Multigrid Algorithms for Inverse Problems with Linear Parabolic PDE Constraints

We present a multigrid algorithm for the solution of distributed parameter 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; for stability we use an L2 Tikhonov regularization. The main feature of our algorithm is that its convergence rate 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 a high-fidelity reconstruction. The problem is formulated as a PDE-constrained optimization. We use a reduced space approach. We eliminate the state and adjoint variables and we iterate in the inversion parameter space using Conjugate Gradients. We precondition 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 highpass 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 isO(NtN +N log N), whereN is number of grid points in space andNt 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 conduct numerical experiments for the case with variable coefficients, and partial observations. We observe the same complexity as in the constant-coefficient case. Finally, to avoid exact forward and adjoint solves far from the minimum, we combine the reduced-space algorithm with a full-space method.