.

compression.tex

@@ -539,79 +539,81 @@
         \draw [dashed, line width=1pt] (0, 0.2) -- (-3.55, 0.2);
         \draw [line width=2pt,-{Latex[length=10pt,width=10pt]}] (-3.45,-1.2) -- (-3.45,-0.2);
         \draw [line width=2pt,-{Latex[length=10pt,width=10pt]}] (-3.45, 1.2) -- (-3.45, 0.2);
-        \node at (-4.0,0) {\Huge d};
+        \node at (-4.0,0) {\Huge $d$};
       \end{tikzpicture}}%>>>
 
     \column{0.75\textwidth}
-      {\bf Interpolated Compressed Inverse Preconditioning (ICIP):}
+      {\bf Adaptive discretization:}
       \only<1>{\begin{tabular}{r r r r r r r r}%<<<
         \hline
-          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & 1e-4 & 1e-5 & 1e-6 & 1e-7 \\
+          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & ~1e-4 & ~~~1e-5 & 1e-6 & 1e-7 \\
         \hline
-            2               &       18 &  20  &   21 &   21 &   21 &   21 &   21 \\
+            2               &       15 &  37  &  104 &  337 & 1283 & 1848 & 2344 \\
+            &&&&&&&\\
+            &&&&&&&\\
+            &&&&&&&\\
+            &&&&&&&\\
         %\hline
       \end{tabular}
-      \vspace{5.36em}
       }%>>>
       \only<2>{\begin{tabular}{r r r r r r r r}%<<<
         \hline
-          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & 1e-4 & 1e-5 & 1e-6 & 1e-7 \\
+          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & ~1e-4 & ~~~1e-5 & 1e-6 & 1e-7 \\
         \hline
-            2               &       18 &  20  &   21 &   21 &   21 &   21 &   21 \\
-            4               &       28 &  34  &   36 &   37 &   37 &   37 &   37 \\
+            2               &       15 &  37  &  104 &  337 & 1283 & 1848 & 2344 \\
+            4               &       25 &  75  &  271 & 1134 & 3770 & 5301 & 6620 \\
+            &&&&&&&\\
+            &&&&&&&\\
+            &&&&&&&\\
         %\hline
       \end{tabular}
-      \vspace{4em}
       }%>>>
       \only<3>{\begin{tabular}{r r r r r r r r}%<<<
         \hline
-          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & 1e-4 & 1e-5 & 1e-6 & 1e-7 \\
+          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & ~1e-4 & ~~~1e-5 & 1e-6 & 1e-7 \\
         \hline
-            2               &       18 &  20  &   21 &   21 &   21 &   21 &   21 \\
-            4               &       28 &  34  &   36 &   37 &   37 &   37 &   37 \\
-         %  8               &       38 &  52  &   54 &   53 &   57 &   57 &   57 \\
-           16               &       46 &  71  &   74 &   80 &   86 &   87 &   88 \\
-         % 32               &       48 &  90  &   98 &  113 &  139 &  146 &  150 \\
-           64               &       49 &  96  &  108 &  131 &  186 &  237 &  251 \\
-         %128               &       49 &  98  &  110 &  134 &  215 &  326 &  431 \\
-          256               &       49 &  98  &  110 &  134 &  220 &  371 &  608 \\
+            2               &       15 &   37 &  104 &  337 & 1283 & 1848 & 2344 \\
+            4               &       25 &   75 &  271 & 1134 & 3770 & 5301 & 6620 \\
+         %  8               &       32 &  124 &  494 & 1939 & 7488 &>8000 &>8000 \\
+           16               &       35 &  147 &  629 & 2754 &>8000 &      &      \\
+         % 32               &       36 &  148 &  682 & 3092 &      &      &      \\
+           64               &       36 &  148 &  683 & 3094 &      &      &      \\
+         %128               &       37 &  149 &  683 & 3094 &      &      &      \\
+          256               &       37 &  149 &  683 & 3094 &      &      &      \\
         %\hline
       \end{tabular}}%>>>
 
