A preconditioner for least-squares distributed parameter estimation (Q2744628)

A preconditioner for a system of linear equations \(( {G + \alpha A})q = b\) with a large dimensional, symmetric, dense and ill-conditioned matrix \(G\) and invertible \(A\) is developed. Such systems arise in connection with inverse problems for partial differential equations (PDEs) which are solved by the Tikhonov stabilization method. The matrix \(A\) differs from the unit one when a stabilizing functional is the total variation of \(q\). The suggested preconditioner is \(B = ( {H + \alpha A})^{ - 1}\) where \(H\) is a low-rank approximation of \textit{G} created from the information accumulated during the auxiliary course of the preconditioned conjugate gradient (PCG) iterations with \(B = A^{-1}\). NEWLINENEWLINENEWLINEAfter several iterations of the PCG a sequence of \(M\) pairs of vectors \(( {s_{j} ,y_{j}})\) related by \(y_{j} = Gs_{j} \) is generated. This sequence defines matrices \(S = [ {s_{1} ,\dots ,s_{M}} ]\), \(Y = [ {y_{1} ,\dots ,y_{M}} ]\), \(H = Y( {Y^{T}S})^{ - 1}Y^{T}\). It is shown that the matrix \(H\) with rank equal to \(M\), is an approximation of \(G\). An upper estimation of the spectral condition number of \(B( {G + \alpha A})\) as well as an effective algorithm for the calculation of \(H\) is presented. Theoretical results are demonstrated on a numerical example for a parabolic PDE (coefficient inverse problem).