Variational data assimilation for the wave equation in heterogeneous media: Numerical investigation of stability

Abstract

In recent years, several numerical methods for solving the unique continuation problem for the wave equation in a homogeneous medium with given data on the lateral boundary of the space-time cylinder have been proposed. This problem enjoys Lipschitz stability if the geometric control condition is fulfilled, which allows devising optimally convergent numerical methods. In this article, we investigate whether these results carry over to the case in which the medium exhibits a jump discontinuity. Our numerical experiments suggest a positive answer. However, we also observe that the presence of discontinuities in the medium renders the computations far more demanding than in the homogeneous case.

Figures (7)

• Figure 1: For the approximation of \ref{['eq:1D:exact:simple']} with polynomial degree $k \in \{2,3\}$ and contrast $c_1 = 2.5$ we measure the errors $\Vert u-L_{\Delta t} \ul_1 \Vert_{L^\infty(0,T;L^2(\Omega_R))}$ (left) and $\Vert \partial_t (u-L_{\Delta t} \ul_1) \Vert_{L^2(0,T;L^2(\Omega_R))}$ (right) and compare to the respective error of the $L^2$-best approximation (dashed). For final time $T = 0.5 > 0.35$ (top), we observe optimal convergence rates, while for $T = 0.1 < 0.35$ (bottom) the order of convergence degrades.
• Figure 2: For $k = 3$ and $L = 3$, we consider the errors $\Vert \partial_t (u-L_{\Delta t} \ul_1) \Vert_{L^2(0,T;L^2(\Omega_R))}$ (left) and $\Vert u-L_{\Delta t} \ul_1 \Vert_{L^\infty(0,T;L^2(\Omega_R))}$ (right) and the respective $L^2$-best approximation errors (dashed) for increasing contrast $c_1 \in \{1.0,1.5,...,4.5\}$. We compare the case where $T = 0.5$ is fixed for all choices of $c_1$ and the case where $T$ adapted such that it barely fulfills $T > 0.25(1+c_1^{-1})$.
• Figure 3: At time $t = 0.25$, we compare the exact solution \ref{['eq:1D:exact:simple']} and approximations with polynomial degree $k = 3$ on refinement levels $L \in \{1,2,3,4 \}$ for contrasts $c_1 = 2.5$ (left) and $c_1= 5.5$ (right). The scale in the right plot indicates the mesh size $h = 1/2^{L+1}$ for the different refinement levels.
• Figure 4: Exact solution \ref{['eq:1D:exact:simpleMult']} and approximated solution with $k = 3$ of \ref{['eq:1D:exact:simpleMult']} at $t = 0.25$ with contrast $c_1 = 7.5$ (left) and $c_1 = 11.5$ (right). We compare the approximations on the refinement levels $L = 3$ and $L = 4$, and set $\Delta t = 1/32$ for both cases.
• Figure 5: For fixed time step $\Delta t = 1/32$, refinement level $L = 4$, and polynomial degree $k = 3$, we compare the exact solution \ref{['eq:1D:exact:simpleMult']} and approximated solution computed with final time$T = 1.0$ and $T = 0.5$ at time $t = 0.5$. We set $c_2 = 1.0$ and consider the cases $c_1 = c_3 = 2.5$, $n = 1$ (left) and $c_1 = c_3 = 7.5$, $n = 3$ (right).

Theorems & Definitions (17)

• Theorem 1.1
• proof
• Theorem 1.2: Thm. 5.19 of Filippas22
• Remark 1
• Lemma 3.1
• proof
• Corollary 3.2
• Lemma 3.3: Continuity of $A$
• proof
• Theorem 3.4