       \vspace{1.5em}
-      {\bf Adaptive discretization:}
-
+      {\bf Interpolated Compressed Inverse Preconditioning (ICIP):}
       \only<1>{\begin{tabular}{r r r r r r r r}%<<<
         \hline
-          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & ~1e-4 & ~~~1e-5 & 1e-6 & 1e-7 \\
+          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & 1e-4 & 1e-5 & 1e-6 & 1e-7 \\
         \hline
-            2               &       15 &  37  &  104 &  337 & 1283 & 1848 & 2344 \\
+            2               &       18 &  20  &   21 &   21 &   21 &   21 &   21 \\
         %\hline
       \end{tabular}
-      \vspace{5.36em}
       }%>>>
       \only<2>{\begin{tabular}{r r r r r r r r}%<<<
         \hline
-          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & ~1e-4 & ~~~1e-5 & 1e-6 & 1e-7 \\
+          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & 1e-4 & 1e-5 & 1e-6 & 1e-7 \\
         \hline
-            2               &       15 &  37  &  104 &  337 & 1283 & 1848 & 2344 \\
-            4               &       25 &  75  &  271 & 1134 & 3770 & 5301 & 6620 \\
+            2               &       18 &  20  &   21 &   21 &   21 &   21 &   21 \\
+            4               &       28 &  34  &   36 &   37 &   37 &   37 &   37 \\
         %\hline
       \end{tabular}
-      \vspace{4em}
       }%>>>
       \only<3>{\begin{tabular}{r r r r r r r r}%<<<
         \hline
-          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & ~1e-4 & ~~~1e-5 & 1e-6 & 1e-7 \\
+          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & 1e-4 & 1e-5 & 1e-6 & 1e-7 \\
         \hline
-            2               &       15 &   37 &  104 &  337 & 1283 & 1848 & 2344 \\
-            4               &       25 &   75 &  271 & 1134 & 3770 & 5301 & 6620 \\
-         %  8               &       32 &  124 &  494 & 1939 & 7488 &>8000 &>8000 \\
-           16               &       35 &  147 &  629 & 2754 &>8000 &      &      \\
-         % 32               &       36 &  148 &  682 & 3092 &      &      &      \\
-           64               &       36 &  148 &  683 & 3094 &      &      &      \\
-         %128               &       37 &  149 &  683 & 3094 &      &      &      \\
-          256               &       37 &  149 &  683 & 3094 &      &      &      \\
+            2               &       18 &  20  &   21 &   21 &   21 &   21 &   21 \\
+            4               &       28 &  34  &   36 &   37 &   37 &   37 &   37 \\
+         %  8               &       38 &  52  &   54 &   53 &   57 &   57 &   57 \\
+           16               &       46 &  71  &   74 &   80 &   86 &   87 &   88 \\
+         % 32               &       48 &  90  &   98 &  113 &  139 &  146 &  150 \\
+           64               &       49 &  96  &  108 &  131 &  186 &  237 &  251 \\
+         %128               &       49 &  98  &  110 &  134 &  215 &  326 &  431 \\
+          256               &       49 &  98  &  110 &  134 &  220 &  371 &  608 \\
         %\hline
       \end{tabular}}%>>>
 

