The perturbation index of linear PDAEs with a hyperbolic part

For numerous problems in science and engineering a refined modelling approach leads to initial-boundary value problems for partial differential-algebraic equations (PDAEs). A perturbation index of the variational formulation of the PDAE is introduced. As examples, the compressible Navier–Stokes equations and a model problem from MHD are considered. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational iteration method for solving non-linear partial differential equations

In this paper, we shall use the variational iteration method to solve some problems of non-linear partial differential equations (PDEs) such as the combined KdV–MKdV equation and Camassa–Holm equation. The variational iteration method is superior than the other non-linear methods, such as the perturbation methods where this method does not depend on small parameters, such that it can fined wide application in non-linear problems without linearization or small perturbation. In this method, the problems are initially approximated with possible unknowns, then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory.     

Compromise allocation in multivariate stratified sample surveys under two stage randomized response model

A general model for the randomized response (RR) method was introduced by Warner (J. Am. Stat. Assoc. 60:63–69, 1965) when a single-sensitive question is under study. However, since social surveys are often based on questionnaires containing more than one sensitive question, the analysis of multiple RR data is of considerable interest. In multivariate stratified surveys with multiple RR data the choice of optimum sample sizes from various strata may be viewed as a multiobjective nonlinear programming problem. The allocation thus obtained may be called a “compromise allocation” in sampling literature. This paper deals with the two-stage stratified Warner’s RR model applied to multiple sensitive questions. The problems of obtaining compromise allocations are formulated as multi-objective integer non linear programming problems with linear and quadratic cost functions as two separate problems. The solution to the formulated problems are achieved through goal programming technique. Numerical examples are presented to illustrate the computational details.     

New estimates of the stability of one-dimensional plane-parallel flows of a viscous incompressible fluid

The stability of a number of one-dimensional plane-parallel steady flows of a viscous incompressible fluid is investigated analytically using the method of integral relations. The mathematical formulation is reduced to eigenvalue problems for the Orr–Sommerfeld equation. One of three versions is chosen as the boundary conditions: all the components of the velocity perturbation are equal to zero on both boundaries of the layer (in this case we have the classical Orr–Sommerfeld problem), all the components of the velocity perturbation on one of the boundaries are equal to zero and the perturbations of the shear component of the stress vector and of the normal component of the velocity are equal to zero on the other, and all the components of the velocity perturbation are equal to zero on one boundary and the other boundary should be free. The boundary conditions derived in the latter case, are characterized by the occurrence of a spectral parameter in them. For kinematic conditions the lower estimates of the critical Reynolds number – the Joseph–Yih estimates, are improved. In the remaining cases the technique of the integral-relations method is developed, leading to new estimates of the stability. Analogs of Squire's theorem are derived for the boundary conditions of all the types mentioned above. Upper estimates of the increment of the increase in perturbations in eigenvalue problems for the Rayleigh equation with two types of boundary conditions are given.     

Method for solving nonlinear initial value problems by combining homotopy perturbation and reproducing kernel Hilbert space methods

The nonlinear singular initial value problems including generalized Lane–Emden-type equations are investigated by combining homotopy perturbation method (HPM) and reproducing kernel Hilbert space method (RKHSM). He’s HPM is based on the use of traditional perturbation method and homotopy technique and can reduce a nonlinear problem to some linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can overcome the difficulty at the singular point of non-homogeneous, linear singular initial value problems; especially when the singularity appears on the right-hand side of this type of equations, so it can solve powerfully linear singular initial value problems. Therefore, using advantages of these two methods, more general nonlinear singular initial value problems can be solved powerfully. Some numerical examples are presented to illustrate the strength of the method.     

A new method for homoclinic solutions of ordinary differential equations

Consideration is given to the homoclinic solutions of ordinary differential equations. We first review the Melnikov analysis to obtain Melnikov function, when the perturbation parameter is zero and when the differential equation has a hyperbolic equilibrium. Since Melnikov analysis fails, using Homotopy Analysis Method (HAM, see [Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press; 2003; Liao SJ. An explicit, totally analytic approximation of Blasius’ viscous flow problems. Int J Non-Linear Mech 1999;34(4):759–78; Liao SJ. On the homotopy analysis method for nonlinear problems. Appl Math Comput 2004;147(2):499–513] and others [Abbasbandy S. The application of the homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 2006;360:109–13; Hayat T, Sajid M. On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder. Phys Lett A, in press; Sajid M, Hayat T, Asghar S. Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dyn, in press]), we obtain homoclinic solution for a differential equation with zero perturbation parameter and with hyperbolic equilibrium. Then we show that the Melnikov type function can be obtained as a special case of this homotopy analysis method. Finally, homoclinic solutions are obtained (for nontrivial examples) analytically by HAM, and are presented through graphs.     

