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1. Target signatures for thin surfaces

   https://doi.org/10.1088/1361-6420/ac4154  

   Cakoni, Fioralba; Monk, Peter; Zhang, Yangwen (January 2022, Inverse Problems)

   Abstract We investigate an inverse scattering problem for a thin inhomogeneous scatterer in R m , m = 2, 3, which we model as an m − 1 dimensional open surface. The scatterer is referred to as a screen. The goal is to design target signatures that are computable from scattering data in order to detect changes in the material properties of the screen. This target signature is characterized by a mixed Steklov eigenvalue problem for a domain whose boundary contains the screen. We show that the corresponding eigenvalues can be determined from appropriately modified scattering data by using the generalized linear sampling method. A weaker justification is provided for the classical linear sampling method. Numerical experiments are presented to support our theoretical results. 

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2. A spectral target signature for thin surfaces with higher order jump conditions

   https://doi.org/10.3934/ipi.2022020  

   Cakoni, Fioralba; Lee, Heejin; Monk, Peter; Zhang, Yangwen (January 2022, Inverse Problems and Imaging)

   In this paper we consider the inverse problem of determining structural properties of a thin anisotropic and dissipative inhomogeneity in \begin{document}$$ {\mathbb R}^m $$\end{document}, \begin{document}$ m = 2, 3 $$\end{document} from scattering data. In the asymptotic limit as the thickness goes to zero, the thin inhomogeneity is modeled by an open \begin{document}$$ m-1 $$\end{document}$ dimensional manifold (here referred to as screen), and the field inside is replaced by jump conditions on the total field involving a second order surface differential operator. We show that all the surface coefficients (possibly matrix valued and complex) are uniquely determined from far field patterns of the scattered fields due to infinitely many incident plane waves at a fixed frequency. Then we introduce a target signature characterized by a novel eigenvalue problem such that the eigenvalues can be determined from measured scattering data, adapting the approach in [20]. Changes in the measured eigenvalues are used to identified changes in the coefficients without making use of the governing equations that model the healthy screen. In our investigation the shape of the screen is known, since it represents the object being evaluated. We present some preliminary numerical results indicating the validity of our inversion approach 

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3. Computational approaches for extremal geometric eigenvalue problems

   C.-Y. Kao; B. Osting; and E ́. Oudet (January 2023, Handbook of numerical analysis)

   In an extremal eigenvalue problem, one considers a family of eigenvalue problems, each with discrete spectra, and extremizes a chosen eigenvalue over the family. In this chapter, we consider eigenvalue problems defined on Riemannian manifolds and extremize over the metric structure. For example, we consider the problem of maximizing the principal Laplace–Beltrami eigenvalue over a family of closed surfaces of fixed volume. Computational approaches to such extremal geometric eigenvalue problems present new computational challenges and require novel numerical tools, such as the parameterization of conformal classes and the development of accurate and efficient methods to solve eigenvalue problems on domains with nontrivial genus and boundary. We highlight recent progress on computational approaches for extremal geometric eigenvalue problems, including (i) maximizing Laplace–Beltrami eigenvalues on closed surfaces and (ii) maximizing Steklov eigenvalues on surfaces with boundary. 

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4. Sum rules for energy deposition eigenchannels in scattering systems

   https://doi.org/10.1364/OL.468697  

   Yamilov, Alexey; Bender, Nicholas; Cao, Hui (September 2022, Optics Letters)

   In a random-scattering system, the deposition matrix maps the incident wavefront onto the internal field distribution across a target volume. The corresponding eigenchannels have been used to enhance the wave energy delivered to the target. Here, we find the sum rules for the eigenvalues and eigenchannels of the deposition matrix in any system geometry: including two- and three-dimensional scattering systems, as well as narrow waveguides and wide slabs. We derive a number of constraints on the eigenchannel intensity distributions inside the system as well as the corresponding eigenvalues. Our results are general and applicable to random systems of arbitrary scattering strength as well as different types of waves including electromagnetic waves, acoustic waves, and matter waves. 

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5. Symmetric Stair Preconditioning of Linear Systems for Parallel Trajectory Optimization

   https://doi.org/10.1109/ICRA57147.2024.10610386  

   Bu, Xueyi; Plancher, Brian (May 2024, IEEE)

   There has been a growing interest in parallel strategies for solving trajectory optimization problems. One key step in many algorithmic approaches to trajectory optimization is the solution of moderately-large and sparse linear systems. Iterative methods are particularly well-suited for parallel solves of such systems. However, fast and stable convergence of iterative methods is reliant on the application of a high-quality preconditioner that reduces the spread and increase the clustering of the eigenvalues of the target matrix. To improve the performance of these approaches, we present a new parallel-friendly symmetric stair preconditioner. We prove that our preconditioner has advantageous theoretical properties when used in conjunction with iterative methods for trajectory optimization such as a more clustered eigenvalue spectrum. Numerical experiments with typical trajectory optimization problems reveal that as compared to the best alternative parallel preconditioner from the literature, our symmetric stair preconditioner provides up to a 34% reduction in condition number and up to a 25% reduction in the number of resulting linear system solver iterations. 

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