![]() and Feng and Neilan study for some finite element methods and Oberman’s study for finite difference methods. And the numerical solution of the Monge–Ampère equation has been a subject of increasing interest recently. The Monge–Ampère equation is originated in geometric surface theory and has been widely applied in dynamic meteorology, elasticity, geometric optics, image processing, conformal geometry, optimal transport, and others. When restricting it to, we can rewrite the equation as with a nonlinear map. The Monge–Ampère equation with Dirichlet boundary is a fully nonlinear partial differential equation, which is given by where z has n independent variables in a bounded domain and is the Hessian of the function u. These observations are obtained by taking the direct approach of numerical experimentation. Compared with the first two methods, the hierarchical radial basis function method can not only solve the present problem on a single level with higher accuracy and lower computational cost but also produce highly sparse nonlinear discrete system. ![]() The third is the hierarchical radial basis function method, which is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. The second method is the stationary multilevel method, which was proposed by Floater and Iske (1996), and is used to solve the fully nonlinear partial differential equation in the paper for the first time. The first method is the cascadic meshfree method, which was proposed by Liu and He (2013). We discuss and study the performance of the three kinds of multiscale methods. This paper considers some multiscale radial basis function collocation methods for solving the two-dimensional Monge–Ampère equation with Dirichlet boundary.
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