Boundary layer preconditioners for finite-element discretizations of singularly perturbed reaction-diffusion problems

We consider the iterative solution of linear systems of equations arising from the discretization of singularly perturbed reaction-diffusion differential equations by finite-element methods on boundary-fitted meshes. The equations feature a perturbation parameter, which may be arbitrarily small, and correspondingly, their solutions feature layers: regions where the solution changes rapidly. Therefore, numerical solutions are computed on specially designed, highly anisotropic layer-adapted meshes. Usually, the resulting linear systems are ill-conditioned, and so, careful design of suitable preconditioners is necessary in order to solve them in a way that is robust, with respect to the perturbation parameter, and efficient. We propose a boundary layer preconditioner, in the style of that introduced by MacLachlan and Madden for a finite-difference method (MacLachlan and Madden, SIAM J. Sci. Comput. 35(5), A2225–A2254 2013). We prove the optimality of this preconditioner and establish a suitable stopping criterion for one-dimensional problems. Numerical results are presented which demonstrate that the ideas extend to problems in two dimensions.     

Singular perturbations in one-dimensional unsteady motion of a real gas : PMM vol. 39, n 6, 1975, pp. 1051–1059

Navier-Stokes equations for one-dimensional motion of gas are reduced to a special dimensionless form convenient for investigations involving a perturbation front. In new variables the transition from limit conditions of motion of an inviscid non-heat-conducting gas to the case of small but finite coefficients of viscosity and thermal conductivity, which is simulated by a perfect gas with singular perturbations induced by the indicated dissipative factors. We establish the inevitability of existence of two regions of singular perturbations, the neighborhood of the perturbation front and that of the point (line, surface) where the investigated motion is generated. The derivation of equations for both boundary layers, which is valid for a fairly general statement of problems of this kind, is presented and conditions of merging with the external (adiabatic) flow are formulated. Examples of computation of motion in boundary layers in problems of piston and point explosion are presented.     

Interior-point methods for nonconvex nonlinear programming: cubic regularization

In this paper, we present a primal-dual interior-point method for solving nonlinear programming problems. It employs a Levenberg-Marquardt (LM) perturbation to the Karush-Kuhn-Tucker (KKT) matrix to handle indefinite Hessians and a line search to obtain sufficient descent at each iteration. We show that the LM perturbation is equivalent to replacing the Newton step by a cubic regularization step with an appropriately chosen regularization parameter. This equivalence allows us to use the favorable theoretical results of Griewank (The modification of Newton’s method for unconstrained optimization by bounding cubic terms, 1981), Nesterov and Polyak (Math. Program., Ser. A 108:177–205, 2006), Cartis et al. (Math. Program., Ser. A 127:245–295, 2011; Math. Program., Ser. A 130:295–319, 2011), but its application at every iteration of the algorithm, as proposed by these papers, is computationally expensive. We propose a hybrid method: use a Newton direction with a line search on iterations with positive definite Hessians and a cubic step, found using a sufficiently large LM perturbation to guarantee a steplength of 1, otherwise. Numerical results are provided on a large library of problems to illustrate the robustness and efficiency of the proposed approach on both unconstrained and constrained problems.     

Localization method for lines of discontinuity of an approximately defined function of two variables

A function of two variables with lines of discontinuity of the first kind is considered. It is assumed that outside the discontinuity lines the function is smooth and has a bounded partial derivative. An approximation to the function in L ₂ and a perturbation level are known. The problem in question belongs to a class of nonlinear ill-posed problems, which are solved by constructing some regularizing algorithms. We propose a simple theoretical approach to solving the problem of localizing the discontinuity lines of a function that is noisy in the space L ₂. Some conditions on the exact function are imposed ??in the small.?? Methods of averaging are constructed, and error estimates of localizing the lines (in the small) are obtained.     

