The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the sp...The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the special non-uniform grids. The uniform con vergence of this scheme is proved and some numerical examples are given.展开更多
It is a well-established fact in the scientific literature that Picard iterations of backward stochastic differential equations with globally Lipschitz continuous nonlinearities converge at least exponentially fast to...It is a well-established fact in the scientific literature that Picard iterations of backward stochastic differential equations with globally Lipschitz continuous nonlinearities converge at least exponentially fast to the solution.In this paper we prove that this convergence is in fact at least square-root factorially fast.We show for one example that no higher convergence speed is possible in general.Moreover,if the nonlinearity is zindependent,then the convergence is even factorially fast.Thus we reveal a phase transition in the speed of convergence of Picard iterations of backward stochastic differential equations.展开更多
文摘The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the special non-uniform grids. The uniform con vergence of this scheme is proved and some numerical examples are given.
文摘It is a well-established fact in the scientific literature that Picard iterations of backward stochastic differential equations with globally Lipschitz continuous nonlinearities converge at least exponentially fast to the solution.In this paper we prove that this convergence is in fact at least square-root factorially fast.We show for one example that no higher convergence speed is possible in general.Moreover,if the nonlinearity is zindependent,then the convergence is even factorially fast.Thus we reveal a phase transition in the speed of convergence of Picard iterations of backward stochastic differential equations.
