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A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, and E. Vanden-Eijnden, Comm. Pure Appl. Math., 58(11):1544--1585, 2005]. The class of problems that we consider are SPDEs with quadratic nonlinearities that were studied in [D. Blomker, M. Hairer, and G.A. Pavliotis, Nonlinearity, 20(7):1721--1744, 2007.] For such SPDEs an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales. Our method, based on micro and macro solvers, allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem. Numerical experiments illustrate the behavior of the proposed method.

The physical phenomena described by nonlinear partial differential equations have become at present the central theme of investigations by many researchers. A good understanding of most physical processes requires accounting for nonlinear effects and, consequently, methods for studying nonlinear equations have to be developed. Among nonlinear equations the Dirichlet problem for the Monge-Ampere equation is the model case for fully nonlinear equations.

This effort studied a proposed numerical method for finding elliptic solutions of the Monge-Ampere equations, in the context of hydrodynamic and antenna design applications. Some computational experiments were performed on representative model problems. A theoretical estimate for the convergence rate of one part of the scheme was obtained. For the problem of reflector antenna design, the solvability of the corresponding linearized problem was demonstrated. A geometric interpretation of the solution was obtained which helps in understanding the problem.

The authors have studied a novel numerical solution approach to Monge Ampere type equations. The equations have important applications. In particular, the leading term in the balance equation in dynamic meteorology has the form (1) . In a more complicated form, an equation of this type appears in the von Karman system of equations for elasticity and also in inverse problems of geometric optics. It also turned out that recent progress in the study of fully nonlinear equations became possible after important properties of the equation (1) were discovered.

The physical phenomena described by nonlinear partial differential equations have become at present the central theme of investigation by many researchers. A good understanding of most physical processes requires accounting for nonlinear effects and, consequently, methods for studying nonlinear equations have to be developed. Among nonlinear equations the Dirichlet problem for the Monge-Ampere equation is the model case for fully nonlinear equations.