality problems, which can be viewed as an improvement of the method of Han and Lo [D.R. Han, Hong K. Lo, Two new self-adaptive projection methods for variational inequality problems, Computers & Mathematics with Applications 43 (2002) 1529–1537], by adopting a new step-size rule. The method is as simple as Han and Lo’s methods [D.R. Han, Hong K. Lo, Two new self-adaptive projection methods for variational inequality problems, Computers & Mathematics with Applications 43 (2002) 1529–1537] and other extra-gradient-type methods, which uses only function evolutions and projections onto the feasible set. We prove that under the condition that the underlying function is co-coercive, the sequence generated by the method converges to a solution of the variational inequality problem globally. Some preliminary computational results are reported, which illustrated that the new method is more efficient than Han and Lo’s method [D.R. Han, Hong K. Lo, Two new self-adaptive projection methods for variational inequality problems, Computers & Mathematics with Applications 43 (2002) 1529–1537].

International audience; This paper presents two algorithms for solving the general dynamic equation for aerosols. Coagulation and condensation are solved simultaneously without using splitting. The first method is based on a collocation method for size discretization and a Rosenbrock solver for time integration. In the second method, time discretization is first performed. This leads to a Volterra equation of first kind, which is solved with cubic splines. Both methods are compared to a reference solution for constant coagulation and linear condensation.

In this paper we characterise the weighting subspaces associated with two approximation techniques for solving ordinary differential equations: the Tau Method [E.L. Ortiz, The Tau Method, SIAM J. Numer. Anal. 6 (1969) 480–92] and the orthogonal collocation method. We show that approximations constructed by means of these two methods are always expressible in terms of a prescribed orthogonal polynomials basis, by projection on a suitably chosen finite dimensional weighting subspace. We show, in particular, that collocation is a special Tau Method with a twisted basis.

In this paper, we suggest and analyze a new inexact proximal point method for solving general variational inequalities, which can be considered as an implicit predictor–corrector method. An easily measurable error term is proposed with further relaxed error bound and an optimal step length is obtained by maximizing the profit-function and is dependent on the previous points. Our results include several known and new techniques for solving variational inequalities and related optimization problems. Results obtained in this paper can be viewed as an important improvement and refinement of the previously known results. Preliminary numerical experiments are included to illustrate the advantage and efficiency of the proposed method.

A subgrid stabilization technique is developed for solving the two-dimensional incompressible Navier–Stokes equations at high Reynolds numbers. The time marching algorithm is based on a well-established fractional-step pressure-correction projection method. The advection–diffusion step is enriched by an implicit subgrid stabilizing term and by an explicit dissipative shock capturing term. The former is calculated by means of a hierarchical finite element setting, the latter is included to avoid Gibbs’ phenomenon in the boundary layer. Convergence tests on prototypical two-dimensional examples are reported and the method is used to simulate the viscous incompressible flows around the airfoil NACA0012 at zero incidence and Reynolds numbers ranging from 105 to 106.

In this paper, we consider and analyze some new projection-proximal methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is a weaker condition than monotonicity. The proposed methods include several new and known methods as special cases. Our results can be considered as a novel and important extension of the previously known results. Since the general variational inequalities include the quasi-variational inequalities and implicit complementarity problems as special cases, results proved in this paper continue to hold for these problems.

We study the solution of consistent, semidefinite and symmetric linear systems by iterative techniques. Given a finite sequence of subspaces a block-iterative projection type algorithm is considered. For two specific choices of iteration parameters we show convergence. We apply our results to over and under determined linear equations. These methods are based on decomposing the system matrix into blocks of rows or blocks of columns. Thereby several algorithms, many used in image reconstruction, are presented in a unified way.

In this paper, we first show that the adjustment parameter in the step size choice strategy of the modified Goldstein-Levitin-Polyak projection method proposed by He et al. for asymmetric strongly monotone variational inequality problems can be bounded away from zero by a positive constant. Under this observation, we propose a new step size rule which seems to be more practical and robust than the original one. We show that the new modified method is globally convergent under the same conditions and report some computational results to illustrate the method.

The purpose of this paper is to investigate the problem of iterative computation of approximately balanced reduced order systems. The resulting approach is completely automatic once an error tolerance is specified and also yields an error bound. This is to be contrasted with existing projection methods, namely Pade via Lanczos (PVL) and rational Krylov, which do not satisfy these properties. Our approach is based on the computation and approximation of the cross gramian of the system. The cross gramian is the solution of a Sylvester equation and therefore some effort is dedicated to the study of this equation leading to some new insights. Our method produces a low rank approximation to this gramian in factored form and thus directly provides a reduced order model and a reduced basis for the original system. It is well suited to large scale problems because there are no matrix factorizations of the large (sparse) system matrix. Only matrix–vector products are required.

Consider the linear system Ax=b , where A∈ C N×N is a singular matrix. In the present work we propose a general framework within which Krylov subspace methods for Drazin-inverse solution of this system can be derived in a convenient way. The Krylov subspace methods known to us to date treat only the cases in which A is hermitian and its index ind (A) is unity necessarily. In the present work A is not required to be hermitian. It can have any type of spectrum and ind (A) is arbitrary. We show that, as is the case with nonsingular systems, the Krylov subspace methods developed here terminate in a finite number of steps that is at most N− ind (A) . For one of the methods derived here we also provide an analysis by which we are able to bound the errors, the relevant bounds decreasing with increasing dimension of the Krylov subspaces involved. The results of this paper are applicable to consistent systems as well as to inconsistent ones. An interesting feature of the approach to singular systems presented in this work is that it is formulated as a generalization of the standard Krylov subspace approach to nonsingular systems. Indeed, our approach here reduces to that relevant for nonsingular systems upon setting ind (A)=0 everywhere.

Cataloged from PDF version of article. In this research we use the projection method (reported by Judd) to find numerical solutions to the Euler equations of a stochastic dynamic growth model. The mode1 that we solve is Brock’s asset pricing model for a variety of parameterizations of the production functions. Using simulated data from the model, conjectures (which are not analytically tractable) can be verified. We show that the market portfolio is mean-variance efficient in this dynamic context. We also show a result that is not available from the static CAPM theory: the efficient frontier shifts up and down over the business cycle.

A direct method is described for the approximation of nonsimple turning points, corresponding to cusp points, of nonlinear operator equations depending on two parameters. The procedure is based on the application of a special projection method to the computation of simple turning points of a suitable augmented system. Numerical examples illustrate the features of the proposed algorithm.

In many fields of numerical analysis there appear transformations of the form T k v = Σ v + k i = v α k i,v T i . When v varies, a sequence of transformations is obtained. This approach covers, for example, the E− and Θ-algorithms, the recursion formulae for B-splines, Bernstein polynomials and orthogonal polynomials, Pade approximants, the divided difference scheme and projection methods. In this paper it will be proved that such transformations can be written as a ratio of determinants and can be recursively computed by a triangular recursion scheme. The reciprocal of these results also holds. Furthermore, we will show that T v k can be represented in terms of a complex contour integral. Throughout the paper we will study several examples in some detail, and it will turn out that the application of our general theory leads to interesting new results in the special cases. Among others, we will derive a new determinantal representation formula for B-splines, a recurrence relation for generalized Bernstein polynomials, a generalization of the E-algorithm and we will prove that the Θ-algorithm can be represented as a quotient of determinants.