Data-driven evolutions of critical points

Figure 1.  Reconstructions $ \hat a $ (left) and $ \hat a' $ (right) in dotted red and true $ a $ and $ a' $ in blue. The underlying red circles depict the (adaptive) nodes in $ \Lambda $. Below you see the distribution $ \tilde \eta $ of available data

Figure 2.  Reconstruction of $ a' $ with increasing number $ {N_e} $ of measurements. The true $ a' $ is depicted by the blue curve while the red line depicts the reconstruction $ \hat a' $ following (5.23). The red circles at the bottom of the figures depict the position of nodes of the underlying mesh $ \Lambda_N $. The histogram below describes the density of the available information encoded by the probability measure $ \tilde{\eta} $ for the case $ N_e = 60 $

Figure 3.  Reconstruction of $ a' $ for varying $ N $ and $ D(N) = 4N $ and distribution $ \tilde \eta $ for $ N = 60 $. Graphic as described in Figure 2

Figure 4.  Reconstruction of $ a' $ for varying $ N $ and fixed $ D(N) = 300 $ and distribution $ \tilde \eta $. Graphics as described in Figure 2

Figure 5.  Impact of different constants $ M_2 $ for constraints on $ a'' $ with $ \|a''\| \leq [2,5;20,1000] $ on reconstructs are depicted, showing improved approximations for increasing $ M_2 $. Graphics as in Figure 2

Figure 6.  Impact of changing constraint $ M_1 $ bounding the values of $ a' $, showing a projection-like behaviour of the reconstruction for small $ M_1 $. Graphics as in Figure 2

Figure 7.  Left showing $ \hat x_\varepsilon(t) $ (dark dashed) and $ x_\varepsilon(t) $ (line bright) with data from Section 5.2.2 at times $ t = [0.2 $, $ 0.4 $, $ 0.6 $, $ 0.8 $, $ 1.0] $, right with data from Section 5.2.3