36 research outputs found
Mixed finite element approximation of periodic Hamilton--Jacobi--Bellman problems with application to numerical homogenization
In the first part of the paper, we propose and rigorously analyze a mixed
finite element method for the approximation of the periodic strong solution to
the fully nonlinear second-order Hamilton--Jacobi--Bellman equation with
coefficients satisfying the Cordes condition. These problems arise as the
corrector problems in the homogenization of Hamilton--Jacobi--Bellman
equations. The second part of the paper focuses on the numerical homogenization
of such equations, more precisely on the numerical approximation of the
effective Hamiltonian. Numerical experiments demonstrate the approximation
scheme for the effective Hamiltonian and the numerical solution of the
homogenized problem.Comment: 23 page
Adaptive C\u3csup\u3e0\u3c/sup\u3e interior penalty methods for Hamilton–Jacobi–Bellman equations with Cordes coefficients
In this paper we conduct a priori and a posteriori error analysis of the C interior penalty method for Hamilton–Jacobi–Bellman equations, with coefficients that satisfy the Cordes condition. These estimates show the quasi-optimality of the method, and provide one with an adaptive finite element method. In accordance with the proven regularity theory, we only assume that the solution of the Hamilton–Jacobi–Bellman equation belongs to H . 0
Adaptive interior penalty methods for Hamilton–Jacobi–Bellman equations with Cordes coefficients
In this paper we conduct a priori and a posteriori error analysis of the C0 interior penalty method for Hamilton–Jacobi–Bellman equations, with coefficients that satisfy the Cordes condition. These estimates show the quasi-optimality of the method, and provide one with an adaptive finite element method. In accordance with the proven regularity theory, we only assume that the solution of the Hamilton–Jacobi–Bellman equation belongs to H2
Unified analysis of discontinuous Galerkin and C0-interior penalty finite element methods for Hamilton-Jacobi-Bellman and Isaacs equations
We provide a unified analysis of a posteriori and a priori error bounds for a broad class of discontinuous Galerkin and C0-IP finite element approximations of fully nonlinear second-order elliptic Hamilton–Jacobi–Bellman and Isaacs equations with Cordes coefficients. We prove the existence and uniqueness of strong solutions in H2 of Isaacs equations with Cordes coefficients posed on bounded convex domains. We then show the reliability and efficiency of computable residual-based error estimators for piecewise polynomial approximations on simplicial meshes in two and three space dimensions. We introduce an abstract framework for the a priori error analysis of a broad family of numerical methods and prove the quasi-optimality of discrete approximations under three key conditions of Lipschitz continuity, discrete consistency and strong monotonicity of the numerical method. Under these conditions, we also prove convergence of the numerical approximations in the small-mesh limit for minimal regularity solutions. We then show that the framework applies to a range of existing numerical methods from the literature, as well as some original variants. A key ingredient of our results is an original analysis of the stabilization terms. As a corollary, we also obtain a generalization of the discrete Miranda–Talenti inequality to piecewise polynomial vector fields
Piecewise Constant Policy Approximations to Hamilton-Jacobi-Bellman Equations
An advantageous feature of piecewise constant policy timestepping for
Hamilton-Jacobi-Bellman (HJB) equations is that different linear approximation
schemes, and indeed different meshes, can be used for the resulting linear
equations for different control parameters. Standard convergence analysis
suggests that monotone (i.e., linear) interpolation must be used to transfer
data between meshes. Using the equivalence to a switching system and an
adaptation of the usual arguments based on consistency, stability and
monotonicity, we show that if limited, potentially higher order interpolation
is used for the mesh transfer, convergence is guaranteed. We provide numerical
tests for the mean-variance optimal investment problem and the uncertain
volatility option pricing model, and compare the results to published test
cases
Computational multiscale methods for nondivergence-form elliptic partial differential equations
This paper proposes novel computational multiscale methods for linear
second-order elliptic partial differential equations in nondivergence-form with
heterogeneous coefficients satisfying a Cordes condition. The construction
follows the methodology of localized orthogonal decomposition (LOD) and
provides operator-adapted coarse spaces by solving localized cell problems on a
fine scale in the spirit of numerical homogenization. The degrees of freedom of
the coarse spaces are related to nonconforming and mixed finite element methods
for homogeneous problems. The rigorous error analysis of one exemplary approach
shows that the favorable properties of the LOD methodology known from
divergence-form PDEs, i.e., its applicability and accuracy beyond scale
separation and periodicity, remain valid for problems in nondivergence-form.Comment: 21 page