Abstract
Regularization robust preconditioners for PDE-constrained optimization problems have been successfully developed. These methods, however, typically assume observation data and control throughout the entire domain of the state equation. For many inverse problems, this is an unrealistic assumption. In this paper we propose and analyze preconditioners for PDE-constrained optimization problems with limited observation data, e.g. observations are only available at the boundary of the solution domain. Our methods are robust with respect to both the regularization parameter and the mesh size. That is, the condition number of the preconditioned optimality system is uniformly bounded, independently of the size of these two parameters. The method does, however, require extra regularity. We first consider a prototypical elliptic control problem and thereafter more general PDE-constrained optimization problems. Our theoretical findings are illuminated by several numerical results.
Highlights
Consider the model problem: min f, u 1 2 u−d + α 2 f (1)on a Lipschitz domain Ω ⊂ Rn, subject to−Δu + u + f = 0 in Ω, (2) ∂u ∂n = on ∂Ω. (3)
Parameter robust preconditioners for PDE-constrained optimization problems have been successfully developed, provided that observation data is available throughout the entire domain of the state equation
We have explored the possibility for constructing robust preconditioners for PDE-constrained optimization problems with limited observation data
Summary
For cases with limited observations, for example with cost-functionals of the form (1), efficient preconditioners are available for a rather large class of PDEconstrained optimization problems, see [10,11] These techniques do not yield convergence rates, for the preconditioned KKT-system, that are completely robust with respect to the size of the regularization parameter α. The purpose of solving an inverse problem is typically to use data recorded at the surface of an object to compute internal properties of that object: Impedance tomography, the inverse problem of electrocardiography (ECG), computerized tomography (CT), etc This fact, combined with the discussion above, motivate the need for further improving numerical methods for solving KKT systems arising in connection with PDE-constrained optimization.
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