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

High-order filtered schemes for time-dependent second order HJB equations

In this paper, we present and analyse a class of "filtered" numerical schemes for second order Hamilton-Jacobi-Bellman equations. Our approach follows the ideas introduced in B.D. Froese and A.M. Oberman, Convergent filtered schemes for the Monge-Amp\`ere partial differential equation, SIAM J. Numer. Anal., 51(1):423--444, 2013, and more recently applied by other authors to stationary or time-dependent first order Hamilton-Jacobi equations. For high order approximation schemes (where "high" stands for greater than one), the inevitable loss of monotonicity prevents the use of the classical theoretical results for convergence to viscosity solutions. The work introduces a suitable local modification of these schemes by "filtering" them with a monotone scheme, such that they can be proven convergent and still show an overall high order behaviour for smooth enough solutions. We give theoretical proofs of these claims and illustrate the behaviour with numerical tests from mathematical finance, focussing also on the use of backward difference formulae (BDF) for constructing the high order schemes.Comment: 27 pages, 16 figures, 4 table

Semi-Lagrangian schemes for linear and fully non-linear Hamilton-Jacobi-Bellman equations

We consider the numerical solution of Hamilton-Jacobi-Bellman equations arising in stochastic control theory. We introduce a class of monotone approximation schemes relying on monotone interpolation. These schemes converge under very weak assumptions, including the case of arbitrary degenerate diffusions. Besides providing a unifying framework that includes several known first order accurate schemes, stability and convergence results are given, along with two different robust error estimates. Finally, the method is applied to a super-replication problem from finance.Comment: to appear in the proceedings of HYP201

A semi-lagrangian scheme for hamilton-jacobi-bellman equations on networks

We present a semi-Lagrangian scheme for the approximation of a class of Hamilton- Jacobi-Bellman (HJB) equations on networks. The scheme is explicit, consistent, and stable for large time steps. We prove a convergence result and two error estimates. For an HJB equation with space-independent Hamiltonian, we obtain a first order error estimate. In the general case, we provide, under a hyperbolic CFL condition, a convergence estimate of order one half. The theoretical results are discussed and validated in a numerical tests section

A WENO finite difference scheme for a new class of Hamilton-Jacobi equations in electroelastostatics

Hamilton-Jacobi equations have repeatedly emerged in many fields of physics, most notably, optimal control, differential games, geometric optics, and image processing. This thesis presents a new numerical method to solve a new class of Hamilton-Jacobi equation that has recently appeared in the context of nonlinear electroelastostatics. In a pioneering contribution, Crandall and Lions (1983) proved that a certain type of first-order finite difference method converges to the viscosity solution of a special class of Hamilton-Jacobi equations. From then on several successful methods of high-order approximation have been proposed in the literature, including the so-called WENO finite difference schemes. These schemes, however, were developed and tested for special classes of Hamilton-Jacobi equations, which do not include the general type of Hamilton-Jacobi equation of interest in this work. The objective of this thesis is to extend high-order WENO finite difference schemes to the most general type of Hamilton-Jacobi equations involving non-periodic boundary conditions in the "space" variables. Following its derivation, the proposed WENO scheme is tested for several cases involving one and two "space" variables for which there are analytical solutions available for arbitrarily large values of the "time" variable. These numerical experiments provide insight into the stability and rate of convergence of the method as "time" increases. They also provide insight into how errors propagate into the domain of computation due to non-periodic boundary conditions. This thesis concludes with the application of the method to compute the effective stored-energy function of an elastomer containing an isotropic distribution of vacuous pores under arbitrary 3D deformations

Hausdorff continuous viscosity solutions of Hamilton-Jacobi equations and their numerical analysis

The theory of viscosity solutions was developed for certain types of nonlinear first-order and second-order partial differential equations. It has been particularly useful in describing the solutions of partial differential equations associated with deterministic and stochastic optimal control problems [16], [53]. In its classical formulation, see [16], the theory deals with solutions which are continuous functions. The concept of continuous viscosity solutions was further generalized in various ways to include discontinuous solutions with the definition of Ishii given in [71] playing a pivotal role. In this thesis we propose a new approach for the treatment of discontinuous solutions of first-order Hamilton-Jacobi equations, namely, by involving Hausdorff continuous interval valued functions. The advantages of the proposed approach are justified by demonstrating that the main ideas within the classical theory of continuous viscosity solutions can be extended almost unchanged to the wider space of Hausdorff continuous functions and the existing theory of discontinuous viscosity solutions is a particular case of that developed in this thesis in terms of Hausdorff continuous interval valued functions. Two approaches to numerical solutions for Hamilton-Jacobi equations are presented. The first one is a monotone scheme for Hamilton-Jacobi equations while the second is based on preserving total variation diminishing property for conservation laws. In the first approach, we couple the finite element method with the nonstandard finite difference method which is based on the Mickens’ rule of nonlocal approximation [9]. The scheme obtained in this way is unconditionally monotone. In the second approach, computationally simple implicit schemes are derived by using nonlocal approximation of nonlinear terms. Renormalization of the denominator of the discrete derivative is used for deriving explicit schemes of first or higher order. Unlike the standard explicit methods, the solutions of these schemes have diminishing total variation for any time step size.Thesis (PhD (Mathematical Science))--University of Pretoria, 2007.Mathematics and Applied Mathematicsunrestricte

A Rotating-Grid Upwind Fast Sweeping Scheme for a Class of Hamilton-Jacobi Equations

We present a fast sweeping method for a class of Hamilton-Jacobi equations that arise from time-independent problems in optimal control theory. The basic method in two dimensions uses a four point stencil and is extremely simple to implement. We test our basic method against Eikonal equations in different norms, and then suggest a general method for rotating the grid and using additional approximations to the derivatives in different directions in order to more accurately capture characteristic flow. We display the utility of our method by applying it to relevant problems from engineering

Error estimates for a tree structure algorithm solving finite horizon control problems

In the Dynamic Programming approach to optimal control problems a crucial role is played by the value function that is characterized as the unique viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. It is well known that this approach suffers of the "curse of dimensionality" and this limitation has reduced its practical in real world applications. Here we analyze a dynamic programming algorithm based on a tree structure. The tree is built by the time discrete dynamics avoiding in this way the use of a fixed space grid which is the bottleneck for high-dimensional problems, this also drops the projection on the grid in the approximation of the value function. We present some error estimates for a first order approximation based on the tree-structure algorithm. Moreover, we analyze a pruning technique for the tree to reduce the complexity and minimize the computational effort. Finally, we present some numerical tests

Some non monotone schemes for Hamilton-Jacobi-Bellman equations

We extend the theory of Barles Jakobsen to develop numerical schemes for Hamilton Jacobi Bellman equations. We show that the monotonicity of the schemes can be relaxed still leading to the convergence to the viscosity solution of the equation. We give some examples of such numerical schemes and show that the bounds obtained by the framework developed are not tight. At last we test some numerical schemes.Comment: 24 page