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Recently, Hao and Li [Fully coupled forward-backward SDEs involving the value function. Nonlocal Hamilton–Jacobi–Bellman equations, ESAIM: Control Optim, Calc. Var.22 (2016) 519–538] studied a new kind of forward-backward stochastic differential equations (FBSDEs), namely the fully coupled FBSDEs involving the value function in the case where the diffusion coefficient σ in forward stochastic differential equations depends on control, but does not depend on z. In our paper, we generalize their work to the case where σ depends on both control and z , which is called the general fully coupled FBSDEs involving the value function. The existence and uniqueness theorem of this kind of equations under suitable assumptions is proved. After obtaining the dynamic programming principle for the value function W , we prove that the value function W is the minimum viscosity solution of the related nonlocal Hamilton–Jacobi–Bellman equation combined with an algebraic equation. [ABSTRACT FROM AUTHOR]

We further develop our recent direct method for the optimized effective potential (OEP) in density functional theory (DFT) [Yang and Wu, Phys. Rev. Lett.89, 143002 (2002)]. First, we show that the stationary condition in our optimization approach leads to a proper nonlinear algebraic equation for the OEP in a finite basis set, which differs from other finite basis set approaches. Then by constructing an approximate second derivative matrix of the energy functional in conjunction with the use of the Newton method, we significantly accelerate the convergence of the iterative optimization for OEP. Enhancement of the method is made in using the Tikhonov regularization method for the inversion of the second derivative matrix when it is singular or nearly singular and the direct inversion in the iterative space. It is shown that under a fixed stepsize condition, the optimization approach is equivalent to the self-consistent solution to the nonlinear algebraic equation for OEP. Because the approximate second derivatives are easy to compute and the iteration numbers are small now, the computation costs of OEP become comparable to that of regular DFT calculations as shown by calculations of some molecules, small and larger ones. We show how to find balanced results between energies and potentials when choosing a basis set for potentials. [ABSTRACT FROM AUTHOR]