The Experts below are selected from a list of 67797 Experts worldwide ranked by ideXlab platform
Emmanuel Prados - One of the best experts on this subject based on the ideXlab platform.
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Viscosity Solution
2014Co-Authors: Fabio Camilli, Emmanuel PradosAbstract:Viscosity Solution is a notion of weak Solution for a class of partial differential equations of Hamilton-Jacobi type. The range of applications of the notions of Viscosity Solution and Hamilton-Jacobi equations is enormous, including common class of partial differential equations such as evolutive problems and problems with boundary conditions, equations arising in optimal control theory, differential games, second-order equations arising in stochastic optimal control and stochastic differential games, geometric equations, etc. In computer vision, it also has various applications. In particular, the distance functions which are the Viscosity Solutions of the Eikonal equations are widely used. Also, all these notions have played an important role in shape representation, morphology, tractography and in general, in image processing. Finally Viscosity Solutions provide powerful tools to analyse the Shape from shading problem.
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A Viscosity Solution METHOD FOR SHAPE-FROM-SHADING WITHOUT IMAGE BOUNDARY DATA
ESAIM: Mathematical Modelling and Numerical Analysis, 2006Co-Authors: Emmanuel Prados, Fabio Camilli, Olivier FaugerasAbstract:In this paper we propose a Solution of the Lambertian shape-from-shading (SFS) problem by designing a new mathematical framework based on the notion of Viscosity Solution. The power of our approach is twofolds: (1) it defines a notion of weak Solutions (in the Viscosity sense) which does not necessarily require boundary data. Moreover, it allows to characterize the Viscosity Solutions by their “minimums”; and (2) it unifies the works of [Rouy and Tourin, SIAM J. Numer. Anal. 29 (1992) 867–884], [Lions et al ., Numer. Math. 64 (1993) 323–353], [Falcone and Sagona, Lect. Notes Math. 1310 (1997) 596–603], [Prados et al. , Proc. 7th Eur. Conf. Computer Vision 2351 (2002) 790–804; Prados and Faugeras, IEEE Comput. Soc. Press 2 (2003) 826–831], based on the notion of Viscosity Solutions and the work of [Dupuis and Oliensis, Ann. Appl. Probab. 4 (1994) 287–346] dealing with classical Solutions.
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A Viscosity Solution method for Shape-From-Shading without image boundary data
ESAIM: Mathematical Modelling and Numerical Analysis, 2006Co-Authors: Emmanuel Prados, Fabio Camilli, Olivier FaugerasAbstract:In this paper we propose a Solution of the Lambertian Shape-From-Shading (SFS) problem by designing a new mathematical framework based on the notion of Viscosity Solution. The power of our approach is twofolds: 1) it defines a notion of weak Solutions (in the Viscosity sense) which does not necessarily require boundary data. Moreover, it allows to characterize the Viscosity Solutions by their minimums"; 2) it unifies the works of Rouy and Tourin, Falcone et al., Prados and Faugeras, based on the notion of Viscosity Solutions and the work of Dupuis and Oliensis dealing with classical Solutions.
Imran H. Biswas - One of the best experts on this subject based on the ideXlab platform.
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On Zero-Sum Stochastic Differential Games with Jump-Diffusion Driven State: A Viscosity Solution Framework
SIAM Journal on Control and Optimization, 2012Co-Authors: Imran H. BiswasAbstract:A zero-sum differential game with controlled jump-diffusion driven state is considered and studied by a combination of dynamic programming and Viscosity Solution techniques. We prove, under certain conditions, that the value of the game exists. Moreover, the value function is shown to be the unique Viscosity Solution of a fully nonlinear integro-partial differential equation. In addition, we formulate and prove a verification theorem for such games within the Viscosity Solution framework for nonlocal equations.
