Source identification from line integral measurements and simple atmospheric models (Q1951326)

Mathematically speaking, this paper concerns the problem of reconstruction of the initial condition \[ u(x,0) = u_0(x) \] in some domain \(\Omega\), where \(u(x,t)\) satisfies the advection-diffusion equation \[ u_t = \nabla\cdot (a(x) \nabla u) - \psi(x)\cdot \nabla u \] with Dirichlet or Neumann boundary conditions. Here \(a(x)\) and \(\psi(x)\) are assumed to be given. The data used to reconstruct the initial condition are the line integrals of the solution at a given time \(t = T\) along some lines \(l_i\), \(i = 1, \dots, m\): \[ b_i = \int_{l_i} u(x,T) ds. \] The authors consider the cases when the intial condition are given as single or mutliple delta functions, i.e., the case when the solution \(u\) is generated by point sources. A least squares approach with adjoint problem is used to solve the problem. Several numerical examples with simulated data are presented.