A Reduced Basis Method for the Hamilton-Jacobi-Bellman Equation with Application to the European Union Emission Trading Scheme

TL;DR

This paper develops a reduced basis method for efficiently solving the Hamilton-Jacobi-Bellman equation, especially in the context of the EU Emission Trading Scheme, addressing the challenge of model reduction for hyperbolic PDEs.

Contribution

It introduces the first reduced basis method tailored for the hyperbolic Hamilton-Jacobi-Bellman equation, including an error estimator and numerical algorithms.

Findings

Successfully constructed an online-efficient error estimator.

Demonstrated the method's effectiveness through numerical experiments.

Extended RBM applicability to hyperbolic PDEs like the HJB equation.

Abstract

This paper draws on two sources of motivation: (1) The European Union Emission Trading Scheme (EU-ETS) aims at limiting the overall emissions of greenhouse gases. The optimal abatement strategy of companies for the use of emission permits can be described as the viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. It is a question of general interest, how regulatory constraints can be set within the EU-ETS in order to reach certain political goals such as a good balance of emission reduction and economical growth. Such regulatory constraints can be modeled as parameters within the HJB equation. (2) The EU-ETS is just one example where one is interested in solving a parameterized HJB equation often for different values of the parameters (e.g.\ to optimize their values with respect to a given target functional). The Reduced Basis Method (RBM) is by now a well-established numerical method to efficiently solve parameterized partial differential equations. However, to the best of our knowledge, an RBM for the HJB equation is not known so far and of (mathematical) interest by its own, since the HJB equation is of hyperbolic type which is in general a nontrivial task for model reduction. We analyze and realize a RBM for the HJB equation. In particular, we construct an online-efficient error estimator for this nonlinear problem using the Brezzi-Rapaz-Raviart (RBB) theory as well as numerical algorithms for the involved parameter-dependent constants. Numerical experiments are presented.