International audienceIn this paper we compute the worst-case and average execution time of the Best Response Algorithm (BRA) to compute a pure Nash equilibrium in finite potential games. Our approach is based on a Markov chain model of BRA and a coupling technique that transform the average execution time of this discrete algorithm into the solution of an ordinary differential equation. In a potential game with N players and A strategies per player, we show that the worst case complexity of BRA (number of moves) is exactly N A N −1 , while its average complexity over random potential games is equal to e γ N + O(N), where γ is the Euler constant. We also show that the effective number of states visited by BRA is equal to log N + c + O(1/N) (with c e γ), on average. Finally , we show that BRA computes a pure Nash Equilibrium faster (in the strong stochastic order sense) than any local search algorithm over random potential games

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Purpose -The purpose of this paper is to study the optimization problem of low-frequency magnetic shielding using the adjoint variable method (AVM). This method is compared with conventional methods to calculate the gradient. Design/methodology/approach -The equation for the vector potential (eddy currents model) in appropriate Sobolev spaces is studied to obtain well-posedness. The optimization problem is formulated in terms of a cost functional which depends on the vector potential and its rotation. Convergence of a steepest descent algorithm to a stationary point of this functional is proved. Finally, some numerical results for an axisymmetric induction heater are presented. Findings -Using Friedrichs' inequality, the existence and uniqueness of the vector potential, its gradient and the corresponding adjoint variable can be proved. From the numerical results, it is concluded that the AVM is advantageous if the number of parameters to optimize is larger than two. Research limitations/implications -The AVM is only faster than conventional methods if the gradients can be calculated with sufficient accuracy. Originality/value -Theoretical results for eddy currents model are often based on a non-vanishing conductivity. The theoretical value of this paper is the presence of non-conducting materials in the domain. From a practical viewpoint, it has been demonstrated that the AVM can yield a significant reduction of computational time for advanced optimization problems.

In this paper, we study a numerical scheme to solve coupled Maxwell's equations with a nonlinear conductivity. This model plays an important role in the study of type‐II superconductors. The approximation scheme is based on backward Euler discretization in time and mixed conforming finite elements in space. We will prove convergence of this scheme to the unique weak solution of the problem and develop the corresponding error estimates. As a next step, we study the stability of the scheme in the quasi‐static limit ϵ → 0 and present the corresponding convergence rate. Finally, we support the theory by several numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.

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