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USER’S GUIDE TO VISCOSITY SOLUTIONS OF SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS
, 1992
"... The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking argume ..."
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Cited by 629 (9 self)
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The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a selfcontained exposition of the basic theory of viscosity solutions.
Discrete time highorder schemes for viscosity solutions of HamiltonJacobiBellman equations
, 1994
"... this paper is to derive highorder numerical schemes for the approximation of the value function, this being done using the Dynamic Programming approach. ..."
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Cited by 19 (2 self)
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this paper is to derive highorder numerical schemes for the approximation of the value function, this being done using the Dynamic Programming approach.
Adaptive Choice of Grid and Time in Reinforcement Learning
 IN NIPS ’97: PROCEEDINGS OF THE 1997 CONFERENCE ON ADVANCES IN NEURAL INFORMATION PROCESSING SYSTEMS 10
, 1997
"... We propose local error estimates together with algorithms for adaptive aposteriori grid and time refinement in reinforcement learning. We consider a deterministic system with continuous state and time with infinite horizon discounted cost functional. For grid refinement we follow the procedure of ..."
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Cited by 15 (1 self)
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We propose local error estimates together with algorithms for adaptive aposteriori grid and time refinement in reinforcement learning. We consider a deterministic system with continuous state and time with infinite horizon discounted cost functional. For grid refinement we follow the procedure of numerical methods for the Bellmanequation. For time refinement we propose a new criterion, based on consistency estimates of discrete solutions of the Bellmanequation. We demonstrate, that an optimal ratio of time to space discretization is crucial for optimal learning rates and accuracy of the approximate optimal value function.
Using dynamic programming with adaptive grid scheme for optimal control problems in economics
 JOURNAL OF ECONOMIC DYNAMICS AND CONTROL, ELSEVIER
, 2004
"... The study of the solutions of dynamic models with optimizing agents has often been limited by a lack of available analytical techniques to explicitly find the global solution paths. On the other hand the application of numerical techniques such as dynamic programming (DP) to find the solution in int ..."
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Cited by 14 (8 self)
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The study of the solutions of dynamic models with optimizing agents has often been limited by a lack of available analytical techniques to explicitly find the global solution paths. On the other hand the application of numerical techniques such as dynamic programming (DP) to find the solution in interesting regions of the state state was restricted by the use of fixed grid size techniques. Following Grüne (1997) and (2003) in this paper an adaptive grid scheme is used for finding the global solutions of discrete time HamiltonJacobiBellman (HJB) equations. Local error estimates are established and an adapting iteration for the discretization of the state space is developed. The advantage of the use of adaptive grid scheme is demonstrated by computing the solutions of one and two dimensional economic models which exhibit steep curvature, complicated dynamics due to multiple equilibria, thresholds (Skiba sets) separating domains of attraction and periodic solutions. We consider deterministic and stochastic model variants. The studied examples are from economic growth, investment theory, environmental and resource economics.
Approximate Solutions to the TimeInvariant HamiltonJacobiBellman Equation
, 1998
"... In this paper we develop a new method to approximate the solution to the HamiltonJacobiBellman (HJB) equation which arises in optimal control when the plant is modeled by nonlinear dynamics. The approximation is comprised of two steps. First, successive approximation is used to reduce the HJB equat ..."
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Cited by 11 (4 self)
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In this paper we develop a new method to approximate the solution to the HamiltonJacobiBellman (HJB) equation which arises in optimal control when the plant is modeled by nonlinear dynamics. The approximation is comprised of two steps. First, successive approximation is used to reduce the HJB equation to a sequence of linear partial differential equations. These equations are then approximated via Galerkin's spectral method. The resulting algorithm has several important advantages over previously reported methods. Namely, the resulting control is in feedback form and its associated region of attraction is well defined. In addition, all computations are performed offline and the control can be made arbitrarily close to optimal. Accordingly this paper presents a new tool for designing nonlinear control systems that adhere to a prescribed integral performance criteria. Key Words: Nonlinear control, optimal control, HamiltonJacobiBellman equation, feedback synthesis, successive approxi...
A FAST ITERATIVE METHOD FOR EIKONAL EQUATIONS
, 2008
"... In this paper we propose a novel computational technique to solve the Eikonal equation efficiently on parallel architectures. The proposed method manages the list of active nodes and iteratively updates the solutions on those nodes until they converge. Nodes are added to or removed from the list ba ..."
