Dynamical approximation and sensor placement for filtering problems

Abstract

We consider the inverse problem of reconstructing an unknown function $u$ from a finite set of measurements, under the assumption that $u$ is the trajectory of a transport-dominated problem with unknown input parameters. We propose an algorithm based on the Parameterized Background Data-Weak method (PBDW) where dynamical sensor placement is combined with approximation spaces that evolve in time. We prove that the method ensures an accurate reconstruction at all times and allows to incorporate relevant physical properties in the reconstructed solutions by suitably evolving the dynamical approximation space. As an application of this strategy we consider Hamiltonian systems modeling wave-type phenomena, where preservation of the geometric structure of the flow plays a crucial role in the accuracy and stability of the reconstructed trajectory.

Figures (13)

• Figure 1: 1D Schrödinger equation. Real part $q(x,t)$ (dotted) and imaginary part $p(x,t)$ (solid) of the high-fidelity solution at times $t=0$, $t=5$, $t=15$ and $t=20$ for $\theta=(1.04,1.04)$.
• Figure 2: 1D Schrödinger equation. Static local averages of width $\sigma=0.1$, $m=6$. Left: $x$ coordinate of the sensors over time; the red dashed line denotes the position of the main hump of the high-fidelity solution $\lvert\psi(t,\theta)\rvert$ for $\theta=(1.04,1.04)$. Center: evolution of the inf-sup constant $\beta$. Right: evolution of the numerical errors with $\lvert\Theta_s\rvert=81$ test parameters.
• Figure 3: 1D Schrödinger equation. Dynamic local averages of width $\sigma=0.1$, $m=6$. Left: $x$ coordinate of the sensors over time; the red dashed line denotes the position of the main hump of the high-fidelity solution $\lvert\psi(t,\theta)\rvert$ for $\theta=(1.04,1.04)$. Center: evolution of the inf-sup constant $\beta$. Right: evolution of the numerical errors with $\lvert\Theta_s\rvert=81$ test parameters.
• Figure 4: 1D Schrödinger equation. Numerical solution of the high-fidelity model corresponding to $\theta=(1.0933,1.0933)$ and comparison with the statically and dynamically reconstructed solutions at the final time. For the sake of visualization, only the interval $[10,30]\subset D$ is shown.
• Figure 5: 1D Schrödinger equation. Left: evolution of the Hamiltonian for the test parameter $\theta=(1.0267,0.9867)$. Center: Hamiltonian approximation error \ref{['eq:eHam']}. Right: error \ref{['eq:dHam']} in the conservation of the Hamiltonian. Comparison between the high-fidelity solution and the reconstructed solution obtained with $m=6$ sensors in the static and dynamic case.