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On zero-sum Stochastic Differential Games with Jump-Diffusion driven state: A Viscosity Solution framework
arXiv: Optimization and Control, 2010Co-Authors: Imran H. BiswasAbstract:A zero-sum differential game with controlled jump-diffusion driven state is considered, and studied using a combination of dynamic programming and Viscosity Solution techniques. We prove, under certain conditions, that the value of the game exists and is the unique Viscosity Solution of a fully nonlinear integro-partial differential equation. In addition, we formulate and prove a verification theorem for such games within the Viscosity Solution framework for nonlocal equations
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Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes
Applied Mathematics and Optimization, 2010Co-Authors: Imran H. Biswas, Espen R. Jakobsen, Kenneth H KarlsenAbstract:We develop a Viscosity Solution theory for a system of nonlinear degenerate parabolic integro-partial differential equations (IPDEs) related to stochastic optimal switching and control problems or stochastic games. In the case of stochastic optimal switching and control, we prove via dynamic programming methods that the value function is a Viscosity Solution of the IPDEs. In our setting the value functions or the Solutions of the IPDEs are not smooth, so classical verification theorems do not apply.
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Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes
Applied Mathematics and Optimization, 2009Co-Authors: Imran H. Biswas, Espen R. Jakobsen, Kenneth H KarlsenAbstract:We develop a Viscosity Solution theory for a system of nonlinear degenerate parabolic integro-partial differential equations (IPDEs) related to stochastic optimal switching and control problems or stochastic games. In the case of stochastic optimal switching and control, we prove via dynamic programming methods that the value function is a Viscosity Solution of the IPDEs. In our setting the value functions or the Solutions of the IPDEs are not smooth, so classical verification theorems do not apply.
Hitoshi Ishii - One of the best experts on this subject based on the ideXlab platform.
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vanishing contact structure problem and convergence of the Viscosity Solutions
arXiv: Analysis of PDEs, 2018Co-Authors: Qinbo Chen, Wei Cheng, Hitoshi Ishii, Kai ZhaoAbstract:This paper is devoted to study the vanishing contact structure problem which is a generalization of the vanishing discount problem. Let $H^\lambda(x,p,u)$ be a family of Hamiltonians of contact type with parameter $\lambda>0$ and converges to $G(x,p)$. For the contact type Hamilton-Jacobi equation with respect to $H^\lambda$, we prove that, under mild assumptions, the associated Viscosity Solution $u^{\lambda}$ converges to a specific Viscosity Solution $u^0$ of the vanished contact equation. As applications, we give some convergence results for the nonlinear vanishing discount problem.
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on Viscosity Solution of hjb equations with state constraints and reflection control
Siam Journal on Control and Optimization, 2017Co-Authors: Anup Biswas, Hitoshi Ishii, Subhamay Saha, Lin WangAbstract:Motivated by a control problem of a certain queueing network we consider a control problem where the dynamics is constrained in the nonnegative orthant $\mathbb{R}^{d}_{+}$ of the $d$-dimensional Euclidean space and controlled by the reflections at the faces/boundaries. We define a discounted value function associated to this problem and show that the value function is a Viscosity Solution to a certain HJB equation in $\mathbb{R}^{d}_{+}$ with nonlinear Neumann type boundary condition. Under certain conditions, we also characterize this value function as the unique Solution to this HJB equation.
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Viscosity Solutions for a Class of Hamilton-Jacobi Equations in Hilbert Spaces
Journal of Functional Analysis, 1992Co-Authors: Hitoshi IshiiAbstract:We study Hamilton-Jacobi equations with an unbounded term in Hilbert spaces. We introduce a new variant of the notion of Viscosity Solution for a class of Hamilton-Jacobi equations, and obtain comparison and existence results for Viscosity Solutions. We also examine the usefulness of the notion of Viscosity Solution in optimal control.
Xiaonyu Xia - One of the best experts on this subject based on the ideXlab platform.
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Continuous Viscosity Solutions to Linear-Quadratic Stochastic Control Problems with Singular Terminal State Constraint
Applied Mathematics & Optimization, 2020Co-Authors: Ulrich Horst, Xiaonyu XiaAbstract:This paper establishes the existence of a unique nonnegative continuous Viscosity Solution to the HJB equation associated with a linear-quadratic stochastic control problem with singular terminal state constraint and possibly unbounded cost coefficients. The existence result is based on a novel comparison principle for semi-continuous Viscosity sub- and superSolutions for PDEs with singular terminal value. Continuity of the Viscosity Solution is enough to carry out the verification argument.
