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We develop a linearized boundary control method for the inverse boundary value problem of determining a density in the acoustic wave equation. The objective is to reconstruct an unknown perturbation in a known background density from the linearized Neumann-to-Dirichlet map. A key ingredient in the derivation is a linearized Blagoves̆c̆enskiĭ’s identity with a free parameter. When the linearization is at a constant background density, we derive two reconstructive algorithms with stability estimates based on the boundary control method. When the linearization is at a non-constant background density, we establish an increasing stability estimate for the recovery of the density perturbation. The proposed reconstruction algorithms are implemented and validated with several numerical experiments to demonstrate the feasibility.

 

Abstract We develop a linearized boundary control method for the inverse boundary value problem of determining a potential in the acoustic wave equation from the Neumann-to-Dirichlet map. When the linearization is at the zero potential, we derive a reconstruction formula based on the boundary control method and prove that it is of Lipschitz-type stability. When the linearization is at a nonzero potential, we prove that the problem is of Hölder-type stability in two and higher dimensions. The proposed reconstruction formula is implemented and evaluated using several numerical experiments to validate its feasibility. 

Many recently developed classes of wireless, skin‐interfaced bioelectronic devices rely on conventional thermoset silicone elastomer materials, such as poly(dimethylsiloxane) (PDMS), as soft encapsulating structures around collections of electronic components, radio frequency antennas and, commonly, rechargeable batteries. In optimized layouts and device designs, these materials provide attractive features, most prominently in their gentle, noninvasive interfaces to the skin even at regions of high curvature and large natural deformations. Past studies, however, overlook opportunities for developing variants of these materials for multimodal means to enhance the safety of the devices against failure modes that range from mechanical damage to thermal runaway. This study presents a self‐healing PDMS dynamic covalent matrix embedded with chemistries that provide thermochromism, mechanochromism, strain‐adaptive stiffening, and thermal insulation, as a collection of attributes relevant to safety. Demonstrations of this materials system and associated encapsulation strategy involve a wireless, skin‐interfaced device that captures mechanoacoustic signatures of health status. The concepts introduced here can apply immediately to many other related bioelectronic devices.