# Finite Differences on Sparse Grids for Continuous-Time Heterogeneous Agent Models
## 6 Conclusions and Outlook  
In this work we introduced a finite difference approach using interpolations on an adaptive sparse grid for solving economic models following the numerical scheme of [3]. We analyzed the accuracy for our approach for economic models ranging from dimension \(d=2\) to dimension \(d=6\) and achieved good results for the used model parameters.  
We can extract multiple results from our numerical studies. First, sparse grid finite differences work quite well in practice for solving continuous-time economic models. For a two-dimensional model, we showed that our numerical scheme converges to the full grid solution, for which it is proven that it converges to the correct solution. Second, the experiments with different types of adaptivity indicate that value function adaptivity performs well for approximating the value function. To achieve a good approximation of the policy functions, it can sometimes be better to use a criterion suited to this function or combined criteria. Note though that for policy functions it strongly depends on the choice of parameters, such as the starting level of the sparse grid or the starting refinement threshold, to observe how well it performs. Nevertheless, we recommend to use value function adaptivity since it leads to the best results in most cases.  
For a convergence result for sparse grids finite difference schemes for solving the HJB equation according to [11], we would need monotone sparse grid interpolation. However, we showed that interpolation on sparse grids is not monotone in general, even if we restrict ourselves to one-dimensional monotonicity for concave monotonically increasing functions. A general theoretical result based on assumptions that are fulfilled by most economic models is therefore hardly possible, since it depends on model parameters if the arising interpolations on the value functions are monotone for the used adaptive sparse grid. Thus, it depends on the model parameters if our approach works correctly without specific approaches to overcome non-monotonicity. Nevertheless, the numerical results indicate that a non-monotone discretization approach can achieve convergence, which highlights the need for additional theoretical investigations.  
While adaptive sparse grids allow the discretization of higher dimensional problems, we note that the computational costs for solving the arising linear equation system do increase with the number of dimensions. For the purpose of this study we used a standard ILUC preconditioned BiCGSTAB iterative solver, but we expect that there are preconditioners available that are more suitable for this problem class, which would be one aspect of future research. Additionally, parallelization can further improve the runtime.  
Moreover, the computational performance of the different sparse grid approaches for solving Hamilton-Jacobi-Bellman equations could be compared. Besides the one presented in this work, this would involve, among others, parallel adaptive sparse grids from [23], finite difference operators on sparse grids [26], and sparse grid semi-Lagrangian approaches [34; 35]. Furthermore, there are investigations on using deep neural networks for solving dynamic economic models, e.g., [39] casts these into nonlinear regression equations. An investigation on which numerical approach is better suited for which economic scenario is warranted.  
Conceptualization, J.G. and S.R.; methodology, J.G. and S.R.; software, S.R.; validation, J.G. and S.R.; writing--original draft preparation, J.G. and S.R.; writing--review and editing, J.G.; visualization, J.G. and S.R.; supervision, J.G. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Data Availability Statement: Data are contained within the article.  
We thank [PERSON] and [PERSON] for their fruitful discussions, and [PERSON] for help with the Matlab implementation, which is based on his code written for [9].