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arXiv · 2512.09609

On the Fr\'echet interaction density of certain wave equations

Rafael Abreu·Chahana Nagesh
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Abstract

We extend the adjoint method to complex-valued PDEs and introduce the \emph{Fr\'echet interaction density}, as the most fundamental interaction from which Fr\'echet sensitivity kernels can be derived. We apply this framework to four representative equations: two real-valued PDEs (the second-order wave equation and the Euler--Bernoulli beam equation) and two complex-valued PDEs (the complex transport equation and the Schr\"odinger equation with zero potential). We compute and analyze the Fr\'echet interaction densities for all four PDEs and show that the interaction shows consistent structure, with a waveform that depends on the initial conditions. For the Schr\"odinger equation, when the adjoint field is chosen as the complex conjugate of the forward wavefunction, the interaction density reduces algebraically to the Born probability density. Our results establish a unified approach to sensitivity analysis for real- and complex-valued PDEs.

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Rafael Abreu, Chahana Nagesh. 2025-12-10. On the Fr\'echet interaction density of certain wave equations. https://arxiv.org/abs/2512.09609

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