A counter-example to the characterization of the discontinuous value function of control problems with reflectionUn contre-exemple à la caractérisation de la fonction-valeur discontinue d'un problème de contrôle réfléchi

We consider a finite horizon deterministic optimal control problem with reflection. The final cost is assumed to be merely a locally bounded function which leads to a discontinuous value function. We address the question of the characterization of the value function as the unique solution of an Hamilton–Jacobi equation with Neumann boundary conditions. We follow the discontinuous approach developed by Barles and Perthame for problems set in the whole space. We prove that the minimal and maximal discontinuous viscosity solutions of the associated Hamilton–Jacobi can be written in terms of value functions of control problems with reflection. Nethertheless, we construct a counter-example showing that the value function is not the unique solution of the equation. To cite this article: O. Ley, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 469–473.     

Ordinary differential equation system for population of individuals and the corresponding probabilistic model

The key model for particle populations in statistical mechanics is the Bogolyubov–Born–Green–Kirkwood–Yvon (BBGKY) equation chain. It is derived mainly from the Hamilton ordinary differential equation (ODE) system for the particle states in the position-momentum phase space. Many problems beyond physics or chemistry, for instance, in the living-matter sciences (biology, medicine, ecology, and sociology) make it necessary to extend the notion of a particle to an individual, or active particle. This challenge is met by the generalized kinetic theory. The corresponding dynamics of the state vector can also be regarded to be described by an ODE system. The latter, however, need not be the Hamilton one. The question is how one can derive the analogue of the BBGKY paradigm for the new settings. The present work proposes an answer to this question. It applies a very limited number of carefully selected tools of probability theory and common statistical mechanics. It also uses the well-known feature that the maximum number of the individuals which can mutually interact directly is bounded by a fixed value of a few units. The proposed approach results in the finite system of equations for the reduced many-individual distribution functions thereby eliminating the so-called closure problem inevitable in the BBGKY theory. The thermodynamic-limit assumption is not needed either. The system includes consistently derived terms of all of the basic types known in kinetic theory, in particular, both the “mean-field” and scattering-integral terms, and admits the kinetic equation of the form allowing a direct chemical-reaction reading. The approach can deal with Hamilton’s model which is nonmonogenic. The results may serve as the basis of the generalized kinetic theory and contribute to stochastic mechanics of populations of individuals.     

On sensitivity of Gauss–Christoffel quadrature

In numerical computations the question how much does a function change under perturbations of its arguments is of central importance. In this work, we investigate sensitivity of Gauss–Christoffel quadrature with respect to small perturbations of the distribution function. In numerical quadrature, a definite integral is approximated by a finite sum of functional values evaluated at given quadrature nodes and multiplied by given weights. Consider a sufficiently smooth integrated function uncorrelated with the perturbation of the distribution function. Then it seems natural that given the same number of function evaluations, the difference between the quadrature approximations is of the same order as the difference between the (original and perturbed) approximated integrals. That is perhaps one of the reasons why, to our knowledge, the sensitivity question has not been formulated and addressed in the literature, though several other sensitivity problems, motivated, in particular, by computation of the quadrature nodes and weights from moments, have been thoroughly studied by many authors. We survey existing particular results and show that even a small perturbation of a distribution function can cause large differences in Gauss–Christoffel quadrature estimates. We then discuss conditions under which the Gauss–Christoffel quadrature is insensitive under perturbation of the distribution function, present illustrative examples, and relate our observations to known conjectures on some sensitivity problems. The work of the first author was supported by the National Science Foundation under Grants CCR-0204084 and CCF-0514213. The work of the other two authors was supported by the Program Information Society under project 1ET400300415 and by the Institutional Research Plan AV0Z100300504. P. Tichy in the years 2003–2006 on leave at the Institute of Mathematics, TU Berlin, Germany.     

He’s homotopy perturbation method for systems of integro-differential equations

In this article, the homotopy perturbation method [He JH. Homotopy perturbation technique. Comput Meth Appl Mech Eng 1999;178:257–62; He JH. A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int J Non-Linear Mech 2000;35(1):37–43; He JH. Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 2004;156:527–39; He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003;135:73–79; He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004;151:287–92; He JH. Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons & Fractals 2005;26:695–700] is applied to solve linear and nonlinear systems of integro-differential equations. Some nonlinear examples are presented to illustrate the ability of the method for such system. Examples for linear system are so easy that has been ignored.     

