An important test of the quality of numerical methods developed to track the interface between two fluids is their ability to reproduce test cases or benchmarks. However, benchmark solutions are scarce and virtually nonexistent for complex geometries. We propose a simple method to generate benchmark solutions in the context of the two-layer flow problem, a classical multiphase flow problem. The solutions are obtained by considering the inverse problem of finding the required channel geometry to obtain a prescribed interface profile. This viewpoint shift transforms the problem from that of having to solve a complex differential equation to the much easier one of finding the roots of a quartic polynomial.

We propose a new primal-dual interior-point algorithm based on a new kernel function for linear optimization problems. New search directions and proximity functions are proposed based on the kernel function. We show that the new algorithm has and iteration bounds for large-update and small-update methods, respectively, which are currently the best known bounds for such methods.

This article considers the linear 1-d Schrödinger equation in (0,π)perturbed by a vanishing viscosity term depending on a small parameterε > 0. We study the boundary controllability properties of thisperturbed equation and the behavior of its boundary controlsvε as ε goes to zero. Itis shown that, for any time T sufficiently large but independent ofε and for each initial datum inH−1(0,π), there exists a uniformly boundedfamily of controls(vε)ε inL2(0, T) acting on the extremityx = π. Any weak limit of this family is a control forthe Schrödinger equation.

We consider an equilibrium problem with equilibrium constraints (EPEC) arising from themodeling of competition in an electricity spot market (under ISO regulation). For acharacterization of equilibrium solutions, so-called M-stationarity conditions arederived. This first requires a structural analysis of the problem, e.g.,verifying constraint qualifications. Second, the calmness property of a certainmultifunction has to be verified in order to justify using M-stationarity conditions.Third, for stating the stationarity conditions, the coderivative of a normal cone mappinghas to be calculated. Finally, the obtained necessary conditions are made fully explicitin terms of the problem data for one typical constellation. A simple two-settlementexample serves as an illustration.

In this paper, we consider the back and forth nudging algorithm that has been introducedfor data assimilation purposes. It consists of iteratively and alternately solving forwardand backward in time the model equation, with a feedback term to the observations. Weconsider the case of 1-dimensional transport equations, either viscous or inviscid, linearor not (Burgers’ equation). Our aim is to prove some theoretical results on theconvergence, and convergence properties, of this algorithm. We show that for non viscousequations (both linear transport and Burgers), the convergence of the algorithm holdsunder observability conditions. Convergence can also be proven for viscous lineartransport equations under some strong hypothesis, but not for viscous Burgers’ equation.Moreover, the convergence rate is always exponential in time. We also notice that theforward and backward system of equations is well posed when no nudging term is considered.

We present an application of optimal control theory to a simple SIR disease model of avian influenza transmission dynamics in birds. Basic properties of the model, including the epidemic threshold, are obtained. Optimal control theory is adopted to minimize the density of infected birds subject to an appropriate system of ordinary differential equations. We conclude that an optimally controlled seasonal vaccination strategy saves more birds than when there is a low uniform vaccination rate as in resource-limited places.

We consider the optimal proportional reinsurance from an insurer’s point of view to maximize the expected utility and minimize the value at risk. Under the general premium principle, we prove the existence and uniqueness of the optimal strategies and Pareto optimal solution, and give the relationship between the optimal strategies. Furthermore, we study the optimization problem with the variance premium principle. When the total claim sizes are normally distributed, explicit expressions for the optimal strategies and Pareto optimal solution are obtained. Finally, some numerical examples are presented to show the impact of the major model parameters on the optimal results.