data/sed2

@@ -0,0 +1,486 @@
+      t      dt  noprecond  precond       N      invdt     KSPprecond
+  0.000   0.100         13        7   88320    10.0000     6.3846
+  0.100   0.416         15        9   39840     2.4038     8.0000
+  0.516   0.735         16       10   22080     1.3605     7.2308
+  1.251   0.923         16       10   17760     1.0834     5.9412
+  2.174   1.877         16       10   15360     0.5328     6.7647
+  4.051   2.468         17       10   14880     0.4052     7.7619
+  6.519   2.590         17       11   14880     0.3861     8.1905
+  9.108   2.579         18       11   14880     0.3877     8.3333
+ 11.690   2.302         19       12   14880     0.4344     6.7619
+ 13.990   1.934         20       14   15840     0.5171     10.0000
+ 15.920   1.650         22       16   16320     0.6061     10.1765
+ 17.570   1.459         24       18   22080     0.6854     10.2941
+ 19.030   1.339         27       20   23520     0.7468     10.8824
+ 20.370   1.278         30       23   24960     0.7825     12.1765
+ 21.650   1.293         34       27   26880     0.7734     11.1905
+ 22.940   1.406         39       32   36960     0.7112     12.2381
+ 24.350   1.369         46       38   38400     0.7305     13.8095
+ 25.720   1.298         54       46   43200     0.7704     18.3529
+ 27.020   1.202         61       52   44640     0.8319     19.5882
+ 28.220   1.161         65       56   45600     0.8613     19.7059
+ 29.380   1.163         66       59   47040     0.8598     19.9412
+ 30.540   1.191         66       57   47040     0.8396     11.1176
+ 31.730   1.232         65       56   47040     0.8117     15.4118
+ 32.960   1.282         61       53   48000     0.7800     17.7647
+ 34.250   1.350         56       47   47040     0.7407     16.2353
+ 35.600   1.464         50       41   46560     0.6831     15.1765
+ 37.060   1.618         43       37   45600     0.6180     13.5294
+ 38.680   1.685         38       31   43200     0.5935     11.8824
+ 40.360   1.740         33       27   40320     0.5747     11.0000
+ 42.100   1.823         29       23   29760     0.5485     10.5882
+ 43.930   1.950         26       20   27840     0.5128     9.9412
+ 45.880   2.140         23       18   25920     0.4673     9.5294
+ 48.020   2.427         22       16   20640     0.4120     9.2353
+ 50.440   2.776         20       15   19680     0.3602     9.1176
+ 53.220   2.953         18       14   18720     0.3386     9.0588
+ 56.170   3.105         17       13   14400     0.3221     8.5294
+ 59.280   3.287         16       12   12480     0.3042     8.4118
+ 62.560   3.525         16       11   12000     0.2837     8.2941
+ 66.090   3.679         15       11   11520     0.2718     8.4118
+ 69.770   3.763         15       10    8640     0.2657     8.0000
+ 73.530   3.830         14        9    8640     0.2611     6.8235
+ 77.360   3.876         13        9    9600     0.2580     7.7059
+ 81.240   3.888         14       10    9600     0.2572     6.8235
+ 85.130   3.866         15       10    8640     0.2587     7.1429
+ 88.990   3.838         16       11    8640     0.2606     6.2857
+ 92.830   3.778         16       12    9120     0.2647     7.7143
+ 96.610   3.551         17       13    9120     0.2816     6.6667
+100.200   3.327         19       14   12960     0.3006     7.8571
+103.500   3.182         21       16   12480     0.3143     8.5238
+106.700   2.801         24       18   17280     0.3570     11.2353
+109.500   2.364         28       22   18240     0.4230     11.8824
+111.800   2.051         33       27   24960     0.4876     12.5294
+113.900   1.863         39       33   26880     0.5368     13.0588
+115.700   1.777         46       39   29280     0.5627     16.1176
+117.500   1.640         55       45   42720     0.6098     18.1176
+119.200   1.436         60       52   43680     0.6964     18.8824
+120.600   1.295         64       55   45600     0.7722     18.6471
+121.900   1.204         64       58   45600     0.8306     19.1765
+123.100   1.142         65       57   45120     0.8757     19.2353
+124.200   1.099         63       54   44640     0.9099     18.4706
+125.300   1.077         58       50   44160     0.9285     17.4118
+126.400   1.087         52       45   44160     0.9200     10.5882
+127.500   1.155         47       38   43680     0.8658     14.7059
+128.700   1.315         41       34   42240     0.7605     11.5714
+130.000   1.403         36       28   38880     0.7128     10.3810
+131.400   1.333         31       24   36000     0.7502     11.2941
+132.700   1.249         28       20   33120     0.8006     10.4118
+134.000   1.223         25       18   31200     0.8177     9.4706
+135.200   1.244         23       17   22560     0.8039     9.4706
+136.400   1.296         21       15   21600     0.7716     9.2941
+137.700   1.375         21       14   20640     0.7273     9.1176
+139.100   1.483         20       13   19680     0.6743     9.0588
+140.600   1.630         19       12   18720     0.6135     8.8824
+142.200   1.835         18       11   14400     0.5450     9.0588
+144.000   2.134         18       11   14400     0.4686     7.5882
+146.200   2.474         18       10   13440     0.4042     8.0000
+148.700   2.619         17       10   13440     0.3818     6.4286
