Stable explicit stepwise marching scheme in ill-posed time-reversed 2D Burgers' equation

Figures & data

Figure 1. Using precomputed input data at time $T=2.5\times {10}^{-3}$, shown in middle column, nonlinear explicit scheme in Equation (8(8) $\begin{array}{rl}{\tilde{u}}^{n+1}& =S{\tilde{u}}^n+\mathrm{\Delta }tS{L}_1{\tilde{u}}^n,\\ {\tilde{v}}^{n+1}& =S{\tilde{v}}^n+\mathrm{\Delta }tS{L}_2{\tilde{v}}^n,n=0,1,\dots ,N.\end{array}$(8) ), run backward in time, seeks to recover true initial data shown in leftmost column. Actually recovered data are shown in rightmost column. Note that maximum values in middle column are much larger than in leftmost column, and maximum values in the rightmost column are slightly smaller than in leftmost column.

Figure 2. Using precomputed input data at time $T=2.5\times {10}^{-4}$, shown in middle column, nonlinear explicit scheme in Equation (8(8) $\begin{array}{rl}{\tilde{u}}^{n+1}& =S{\tilde{u}}^n+\mathrm{\Delta }tS{L}_1{\tilde{u}}^n,\\ {\tilde{v}}^{n+1}& =S{\tilde{v}}^n+\mathrm{\Delta }tS{L}_2{\tilde{v}}^n,n=0,1,\dots ,N.\end{array}$(8) ), run backward in time, seeks to recover the true images shown in leftmost column. Actually recovered mages are shown in rightmost column. Notice severe wavy distortion of sea wall in blurred Sydney Opera House image, shown at bottom in middle column, and its successful reconstruction in bottom rightmost image.

Figure 3. Backward recovery of the underlying intensity data that generate the images in the reconstruction experiment shown in Figure 2.

Figure 4. Using precomputed input data at time $T=2.5\times {10}^{-4}$, shown in middle column, nonlinear explicit scheme in Equation (8(8) $\begin{array}{rl}{\tilde{u}}^{n+1}& =S{\tilde{u}}^n+\mathrm{\Delta }tS{L}_1{\tilde{u}}^n,\\ {\tilde{v}}^{n+1}& =S{\tilde{v}}^n+\mathrm{\Delta }tS{L}_2{\tilde{v}}^n,n=0,1,\dots ,N.\end{array}$(8) ), run backward in time, seeks to recover the true images shown in leftmost column. Actually recovered images are shown in rightmost column.

Figure 5. Backward recovery of the underlying intensity data that generate the images in the reconstruction experiment shown in Figure 4.