Output–input ratio analysis and DEA frontier

Data envelopment analysis (DEA) is a mathematical programming technique for identifying efficient frontiers for peer decision making units (DMUs). The ability of identifying frontier DMUs prior to the DEA calculation is of extreme importance to an effective and efficient DEA computation. In this paper, we present mathematical properties which characterize the inherent relationships between DEA frontier DMUs and output–input ratios. It is shown that top-ranked performance by ratio analysis is a DEA frontier point. This in turn allows identification of membership of frontier DMUs without solving a DEA program. Such finding is useful in streamlining the solution of DEA.     

Stability analysis of exponential Runge–Kutta methods for delay differential equations

A sufficient condition of stability of exponential Runge–Kutta methods for delay differential equations is obtained. Furthermore, a relationship between P-stability and GP-stability is established. It is proved that the numerical methods can preserve the analytical stability for a class of test problems.     

Convergence analysis of high-order time-splitting pseudo-spectral methods for rotational Gross–Pitaevskii equations

A convergence analysis of time-splitting pseudo-spectral methods adapted for time-dependent Gross–Pitaevskii equations with additional rotation term is given. For the time integration high-order exponential operator splitting methods are studied, and the space discretization relies on the generalized-Laguerre–Fourier spectral method with respect to the $(x,y)$ -variables as well as the Hermite spectral method in the $z$ -direction. Essential ingredients in the stability and error analysis are a general functional analytic framework of abstract nonlinear evolution equations, fractional power spaces defined by the principal linear part, a Sobolev-type inequality in a curved rectangle, and results on the asymptotical distribution of the nodes and weights associated with Gauß–Laguerre quadrature. The obtained global error estimate ensures that the nonstiff convergence order of the time integrator and the spectral accuracy of the spatial discretization are retained, provided that the problem data satisfy suitable regularity requirements. A numerical example confirms the theoretical convergence estimate.     

Formulating diets for growing pigs: economic and?environmental considerations

We look at the environmental impact of formulating diets for an animal production system, namely the case of growing pigs. The classic approach in animal production is to use a growth model based on the least-cost diet formulation. This optimal diet is generally established by linear programming. Such an approach can lead to adverse environmental effects in the form of nitrogen and phosphorus excretions. Multi-criteria (two and three criteria) models are proposed with the aim of addressing both economic and environmental considerations. We apply the models to two real-life contexts: Qu??bec (Canada) and France, and make some comparisons. We show that important reductions in nitrogen and phosphorus excretions can be achieved at relatively low costs in both contexts.     

Construction and analysis of some nonstandard finite difference methods for the FitzHugh–Nagumo equation

In this work, we construct four versions of nonstandard finite difference schemes in order to solve the FitzHugh–Nagumo equation with specified initial and boundary conditions under three different regimes giving rise to three cases. The properties of the methods such as positivity and boundedness are studied. The numerical experiment chosen is quite challenging due to shock-like profiles. The performance of the four methods is compared by computing L₁, L_∞ errors, rate of convergence with respect to time and central processing unit time at given time, T = 0.5. Error estimates have also been studied for the most efficient scheme.     

Runge–Kutta methods for jump–diffusion differential equations

In this paper we consider Runge–Kutta methods for jump–diffusion differential equations. We present a study of their mean-square convergence properties for problems with multiplicative noise. We are concerned with two classes of Runge–Kutta methods. First, we analyse schemes where the drift is approximated by a Runge–Kutta ansatz and the diffusion and jump part by a Maruyama term and second we discuss improved methods where mixed stochastic integrals are incorporated in the approximation of the next time step as well as the stage values of the Runge–Kutta ansatz for the drift. The second class of methods are specifically developed to improve the accuracy behaviour of problems with small noise. We present results showing when the implicit stochastic equations defining the stage values of the Runge–Kutta methods are uniquely solvable. Finally, simulation results illustrate the theoretical findings.     