+151.300   2.657         16       10   13440     0.3764     6.5238
+153.900   2.662         16       10   13440     0.3757     6.5238
+156.600   2.649         17       10   13440     0.3775     6.9524
+159.200   2.621         18       11   13440     0.3815     7.8095
+161.900   2.575         18       11   13440     0.3883     7.2857
+164.400   2.215         20       13   13920     0.4515     8.3333
+166.600   1.852         21       14   14400     0.5400     9.8235
+168.500   1.591         22       16   19200     0.6285     9.9412
+170.100   1.420         25       18   20640     0.7042     10.5294
+171.500   1.317         28       21   22080     0.7593     11.1765
+172.800   1.274         31       24   23520     0.7849     11.9412
+174.100   1.322         35       28   33600     0.7564     11.2381
+175.400   1.395         41       34   35520     0.7168     12.6190
+176.800   1.367         49       41   37920     0.7315     14.2857
+178.200   1.268         56       48   41760     0.7886     18.5882
+179.500   1.186         62       53   42720     0.8432     19.2353
+180.600   1.158         65       57   44640     0.8636     18.6471
+181.800   1.168         66       59   45120     0.8562     19.0000
+183.000   1.200         65       56   45120     0.8333     11.1765
+184.200   1.244         64       55   44640     0.8039     18.1765
+185.400   1.298         58       51   45120     0.7704     16.8824
+186.700   1.376         55       45   45600     0.7267     15.8824
+188.100   1.513         48       39   45120     0.6609     14.2941
+189.600   1.638         41       35   42720     0.6105     12.4706
+191.200   1.698         37       30   40320     0.5889     11.2353
+192.900   1.760         32       25   37440     0.5682     10.8235
+194.700   1.855         28       22   27360     0.5391     10.1765
+196.600   1.998         25       19   25920     0.5005     9.7059
+198.500   2.212         23       17   24000     0.4521     9.5882
+200.800   2.540         21       16   18720     0.3937     9.0588
+203.300   2.828         20       14   17760     0.3536     8.8235
+206.100   2.993         18       13   13440     0.3341     8.8235
+209.100   3.152         17       13   12480     0.3173     8.7647
+212.300   3.348         16       12   12000     0.2987     8.5294
+215.600   3.576         16       11   12000     0.2796     7.1765
+219.200   3.703         15       10    9120     0.2701     8.0588
+222.900   3.782         14       10    8640     0.2644     7.7059
+226.700   3.845         13        9    8640     0.2601     6.7059
+230.500   3.883         14        9    9600     0.2575     7.7647
+234.400   3.885         15       10    9600     0.2574     7.2353
+238.300   3.857         15       11    8640     0.2593     7.1429
+242.200   3.836         16       12    8640     0.2607     6.3333
+246.000   3.713         16       12    9120     0.2693     7.4762
+249.700   3.481         18       13   12000     0.2873     7.8095
+253.200   3.275         20       15   12480     0.3053     8.2381
+256.500   3.166         22       16   12480     0.3159     7.0952
+259.600   2.686         25       19   16800     0.3723     10.9412
+262.300   2.264         30       23   18240     0.4417     12.1765
+264.600   1.984         35       28   26400     0.5040     13.2353
+266.600   1.828         41       34   28800     0.5470     14.4118
+268.400   1.768         49       40   37920     0.5656     16.5882
+270.200   1.570         56       46   40800     0.6369     17.9412
+271.700   1.383         62       53   44640     0.7231     18.4118
+273.100   1.260         64       55   45120     0.7937     19.2353
+274.400   1.180         65       57   45120     0.8475     18.7059
+275.500   1.125         64       55   44640     0.8889     19.6471
+276.700   1.089         61       52   44640     0.9183     17.8235
+277.800   1.077         56       48   44160     0.9285     16.8235
+278.800   1.103         50       43   43680     0.9066     15.1176
+279.900   1.209         45       37   42240     0.8271     14.4706
+281.100   1.345         39       32   40320     0.7435     11.3810
+282.500   1.447         34       26   38400     0.6911     10.1429
+283.900   1.316         30       23   35040     0.7599     10.7059
+285.300   1.241         26       19   32160     0.8058     9.8235
+286.500   1.231         24       17   23040     0.8123     9.2353
+287.700   1.263         22       16   22080     0.7918     9.1765
+289.000   1.326         22       15   21120     0.7541     9.1765
+290.300   1.415         20       14   20640     0.7067     9.1176
+291.700   1.537         20       13   19680     0.6506     9.0000
+293.300   1.704         18       12   18240     0.5869     9.1176
+295.000   1.941         18       11   14400     0.5152     9.0000
+296.900   2.297         18       11   13920     0.4354     9.2353
+299.200   2.573         17       10   13440     0.3887     8.0000
+301.800   2.646         16       10   13440     0.3779     6.5714
+304.400   2.664         15        9   13440     0.3754     6.6667
+307.100   2.661         16       10   13440     0.3758     6.7143
+309.800   2.642         17       10   13440     0.3785     6.7619
+312.400   2.607         18       11   13440     0.3836     8.0476