The perturbation on initial binding energy for a Majda-CJ combustion model

We investigate the perturbed Riemann problem for a scalar Chapman–Jouguet combustion model – the perturbation on initial binding energy. Under the entropy conditions, we obtain the unique solutions in a neighbourhood of the origin (t?>?0) on the (x,?t) plane. It is found that, for some cases, the perturbed Riemann solutions are essentially different from the corresponding Riemann solutions. That is, the perturbation may transform a Chapman–Jouguet detonation into a strong detonation or a weak deflagration following a shock wave; a strong detonation into a weak deflagration following a shock wave; a Chapman–Jouguet deflagration into a weak deflagration.     

Forward–Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty

We study optimal stochastic control problems with jumps under model uncertainty. We rewrite such problems as stochastic differential games of forward–backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the nonzero sum games (finding conditions for Nash equilibria). We then apply these results to study robust optimal portfolio-consumption problems with penalty. We establish a connection between market viability under model uncertainty and equivalent martingale measures. In the case with entropic penalty, we prove a general reduction theorem, stating that a optimal portfolio-consumption problem under model uncertainty can be reduced to a classical portfolio-consumption problem under model certainty, with a change in the utility function, and we relate this to risk sensitive control. In particular, this result shows that model uncertainty increases the Arrow–Pratt risk aversion index.     

Perturbation method for determining the group of invariance of hierarchical models

We propose a perturbation method for determining the (largest) group of invariance of a toric ideal defined in [S. Aoki, A. Takemura, The largest group of invariance for Markov bases and toric ideals, J. Symbolic Comput. 43 (5) (2008) 342–358]. In the perturbation method, we investigate how a generic element in the row space of the configuration defining a toric ideal is mapped by a permutation of the indeterminates. Compared to the proof by Aoki and Takemura which was based on stabilizers of a subset of indeterminates, the perturbation method gives a much simpler proof of the group of invariance. In particular, we determine the group of invariance for a general hierarchical model of contingency tables in statistics, under the assumption that the numbers of the levels of the factors are generic. We prove that it is a wreath product indexed by a poset related to the intersection poset of the maximal interaction effects of the model.     

Analytic perturbation of generalized inverses

We investigate the analytic perturbation of generalized inverses. Firstly we analyze the analytic perturbation of the Drazin generalized inverse (also known as reduced resolvent in operator theory). Our approach is based on spectral theory of linear operators as well as on a new notion of group reduced resolvent. It allows us to treat regular and singular perturbations in a unified framework. We provide an algorithm for computing the coefficients of the Laurent series of the perturbed Drazin generalized inverse. In particular, the regular part coefficients can be efficiently calculated by recursive formulae. Finally we apply the obtained results to the perturbation analysis of the Moore–Penrose generalized inverse in the real domain.     

Analytical modelling of fractional advection–dispersion equation defined in a bounded space domain

The fractional advection–dispersion equation (FADE) known as its non-local dispersion, is used in groundwater hydrology and has been proven to be a reliable tool to model the transport of passive tracers carried by fluid flow in a porous media. In this paper, compact structures of FADE are investigated by means of the homotopy perturbation method with consideration of a promising scheme to calculate nonlinear terms. The problems are formulated in the Jumarie sense. Analytical and numerical results are presented.     

Warmstarting the homogeneous and self-dual interior point method for linear and conic quadratic problems

We present two strategies for warmstarting primal-dual interior point methods for the homogeneous self-dual model when applied to mixed linear and quadratic conic optimization problems. Common to both strategies is their use of only the final (optimal) iterate of the initial problem and their negligible computational cost. This is a major advantage when compared to previously suggested strategies that require a pool of iterates from the solution process of the initial problem. Consequently our strategies are better suited for users who use optimization algorithms as black-box routines which usually only output the final solution. Our two strategies differ in that one assumes knowledge only of the final primal solution while the other assumes the availability of both primal and dual solutions. We analyze the strategies and deduce conditions under which they result in improved theoretical worst-case complexity. We present extensive computational results showing work reductions when warmstarting compared to coldstarting in the range 30–75% depending on the problem class and magnitude of the problem perturbation. The computational experiments thus substantiate that the warmstarting strategies are useful in practice.