Multigrid Algorithms for Inverse Problems with Linear Parabolic PDE Constraints

Abstract. 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 ...

• Type: Journal
• Year: 2008
• Where: SIAMSC
• Authors: Santi S. Adavani, George Biros
• Keywords: Multigrid, Parameter Inverse Problems, Regularization Parameter, SIAMSC 2008