Local linear convergence analysis of Primal–Dual splitting methods

In this paper, we study the local linear convergence properties of a versatile class of Primal–Dual splitting methods for minimizing composite non-smooth convex optimization problems. Under the assumption that the non-smooth components of the problem are partly smooth relative to smooth manifolds, we present a unified local convergence analysis framework for these methods. More precisely, in our framework, we first show that (i) the sequences generated by Primal–Dual splitting methods identify a pair of primal and dual smooth manifolds in a finite number of iterations, and then (ii) enter a local linear convergence regime, which is characterized based on the structure of the underlying active smooth manifolds. We also show how our results for Primal–Dual splitting can be specialized to cover existing ones on Forward–Backward splitting and Douglas–Rachford splitting/ADMM (alternating direction methods of multipliers). Moreover, based on these obtained local convergence analysis result, several practical acceleration techniques are discussed. To exemplify the usefulness of the obtained result, we consider several concrete numerical experiments arising from fields including signal/image processing, inverse problems and machine learning. The demonstration not only verifies the local linear convergence behaviour of Primal–Dual splitting methods, but also the insights on how to accelerate them in practice.     

Conservative difference methods for the Klein–Gordon–Zakharov equations

Firstly an implicit conservative finite difference scheme is presented for the initial-boundary problem of the one space dimensional Klein–Gordon–Zakharov (KGZ) equations. The existence of the difference solution is proved by Leray–Schauder fixed point theorem. It is proved by the discrete energy method that the scheme is uniquely solvable, unconditionally stable and second order convergent for U   in _l∞$l_{\infty }$ norm, and for N   in _l2$l_2$ norm on the basis of the priori estimates. Then an explicit difference scheme is proposed for the KGZ equations, on the basis of priori estimates and two important inequalities about norms, convergence of the difference solutions is proved. Because it is explicit and not coupled it can be computed by a parallel method. Numerical experiments with the two schemes are done for several test cases. Computational results demonstrate that the two schemes are accurate and efficient.     

Stability analysis of Runge-Kutta methods for nonlinear neutral delay integro-differential equations

The sufficient conditions for the stability and asymptotic stability of Runge-Kutta methods for nonlinear neutral delay integro-differential equations are derived. A numerical test that confirms the theoretical results is given in the end.     

Settings and methods for global sensitivity analysis – a short guide

Global sensitivity analysis offers a set of tools tailored to the impact assessment of certain assumptions on a models output. A recent book on the topic covers those issues [1]. Given the limited space for discussing thoroughly any of those methods, we will next summarize the main conclusions that derive from the application of various global sensitivity analysis methods on chemical models [2], econometric studies [3] financial models [4] and composite indicators [5, 6]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Decent flow methods for inverse Sturm–Liouville problem

In this paper, we propose three numerical methods for the inverse Sturm–Liouville operator in impedance form. We use a finite difference method to discretize the Sturm–Liouville operator and expand the impedance function with some basis functions. The correction technique is discussed. By solving an un-weighted least squares problem, we find an approximation to the impedance function. Numerical experiments are presented to show the accuracy and stability of the numerical methods.     

Variational methods for the fractional Sturm–Liouville problem

This article is devoted to the regular fractional Sturm–Liouville eigenvalue problem. By applying the methods of fractional variational analysis, we prove the existence of a countable set of orthogonal solutions and corresponding eigenvalues. Moreover, we formulate two results showing that the lowest eigenvalue is the minimum value for a certain variational functional.     

Lagrangian–Eulerian methods for uniqueness in hydrodynamic systems

We present a Lagrangian–Eulerian strategy for proving uniqueness and local existence of solutions of limited smoothness for a class of incompressible hydrodynamic models including Oldroyd-B type complex fluid models and zero magnetic resistivity magneto-hydrodynamics equations.     

Forward–backward quasi-Newton methods for nonsmooth optimization problems

The forward–backward splitting method (FBS) for minimizing a nonsmooth composite function can be interpreted as a (variable-metric) gradient method over a continuously differentiable function which we call forward–backward envelope (FBE). This allows to extend algorithms for smooth unconstrained optimization and apply them to nonsmooth (possibly constrained) problems. Since the FBE can be computed by simply evaluating forward–backward steps, the resulting methods rely on a similar black-box oracle as FBS. We propose an algorithmic scheme that enjoys the same global convergence properties of FBS when the problem is convex, or when the objective function possesses the Kurdyka–?ojasiewicz property at its critical points. Moreover, when using quasi-Newton directions the proposed method achieves superlinear convergence provided that usual second-order sufficiency conditions on the FBE hold at the limit point of the generated sequence. Such conditions translate into milder requirements on the original function involving generalized second-order differentiability. We show that BFGS fits our framework and that the limited-memory variant L-BFGS is well suited for large-scale problems, greatly outperforming FBS or its accelerated version in practice, as well as ADMM and other problem-specific solvers. The analysis of superlinear convergence is based on an extension of the Dennis and Moré theorem for the proposed algorithmic scheme.     