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Continuous Viscosity Solutions to linear-quadratic stochastic control problems with singular terminal state constraint
2018Co-Authors: Ulrich Horst, Xiaonyu XiaAbstract:This paper establishes the existence of a unique nonnegative continuous Viscosity Solution to the HJB equation associated with a Markovian linear-quadratic control problems with singular terminal state constraint and possibly unbounded cost coefficients. The existence result is based on a novel comparison principle for semi-continuous Viscosity sub- and superSolutions for PDEs with singular terminal value. Under a mild additional assumption on the model parameters we show that the Viscosity Solution is in fact a $\pi$-strong Solution to the HJB equation and can hence be compactly approximated by smooth functions.
Fabio Camilli - One of the best experts on this subject based on the ideXlab platform.
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A unified approach to the well-posedness of some non-Lambertian models in Shape-from-Shading theory
arXiv: Analysis of PDEs, 2016Co-Authors: Fabio Camilli, Silvia TozzaAbstract:In this paper we show that the introduction of an attenuation factor in the %image irradiance brightness equations relative to various perspective Shape from Shading models allows to make the corresponding differential problems well-posed. We propose a unified approach based on the theory of Viscosity Solution and we show that the brightness equations with the attenuation term admit a unique Viscosity Solution. We also discuss in detail the possible boundary conditions that we can use for the Hamilton-Jacobi equations associated to these models.
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Viscosity Solution
2014Co-Authors: Fabio Camilli, Emmanuel PradosAbstract:Viscosity Solution is a notion of weak Solution for a class of partial differential equations of Hamilton-Jacobi type. The range of applications of the notions of Viscosity Solution and Hamilton-Jacobi equations is enormous, including common class of partial differential equations such as evolutive problems and problems with boundary conditions, equations arising in optimal control theory, differential games, second-order equations arising in stochastic optimal control and stochastic differential games, geometric equations, etc. In computer vision, it also has various applications. In particular, the distance functions which are the Viscosity Solutions of the Eikonal equations are widely used. Also, all these notions have played an important role in shape representation, morphology, tractography and in general, in image processing. Finally Viscosity Solutions provide powerful tools to analyse the Shape from shading problem.
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Hamilton-Jacobi equations constrained on networks
Nonlinear Differential Equations and Applications, 2013Co-Authors: Yves Achdou, Fabio Camilli, Alessandra Cutrì, Nicoletta TchouAbstract:We consider continuous-state and continuous-time control problems where the admissible trajectories of the system are constrained to remain on a network. In our setting, the value function is continuous. We de ne a notion of constrained Viscosity Solution of Hamilton-Jacobi equations on the network and we study related comparison principles. Under suitable assumptions, we prove in particular that the value function is the unique constrained Viscosity Solution of the Hamilton-Jacobi equation on the network.
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Hamilton–Jacobi equations constrained on networks
Nodea-nonlinear Differential Equations and Applications, 2012Co-Authors: Yves Achdou, Fabio Camilli, Alessandra Cutrì, Nicoletta TchouAbstract:We consider continuous-state and continuous-time control problems where the admissible trajectories of the system are constrained to remain on a network. In our setting, the value function is continuous. We de ne a notion of constrained Viscosity Solution of Hamilton-Jacobi equations on the network and we study related comparison principles. Under suitable assumptions, we prove in particular that the value function is the unique constrained Viscosity Solution of the Hamilton-Jacobi equation on the network.
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Hamilton-Jacobi equations on networks
IFAC Proceedings Volumes, 2011Co-Authors: Yves Achdou, Fabio Camilli, Alessandra Cutrì, Nicoletta TchouAbstract:We consider continuous-state and continuous-time control problem where the admissible trajectories of the system are constrained to remain on a network. Under suitable assumptions, we prove that the value function is continuous. We define a notion of Viscosity Solution of Hamilton-Jacobi equations on the network for which we prove a comparison principle. The value function is thus the unique Viscosity Solution of the Hamilton-Jacobi equation on the network.