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Cited by 10 (3 self)
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In this paper we propose a novel computational technique to solve the Eikonal equation efficiently on parallel architectures. The proposed method manages the list of active nodes and iteratively updates the solutions on those nodes until they converge. Nodes are added to or removed from the list based on a convergence measure, but the management of this list does not entail an extra burden of expensive ordered data structures or special updating sequences. The proposed method has suboptimal worstcase performance but, in practice, on real and synthetic datasets, runs faster than guaranteedoptimal alternatives. Furthermore, the proposed method uses only local, synchronous updates and therefore has better cache coherency, is simple to implement, and scales efficiently on parallel architectures. This paper describes the method, proves its consistency, gives a performance analysis that compares the proposed method against the stateoftheart Eikonal solvers, and describes the implementation on a single instruction multiple datastream (SIMD) parallel architecture.
Optimal Control Using Bisimulations: Implementation
, 2001
"... We consider the synthesis of optimal controls for continuous feedback systems by recasting the problem to a hybrid optimal control problem which is to synthesize optimal enabling conditions for switching between locations in which the control is constant. We provide a singlepass algorithm to sol ..."
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Cited by 6 (0 self)
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We consider the synthesis of optimal controls for continuous feedback systems by recasting the problem to a hybrid optimal control problem which is to synthesize optimal enabling conditions for switching between locations in which the control is constant. We provide a singlepass algorithm to solve the dynamic programming problem that arises, with added constraints to ensure nonZeno trajectories.
Numerical approximation of the maximal solutions for a class of degenerate HamiltonJacobi equations
 SIAM Journal of Numerical Analysis
, 2000
"... : In this paper we study an approximation scheme for a class of HamiltonJacobi problems for which uniqueness of the viscosity solution does not hold. This class includes the Eikonal equation arising in the Shape from Shading problem. We show that, if an appropriate stability condition is satisfied, ..."
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Cited by 5 (0 self)
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: In this paper we study an approximation scheme for a class of HamiltonJacobi problems for which uniqueness of the viscosity solution does not hold. This class includes the Eikonal equation arising in the Shape from Shading problem. We show that, if an appropriate stability condition is satisfied, the scheme converges to the maximal viscosity solution of the problem. Furthermore we give an estimate for the discretization error. Keywords: Singular HamiltonJacobi equations, maximal solution, regularization, numerical approximation, discretization error AMS Classification: 65M12, 65M15, 49L25 1 Introduction Given a HamiltonJacobi equation, a general result due to BarlesSouganidis [3] says that any "reasonable" approximation scheme (based f.e. on finite differences, finite elements, finite volumes, discretization of characteristics, etc.) converges to the viscosity solution of the equation. Besides some simple properties that the approximation scheme has to satisfy, it is only requ...
A maxplus finite element method for solving finite horizon deterministic optimal control problems
 in "Proceedings of MTNS’04, Louvain, Belgique", Also arXiv:math.OC/0404184, 2004, http://hal.inria.fr/inria00071426. Maxplus 31
"... Abstract. We introduce a maxplus analogue of the PetrovGalerkin finite element method, to solve finite horizon deterministic optimal control problems. The method relies on a maxplus variational formulation, and exploits the properties of projectors on maxplus semimodules. We obtain a nonlinear d ..."
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Cited by 5 (2 self)
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Abstract. We introduce a maxplus analogue of the PetrovGalerkin finite element method, to solve finite horizon deterministic optimal control problems. The method relies on a maxplus variational formulation, and exploits the properties of projectors on maxplus semimodules. We obtain a nonlinear discretized semigroup, corresponding to a zerosum two players game. We give an error estimate of order √ ∆t + ∆x(∆t) −1, for a subclass of problems in dimension 1. We compare our method with a maxplus based discretization method previously introduced by Fleming and McEneaney.
Asymptotic Controllability And Exponential Stabilization Of Nonlinear Control Systems At Singular Points
, 1998
"... We discuss the relation between exponential stabilization and asymptotic controllability of nonlinear control systems with constrained control range at singular points. Using a discounted optimal control approach we construct discrete feedback laws minimizing the Lyapunov exponent of the linearizati ..."
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Cited by 4 (2 self)
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We discuss the relation between exponential stabilization and asymptotic controllability of nonlinear control systems with constrained control range at singular points. Using a discounted optimal control approach we construct discrete feedback laws minimizing the Lyapunov exponent of the linearization. Thus we obtain an equivalence result between uniform exponential controllability and uniform exponential stabilizability by means of a discrete feedback law.