+315.000   2.439         18       12   13440     0.4100     7.2857
+317.400   2.057         20       13   14400     0.4861     8.4762
+319.500   1.736         22       15   14880     0.5760     9.8235
+321.200   1.515         24       17   20640     0.6601     10.2353
+322.800   1.374         26       19   21600     0.7278     10.6471
+324.100   1.293         29       22   22560     0.7734     11.5882
+325.400   1.279         33       25   24960     0.7819     12.7059
+326.700   1.391         37       30   33600     0.7189     11.6667
+328.100   1.392         44       36   35520     0.7184     13.0952
+329.500   1.368         51       43   38880     0.7310     14.8095
+330.900   1.241         58       50   42720     0.8058     18.1765
+332.100   1.174         63       55   44160     0.8518     18.8824
+333.300   1.161         65       57   44640     0.8613     19.0588
+334.400   1.181         62       58   45120     0.8467     18.5882
+335.600   1.218         64       56   45600     0.8210     17.8824
+336.800   1.265         62       53   45600     0.7905     11.1176
+338.100   1.328         57       47   45600     0.7530     18.7059
+339.400   1.427         52       43   45120     0.7008     15.0000
+340.800   1.604         44       38   43200     0.6234     13.4706
+342.500   1.677         39       32   41280     0.5963     12.2353
+344.100   1.728         34       27   38880     0.5787     11.2941
+345.900   1.802         30       24   28320     0.5549     10.4706
+347.700   1.917         27       20   26400     0.5216     9.8824
+349.600   2.091         24       18   24480     0.4782     9.7647
+351.700   2.350         22       17   18720     0.4255     9.4118
+354.000   2.724         21       15   17280     0.3671     9.2353
+356.700   2.921         19       14   16800     0.3423     8.7647
+359.700   3.071         17       13   12480     0.3256     8.4706
+362.700   3.243         16       12   12480     0.3084     6.9412
+366.000   3.467         16       12   12000     0.2884     8.4118
+369.400   3.638         16       11   11520     0.2749     8.0588
+373.100   3.739         15       10    8640     0.2675     7.8824
+376.800   3.813         14        9    8640     0.2623     6.9412
+380.600   3.867         13        8    9120     0.2586     7.9412
+384.500   3.888         14        9    9600     0.2572     7.8235
+388.400   3.873         15       10    8640     0.2582     8.0588
+392.300   3.842         16       11    8640     0.2603     6.2381
+396.100   3.835         16       12    8640     0.2608     6.5238
+399.900   3.614         17       13    9120     0.2767     7.9048
+403.600   3.376         19       14   12000     0.2962     8.1905
+406.900   3.206         21       15   12480     0.3119     8.3333
+410.100   2.905         23       18   17280     0.3442     11.0588
+413.000   2.452         27       21   17760     0.4078     11.3529
+415.500   2.109         32       25   18720     0.4742     12.4118
+417.600   1.896         38       31   27360     0.5274     13.0588
+419.500   1.789         44       37   28800     0.5590     15.5294
+421.300   1.700         52       44   40800     0.5882     18.0000
+423.000   1.479         57       50   42720     0.6761     17.6471
+424.500   1.320         63       53   45120     0.7576     18.7059
+425.800   1.219         63       56   45600     0.8203     19.6471
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+928.000   1.738         21       15   14880     0.5754     9.8235
+929.700   1.517         23       17   20640     0.6592     10.1176
+931.200   1.377         26       19   21600     0.7262     10.4706
+932.600   1.298         29       22   22560     0.7704     11.4118
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+936.600   1.417         43       35   35520     0.7057     13.2381
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+944.200   1.216         62       53   45600     0.8224     18.0588
+945.400   1.264         57       51   45600     0.7911     10.8235
+946.700   1.329         55       46   44640     0.7524     15.3529
+948.000   1.434         50       41   44640     0.6974     9.7059
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+952.700   1.733         34       27   38400     0.5770     11.1765
+954.400   1.810         30       23   27360     0.5525     10.7059
+956.300   1.929         26       20   25920     0.5184     9.8824
+958.200   2.107         23       18   24000     0.4746     9.4706
+960.300   2.374         22       16   18720     0.4212     9.2353
+962.700   2.756         20       15   17280     0.3628     8.9412
+965.400   2.939         18       14   16800     0.3403     8.8824
+968.400   3.087         17       13   12480     0.3239     8.5882
+971.400   3.262         16       12   12480     0.3066     7.2941
+974.700   3.490         16       11   12000     0.2865     8.2941
+978.200   3.652         15       11   11520     0.2738     8.0000
+981.800   3.747         15       10    8640     0.2669     7.8824
+985.600   3.819         14        9    8640     0.2618     6.8824
+989.400   3.870         13        9    9120     0.2584     7.8824
+993.300   3.887         14        9    9600     0.2573     7.8824
+997.200   2.831         15       10    8640     0.3532     7.7647

