This paper is concerned with the boundary behavior of strictly convex large solutions to the Monge–Ampère equation det D^2u(x) = b(x)f(u(x)), u >0, x∈Ω, where Ω is a strictly convex and bounded smooth doma...This paper is concerned with the boundary behavior of strictly convex large solutions to the Monge–Ampère equation det D^2u(x) = b(x)f(u(x)), u >0, x∈Ω, where Ω is a strictly convex and bounded smooth domain in R^N with N ≥ 2, f is normalized regularly varying at infinity with the critical index N and has a lower term, and b∈C~∞(Ω) is positive in Ω, but may be appropriate singular on the boundary.展开更多
We consider the problem of minimizing the average of a large number of smooth component functions over one smooth inequality constraint.We propose and analyze a stochastic Moving Balls Approximation(SMBA)method.Like s...We consider the problem of minimizing the average of a large number of smooth component functions over one smooth inequality constraint.We propose and analyze a stochastic Moving Balls Approximation(SMBA)method.Like stochastic gradient(SG)met hods,the SMBA method's iteration cost is independent of the number of component functions and by exploiting the smoothness of the constraint function,our method can be easily implemented.Theoretical and computational properties of SMBA are studied,and convergence results are established.Numerical experiments indicate that our algorithm dramatically outperforms the existing Moving Balls Approximation algorithm(MBA)for the structure of our problem.展开更多
基金supported by NSF of P.R.China(Grant No.11571295)
文摘This paper is concerned with the boundary behavior of strictly convex large solutions to the Monge–Ampère equation det D^2u(x) = b(x)f(u(x)), u >0, x∈Ω, where Ω is a strictly convex and bounded smooth domain in R^N with N ≥ 2, f is normalized regularly varying at infinity with the critical index N and has a lower term, and b∈C~∞(Ω) is positive in Ω, but may be appropriate singular on the boundary.
文摘We consider the problem of minimizing the average of a large number of smooth component functions over one smooth inequality constraint.We propose and analyze a stochastic Moving Balls Approximation(SMBA)method.Like stochastic gradient(SG)met hods,the SMBA method's iteration cost is independent of the number of component functions and by exploiting the smoothness of the constraint function,our method can be easily implemented.Theoretical and computational properties of SMBA are studied,and convergence results are established.Numerical experiments indicate that our algorithm dramatically outperforms the existing Moving Balls Approximation algorithm(MBA)for the structure of our problem.
