On the Frechet Root Kernel of Certain Wave Equations

Abstract

We extend the adjoint method to complex-valued PDEs and introduce the Fréchet root sensitivity kernel, as the most fundamental kernel from which all other material-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 Schroedinger equation with zero potential). We compute and analyze the Frechet root kernels for all four PDEs and show that, for constant material parameters, the kernel exhibits a consistent structure across systems, while its instantaneous form propagates as a wave whose shape depends on the initial conditions. For the Schroedinger equation, we find an especially notable result: the integrand of the Frechet root kernel coincides with the Born rule of quantum mechanics, suggesting that the probabilistic interpretation of the wavefunction may arise naturally from a general sensitivity-analysis framework rather than from an independent postulate. Our results establish a unified approach to sensitivity analysis for real- and complex-valued PDEs, provide a new perspective on the origin of the Born rule.

Figures (7)

• Figure 1: (a) Gaussian pulse (eq. \ref{['eq.source_time_function_Gaussian']}) centered at $t_0=2$s and with a dominant period of 2s ($f_0=0.5$Hz). (b) The real and imaginary parts of the oscillatory term of the modulated Gaussian pulse (imaginary exponential in eq. \ref{['eq.complex_source_time_function_Gaussian']}) with the same dominant frequency of $f_0=0.5$Hz. (c) Real and (d) Imaginary parts of the modulated Gaussian pulse (eq. \ref{['eq.complex_source_time_function_Gaussian']}).
• Figure 2: (a) and (b): Snapshots of the forward $u$ and adjoint $u^{\dagger}$ wavefield interactions $(uu^{\dagger})$ considering two different source time functions: (a) first-order and (b) second-order derivatives of a Gaussian (eq. \ref{['eq.source_time_function_Gaussian']}). (c) and (d): Fréchet root kernels (eq. \ref{['eq.1D_Root_Kernel']}) corresponding to two different source time functions: (c) the first-order and (d) the second-order derivative of a Gaussian. Source and receiver locations are displayed by a star and a triangle respectively.
• Figure 3: Two-dimensional sketch of lithospheric bending due to the weight of a mountain and transient deformation produced by an earthquake.
• Figure 4: Euler-Bernoulli beam equation: (a) Snapshot of the forward $u$ and adjoint $u^{\dagger}$ wavefield interaction $(uu^{\dagger})$ at the specified time. (b) Corresponding Fréchet root kernel (eq. \ref{['eq.1D_Root_Kernel']}). Source and receiver locations are displayed by a star and a triangle respectively.
• Figure 5: Snapshots of the forward $\textbf{u}$ and adjoint $\textbf{u}^{\dagger}$ wavefield interactions $(\textbf{u}\textbf{u}^{\dagger})$ considering two different source time functions for the adjoint wavefield: (a) a Dirac delta pulse at the arrival time of the forward wavefield at the receiver location and (b) the complex conjugate $\bar{\textbf{u}}$ of the forward wavefield. Source and receiver locations are displayed by a star and a triangle respectively.