Error Plots Under Grid Refinement
Here, we present numerical evidence of the convergence of the value function under grid refinement. To estimate the error, we treat the numerical solution \(w^*(r,\theta,q,s)\) computed on a fine grid \(\Big((N_r,N_{\theta})=(1601,1601)\) with \(\Delta s = 0.025 \Big)\) as the "ground truth". We then measure how much the numerical solution on coarser grids deviates from the ground truth.
\(L_1\)-Error
Let \(N_{\mathrm{tot}}\) denote the total number of points of the grid.
- We show the discrete \(l_1\)-error \(\frac{1}{N_{\mathrm{tot}}} \sum_{r,\theta,s}|w(r,\theta,q,s)−w^*(r,\theta,q,s)|\) (for each tack \( q \in \{1,2\}\)) as a function of the grid refinement.
- We plot the error as a function of \(\Delta s\) indicating the grid refinement. In fact, \(\Delta r\) and \(\Delta \theta\) both change proportionally to \(\Delta s\) in the refinement study.
- The solid red line is the \(l_1\)-error while the dashed black line is the slope-1 (linear convergence) reference line in the log-log space.
For the starboard tack \(q = 1\):
For \(a=0\) and \(\sigma = 0.05\):
For \(a=0.05\) and \(\sigma = 0.05\):
For \(a=0.15\) and \(\sigma = 0.05\):
For the port tack \(q = 2\):
For \(a=0\) and \(\sigma = 0.05\):
For \(a=0.05\) and \(\sigma = 0.05\):
For \(a=0.15\) and \(\sigma = 0.05\):
