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Preconditioned all-at-once methods for large, sparse parameter estimation problems. (English) Zbl 0995.65110

The inverse problem about recovering a parameter function by measurements of solutions of the system partial differential equations is considered. A typical formulation of this inverse problem consists minimization of the sum of a data fitting error term and a regularization term, subject to the forward problem being satisfied. The problem is typically ill-posed without regularization and it is ill-conditioned with it, since the regularization term is aimed at removing noise without overshadowing the data.

Let the forward problem be a linear elliptic differential equation $A\left(m\right)u=q$ where $A$ refers a differential operator depending on a parameter vector function $m$, defined on an appropriate domain and equipped with suitable boundary conditions. The discrimination of this problem is studied and for regularization the Tikhonov method with introducing the Lagrangian approach is applied. Finally the problem is numerically solved by the Gauss-Newton method, and a preconditioned conjugate gradient algorithm is applied at each iteration for the resulting reduced Hessian system. Alternatively, a preconditioned Krylov method is applied to arising system.

The considered problem the arises in many applications. The results are illustrated by different computational examples.

MSC:
 65N21 Inverse problems (BVP of PDE, numerical methods) 35R30 Inverse problems for PDE 65N06 Finite difference methods (BVP of PDE) 65F10 Iterative methods for linear systems 65F35 Matrix norms, conditioning, scaling (numerical linear algebra)
KELLEY