Convergence of Implicit Schemes for Hamilton--Jacobi--Bellman Quasi-Variational Inequalities

Abstract

In [Azimzadeh, P., and P. A. Forsyth. "Weakly chained matrices, policy iteration, and impulse control." SIAM J. Num. Anal. 54.3 (2016): 1341-1364], we outlined the theory and implementation of computational methods for implicit schemes for Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs). No convergence proofs were given therein. This work closes the gap by giving rigorous proofs of convergence. We do so by introducing the notion of nonlocal consistency and appealing to a Barles-Souganidis type analysis. Our results rely only on a well-known comparison principle and are independent of the specific form of the intervention operator.

A Penalty Scheme for Monotone Systems with Interconnected Obstacles: Convergence and Error Estimates

“…Assumption 1 will only be used in Section 3 to quantify the regularization errors (not for the well-posedness or the monotone convergence of the regularization procedures). It is well-known that a concave function can be equivalently represented as the infimum of a family of affine functions, i.e., F i (u) = inf α∈Ai B i (α)u − b i (α) for some set A i and coefficients B i : A i → R N d×N d and b i : A i → R N d , hence our error estimates apply to the HJBQVIs studied in [7,23,2,12,1]. However, our setting significantly extends the classical HJBQVIs in the following important aspects: (1) F i can depend on all components of the solutions to the switching systems, (2) the control set A i can be non-compact and coefficients B i , b i can be discontinuous, (3) b i does not necessarily have a unique sign.…”

“…Due to the fact that the control j takes only d distinct values, we can apply the penalty term finitely many times (once per value). This is not directly possible in the framework of [1,2], where the number of attainable values for the control in the intervention operator grows unbounded as the meshing parameter in the approximation of an infinite control set approaches zero (see [19] for an extension of such a penalty scheme to general intervention operators with an infinite number of control values: the summation is replaced by an integral, which might subsequently be approximated by quadrature).…”

Error Estimates of Penalty Schemes for Quasi-Variational Inequalities Arising from Impulse Control Problems

“…This suggests that the penalty schemes are significantly more efficient than the direct control scheme for solving large-scale discrete QVIs, as pointed out in [3]. In practice, instead of solving the penalized equation (7.2) with a fixed penalty parameter ρ, we shall construct a convergent approximation to the solution of the QVI (7.1) based on the penalized solutions, by letting 1/ρ and h tend to zero simultaneously (see also [3,1]). The first order convergence of both the penalization error and the discretization error (see Figure 7.1 and Table 7.1) suggests us to take ρ = CN , where the constant C = 1/16 was found to achieve the optimal balance between the penalization error and the discretization error.…”

“…Properties (1) and (2) follow directly from the structure of M i . Property (3) is an analogue of [1,Lemma 12] to the present concave intervention operator M i and compact set Z i , whose proof will be given in Appendix A for completeness.…”

“…The condition (2.5) in (H.1) is the standard regularity assumption for the coefficients in viscosity solution theory, while (2.6) in (H.1) implies the monotonicity of the HJB equations. The condition (H.2) on the intervention operator is the same as that in [38,1], which ensures the well-posedness of (2.1) in the class of bounded continuous functions. As we shall show in Section 3, they are sufficient for the monotone convergence of the penalty approximation, even for non-convex/non-concave systems involving Isaacs' equations.…”