Legendre–Gauss collocation methods for ordinary differential equations

In this paper, we propose two efficient numerical integration processes for initial value problems of ordinary differential equations. The first algorithm is the Legendre–Gauss collocation method, which is easy to be implemented and possesses the spectral accuracy. The second algorithm is a mixture of the collocation method coupled with domain decomposition, which can be regarded as a specific implicit Legendre–Gauss Runge–Kutta method, with the global convergence and the spectral accuracy. Numerical results demonstrate the spectral accuracy of these approaches and coincide well with theoretical analysis.      

A-stable Runge–Kutta methods for semilinear evolution equations

We consider semilinear evolution equations for which the linear part generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. In this setting, we prove the existence of solutions which are temporally smooth in the norm of the lowest rung of the scale for an open set of initial data on the highest rung of the scale. Under the same assumptions, we prove that a class of implicit, A-stable Runge–Kutta semidiscretizations in time of such equations are smooth as maps from open subsets of the highest rung into the lowest rung of the scale. Under the additional assumption that the linear part of the evolution equation is normal or sectorial, we prove full order convergence of the semidiscretization in time for initial data on open sets. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schrödinger equation.     

Multiple objectives decomposition–coordination methods for hierarchical organizations

In this paper, the integration of goal programming models and hierarchical programming models is analyzed. The systems under study are assumed to consist of interconnected subsystems with multiple goals in each. Three possible cases regarding the number of decision makers will be considered: (1) one decision maker for the overall goals and one decision maker for each subsystem, (2) conflicting decision makers who are interested in their subsystems, and (3) just one decision maker for the overall system. Next, conditions are stated under which the problem of obtaining satisfying solutions for problems (1) and (3) can be reduced to the problem of obtaining satisfying solutions for the case (2). In order to determine such solutions, hierarchical techniques which exploit the structure of a decomposable system are analyzed. The empirical implementation of the two algorithms proposed shows their efficiency in terms of processing time.     

Tower-of-sets analysis for the Kise–Ibaraki–Mine algorithm

Kise et al. (Oper. Res. 26:121–126, 1978) give an O(n ²) time algorithm to find an optimal schedule for the single-machine number of late jobs problem with agreeable job release dates and due dates. Li et al. (Oper. Res. 58:508–509, 2010a) point out that their proof of optimality for their algorithm is incorrect by giving a counter-example. In this paper, using the concept of “tower-of-sets” from Lawler (Math. Comput. Model. 20:91–106, 1994), we construct the tower-of-sets of the early job set generated by the algorithm. Then we give a correct proof of optimality for the algorithm and show a new result that the early job set by the algorithm obtains not only the maximum number of jobs but also the smallest total processing time among all optimal schedules. The result can be applied to study the problems of the hard real-time systems.     

Stability analysis and classification of Runge–Kutta methods for index 1 stochastic differential-algebraic equations with scalar noise

The problem of solving stochastic differential-algebraic equations (SDAEs) of index 1 with a scalar driving Wiener process is considered. Recently, the authors have proposed a class of stiffly accurate stochastic Runge–Kutta (SRK) methods that do not involve any pseudo-inverses or projectors for the numerical solution of the problem. Based on this class of approximation methods, classifications for the coefficients of stiffly accurate SRK methods attaining strong order 0.5 as well as strong order 1.0 are calculated. Further, the mean-square stability of the considered class of SRK methods is analyzed. As the main result, families of A-stable efficient order 0.5 and 1.0 stiffly accurate SRK methods with a minimal number of stages for SDEs as well as for SDAEs are presented.