main.tex

@@ -110,6 +110,7 @@
   \input{intro.tex}
 
   \input{compression.tex}
+
   \input{timestepping.tex}
 
   \input{conclusions.tex}

timestepping.tex

@@ -31,25 +31,32 @@
   \end{itemize}
 
   \only<4->{
-  \vspace{0.5em}
-  \begin{itemize}
-    \item Re-use Krylov subspace from previous time step?
-
-    \only<5->{
+  %\vspace{0.5em}
+  \begin{columns}
+    \column{0.65\textwidth}
     \begin{itemize}
-      \item Krylov subspace: ~~$X \leftarrow [b, ~A b, ~\cdots, ~A^{k\shortminus\!1} b]$
+      \item Re-use Krylov subspace from previous time step?
+
+      \only<5->{
+      \begin{itemize}
+        \item Krylov subspace: ~~$X \leftarrow [b, ~A b, ~\cdots, ~A^{k\shortminus\!1} b]$
 
-      \item Compute QR decomposition: ~ $QR \leftarrow AX$
+        \item Compute QR decomposition: ~ $QR \leftarrow AX$
 
-      \item Preconditioner: ~ $P \coloneq I - Q Q^{T} + X R^{-1} Q^{T}$
+        \item Preconditioner: ~ $P \coloneq I - Q Q^{T} + X R^{-1} Q^{T}$
 
-        \vspace{0.5em}
-      \item[] \qquad $P \, Ax = x$ \quad for all~~ $x \in span(X)$
-      \item[] \qquad ~\, $P \, y = y$  \quad for all~~ $y \perp span(X)$
+          \vspace{0.5em}
+        \item[] \qquad $P \, Ax = x$ \quad for all~~ $x \in span(X)$
+        \item[] \qquad ~\, $P \, y = y$  \quad for all~~ $y \perp span(X)$
 
+      \end{itemize}
+      }
     \end{itemize}
-    }
-  \end{itemize}}
+    \column{0.35\textwidth}
+      \centering
+      \only<6->{\bf Similar to \\
+      subspace deflation}
+  \end{columns}}
 
