Summary: We present a multigrid algorithm for the solution of source identification inverse problems constrained by variable-coefficient linear parabolic partial differential equations. We consider problems in which the inversion variable is a function of space only. We consider the case of \(L^2\) Tikhonov regularization. The convergence rate of our algorithm is mesh-independent — even in the case of no regularization. This feature makes the method algorithmically robust to the value of the regularization parameter, and thus useful for the cases in which we seek high-fidelity reconstructions.
The inverse problem is formulated as a partial differential equation (PDE)-constrained optimization. We use a reduced-space approach in which we eliminate the state and adjoint variables, and we iterate in the inversion parameter space using conjugate gradients. We precondition the Hessian with a V-cycle multigrid scheme. The multigrid smoother is a two-step stationary iterative solver that inexactly inverts an approximate Hessian by iterating exclusively in the high-frequency subspace (using a high-pass filter). We analyze the performance of the scheme for the constant coefficient case with full observations; we analytically calculate the spectrum of the reduced Hessian and the smoothing factor for the multigrid scheme.
The forward and adjoint problems are discretized using a backward-Euler finite-difference scheme. The overall complexity of our inversion algorithm is \(\mathcal{O}(N_tN+N\log^2N)\), where \(N\) is the number of grid points in space and \(N_t\) is the number of time steps. We provide numerical experiments that demonstrate the effectiveness of the method for different diffusion coefficients and values of the regularization parameter. We also provide heuristics, and we conduct numerical experiments for the case with variable coefficients and partial observations. We observe the same complexity as in the constant-coefficient case. Finally, we examine the effectiveness of using the reduced-space solver as a preconditioner for a full-space solver.