Preconditioned all-at-once methods for large, sparse parameter estimationproblems

Abstract

The problem of recovering a parameter function based on measurements ofsolutions of a system of partial differential equations in several space variablesleads to a number of computational challenges. Upon discretization of aregularized formulation a large, sparse constrained optimization problem isobtained. Typically in the literature, the constraints are eliminated and theresulting unconstrained formulation is solved by some variant of Newton’smethod, usually the Gauss–Newton method. A preconditioned conjugate gradientalgorithm is applied at each iteration for the resulting reduced Hessiansystem.Alternatively, in this paper we apply a preconditioned Krylov method directly tothe KKT system arising from a Newton-type method for the constrainedformulation (an ‘all-at-once’ approach). A variant of symmetric QMR isemployed, and an effective preconditioner is obtained by solving the reducedHessian system approximately. Since the reduced Hessian system presentssignificant expense already in forming a matrix–vector product, the savings indoing so only approximately are substantial. The resulting preconditionermay be viewed as an incomplete block-LU decomposition, and we obtainconditions guaranteeing bounds for the condition number of the preconditionedmatrix.Numerical experiments are performed for the dc-resistivity and the magnetostaticproblems in 3D, comparing the two approaches for solving the linear system ateach Gauss–Newton iteration. A substantial efficiency gain is demonstrated. Therelative efficiency of our proposed method is even higher in the context of inexactNewton-type methods, where the linear system at each iteration is solved lessaccurately.