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Level Set Methods in Segmentation of SDOCT Retinal Images
Published in Ayman El-Baz, Jasjit S. Suri, Level Set Method in Medical Imaging Segmentation, 2019
N Padmasini, R Umamaheswari, Yacin Sikkandar Mohamed, Manavi D Sindal
However, φ is not the only function that satisfies equation (5.11) in the distribution sense. In order to define a unique solution for the equation, the concept of viscosity solution need to introduced. The existence and uniqueness of viscosity solutions for linear and nonlinear partial differential equations constitutes an active field of research with rich literature [30]. One way to introduce the viscosity function is to add an extra time variables t. Let 0˜ be any function such that Γ is the zero level set curve of 0˜ and 0˜ is positive inside Γ and negative outside Γ. Then the distance function φ is the steady state of the following time dependent equation ∂d∂t+sign(d)(|∇d|−1)=0
LP-related representations of Cesàro and Abel limits of optimal value functions
Published in Optimization, 2022
Vladimir Gaitsgory, Ilya Shvartsman
Recall that a continuous function ψ is a viscosity solution of the nonlinear partial differential equation if where It is known ([5], p.5) that for any , function is a viscosity solution of the equation Due to stability of viscosity solutions ([5], Proposition 2.2), we can pass to the limit as in (40) under the assumption of uniform convergence of , which leads to h being a viscosity solution of Since V = h due to Proposition 3.6, the same is true for V. Finally, due to Theorem 2.3 in [10] with , the latter implies that which means that (38) is true with and .
Stochastic recursive optimal control problem of reflected stochastic differential systems
Published in International Journal of Control, 2020
Independently, Duffie and Epstein (1992) introduced BSDEs from economic background and they presented stochastic differential formulation of recursive utility. Recursive utility is an extension of the standard additive utility where the utility at time t is not only a function of the instantaneous consumption rate, but also of the future utility. In El Karoui, Peng, et al. (1997), they found that this kind of recursive utility can be described by BSDEs. The optimal control problem in which the cost functional is defined by the solution of BSDEs is called stochastic recursive optimal control problem. Peng (1992) obtained the dynamic programming principle of recursive optimal control problem and showed that the value function is a viscosity solution of the well-known Hamilton-Jacobi-Bellman equation. Wu and Yu (2008) studied stochastic recursive optimal control with the cost functional described by the solution to a reflected BSDEs and proved that the value function is the unique viscosity solution to the obstacle problem of the corresponding HJB equation. Recently, Li and Tang (2015) extended the stochastic recursive optimal control to the case where the control system is reflected SDEs and the recursive cost functional is defined by the solution of generalised BSDEs introduced in Pardoux and Zhang (1998). They also showed that the value function was the unique viscosity solution to the associated HJB equation with nonlinear Neumann boundary condition.
The maximum principle for partially observed optimal control problems of mean-field FBSDEs
Published in International Journal of Control, 2019
The study for the stochastic maximum principle (SMP) was initiated in the early 1970s by Kushner Kushner (1973) and later on by Haussmann Haussmann (1976), and Haussmann (1986). From then on, a great deal of research has been devoted to different versions of SMP, such as Øksendal and Sulem (2014), Wu (2013), Yong (2010) and so on. Especially in the last few years, since the works of Buckdahn, Djehiche, Li, and Peng (2009), the mean-field backward stochastic differential equations (BSDEs) derived as a limit of some highly dimensional system of forward and backward stochastic differential equations (FBSDEs), have been greatly developed. By using the dynamic programming principle, Buckdahn et al. (2009) proved that this mean-field BSDE gave the viscosity solution of a nonlocal partial differential equation. Then in the subsequent period of time, many authors investigated such kind of McKean–Vlasov type equations, and obtained various SMPs adapted to different frameworks, see Buckdahn, Djehiche, and Li (2011), Carmona and Delarue (2013), Djehiche, Tembine, and Tempone (2015), Shen and Siu (2013) and the references therein.