Abstract
Many imaging modalities, such as ultrasound and radar, rely heavily on the ability to accurately model wave propagation. In most applications, the response of an object to an incident wave is recorded and the goal is to characterize the object in terms of its physical parameters (e.g., density or soundspeed). We can cast this as a joint parameter and state estimation problem. In particular, we consider the case where the inner problem of estimating the state is a weakly constrained data-assimilation problem. In this paper, we discuss a numerical method for solving this variational problem.
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Notes
- 1.
This requires that the solution of \(\mathcal {L}u = q\) can be bounded point-wise as \(|u(t,x)| \leq C \|q\|{ }_{L^2}\). While this is possible in general for d = 1, it perhaps requires more regularity of the source function for d > 1.
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Acknowledgements
The first author acknowledges W.W. Symes for pointing out some issues with the RKHS framework for wave-equations in d > 1 dimensions.
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van Leeuwen, T., van Leeuwen, P.J., Zhuk, S. (2021). Data-Driven Modeling for Wave-Propagation. In: Vermolen, F.J., Vuik, C. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2019. Lecture Notes in Computational Science and Engineering, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-030-55874-1_67
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