Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations

Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained

Equilibrium or Simple Rule at Wimbledon? An Empirical Study

We follow Walker and Wooders’(2001) empirical analysis to collect and study a broader data set in tennis, including male, female and junior matches. We find that there is mixed evidence in support of the minimax hypothesis. Granted, the plays in our data pass all the tests in Walker and Wooders (2001). However, we argue that not only the test on equal winning probabilities may lack power, but also the current serve choices may depend on past serve choices, the performance of past serve choices, or the time that the game has elapsed. We therefore examine the role that simple rules may play in determining the plays. For a significant number of top tennis players, some simple low-information rules outperform the minimax hypothesis. By comparing junior players with adult players, we find that the former tend to adopt simpler rules. The result of comparison between female and male players is inconclusiveminimax, learning, low-information