   % TODO: {Similar idea to subspace deflation}
 
@@ -168,68 +175,44 @@
 
 \begin{FIframe}{Numerical Results}{} %<<< disc-chain
 
-  Iteration counts with and without Krylov preconditioning -- disc-chain
-
-  %\vspace{-2.0em}
-  %\begin{center}
-  %\resizebox{0.85\textwidth}{!}{\begin{tikzpicture}
-  %  \pgfplotsset{
-  %    xmin=0, xmax=250,
-  %    width=12cm, height=7cm,
-  %    xlabel={$T$}, xtick distance=50,
-  %  }
-  %  \begin{axis}[ymin=0, ymax=110, ylabel={$N_{iter}$}, legend style={draw=none,at={(0,1)},anchor=north west}]
-  %    \addplot [thick,color=blue] table [x={t},y={noprecond}] {data/sed2}; \addlegendentry{no-preconditioner};
-  %    \addplot [thick,color=red] table [x={t},y={precond}] {data/sed2}; \addlegendentry{block-preconditioner};
-  %    %\addplot [thick,color=green] table [x={t},y={KSPprecond}] {data/sed2}; \addlegendentry{Krylov-preconditioner};
-  %  \end{axis}
-
-  %  \begin{axis}[axis y line*=right, ymin=0, ymax=65000, ylabel={$N$}, legend style={draw=none,at={(0.97,0.97)},anchor=north east}]
-  %    \addplot [thick,dashed,color=black] table [x={t},y={N}] {data/sed2}; \addlegendentry{$N$};
-  %  \end{axis}
-  %\end{tikzpicture}}
-  %\end{center}
-
-  %\vspace{-1em}
-  %{\bf Close-to-touching:} ~~smaller time-steps, ~~more unknowns ($N$), \\
-  %high GMRES iteration count (one-body preconditioner doesn't help). \\
-  %{\color{red} $\sim 125 \times$ more expensive!}
-\end{FIframe} %>>>
-
+  \centering
+  {\bf Average iterations per GMRES solve: 30 \quad (130 without Krylov preconditioner)}
 
-\begin{FIframe}{High Order vs Low Order Time Stepping}{} %<<<
+  \embedvideo{\includegraphics[width=0.985\textwidth]{videos/chain32}}{videos/chain32.avi}
 
-  TODO: compare high order vs low order
+  {\bf 32 discs (36K unknowns)}, \hfill {\bf minimum distance: 1e-4}, \hfill {\bf 10-th order SDC}
 
-  TODO: explain Krylov preconditioner
+  \vspace{1em}
+  {\bf 7-digits accuracy} in quadratures, GMRES solve, and time-stepping.
 
 \end{FIframe} %>>>
 
 \begin{FIframe}{Numerical Results - Sedimentation Flow}{} %<<<
 
-  \vspace{-2.0em}
-  \begin{center}
-  \resizebox{0.95\textwidth}{!}{\begin{tikzpicture}
-    \pgfplotsset{
-      xmin=0, xmax=250,
-      width=12cm, height=7cm,
-      xlabel={$T$}, xtick distance=50,
-    }
-    \begin{axis}[ymin=0, ymax=110, ylabel={$N_{iter}$}, legend style={draw=none,at={(0,1)},anchor=north west}]
-      \addplot [thick,color=blue] table [x={t},y={noprecond}] {data/sed2}; \addlegendentry{no-preconditioner};
-      \addplot [thick,color=red] table [x={t},y={precond}] {data/sed2}; \addlegendentry{block-preconditioner};
-      \addplot [thick,color=DarkGreen] table [x={t},y={KSPprecond}] {data/sed2}; \addlegendentry{Krylov-preconditioner};
-    \end{axis}
-
-    \begin{axis}[axis y line*=right, ymin=0, ymax=65000, ylabel={$N$}, legend style={draw=none,at={(0.97,0.97)},anchor=north east}]
-      \addplot [thick,dashed,color=black] table [x={t},y={N}] {data/sed2}; \addlegendentry{$N$};
-    \end{axis}
-  \end{tikzpicture}}
-  \end{center}
-
-  %\vspace{-1em}
-  %{\bf Close-to-touching:} ~~smaller time-steps, ~~more unknowns ($N$), \\
-  %high GMRES iteration count (block preconditioner doesn't help). \\
-  %{\color{red} $\sim 125 \times$ more expensive!}
+  \vspace{-1.75em}
+  \begin{columns}
+    \column{0.42\textwidth}
+
+      {\bf 200K unknowns}\\
+      127 discs.
+
+      \vspace{1em}
+      {\bf Minimum distance: 1e-4}
+
+      \vspace{1em}
+      {\bf 10-th order adaptive SDC}
+
+      \vspace{1em}
+      {\bf 5-digits accuracy} in quadratures, GMRES solve, and time-stepping.
+
+    \column{0.583\textwidth}
+      \embedvideo{\includegraphics[width=0.99\textwidth]{videos/sed127_2}}{videos/sed127__.mp4}
+  \end{columns}
+
 \end{FIframe} %>>>
 
+%\begin{FIframe}{High Order vs Low Order Time Stepping}{} %<<<
+%  TODO: compare high order vs low order
+%  TODO: explain Krylov preconditioner
+%\end{FIframe} %>>>
+