Inverse problem in Heat PDE
Governing PDE: \[ u_t = \nabla\cdot(a(x) \nabla u(x)), \] where \(a(x)\) means the physical heat conductivity.
Spectral methods have been adpoted to solve this with final time \(t=T\) and dense observations [Boumenir (2023)].
We further simplify the PDE to be \[ u_t = a(x) u_{xx}, \\ u(t=0,x)=u_0(x), \] with periodic boundary condition \(x=1\sim x=-1\).
Finite difference (Forward-Euler in time, central difference in space) \[ \frac{u(t+\Delta t, x) - u(t,x)}{\Delta t} = a(x)\frac{u(t, x+\Delta x) - 2u(t,x) + u(t,x-\Delta x)}{\Delta x^2} \]
Naive recovery on dense observation
\[ a(x) = \frac{\Delta x^2}{\Delta t} \cdot \frac{u(t+\Delta t, x) - u(t,x)}{u(t, x+\Delta x) - 2u(t,x) + u(t,x-\Delta x)} \]