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A flexible Krylov solver for shifted systems with application to oscillatory hydraulic tomography. (English) Zbl 1288.65041

In the direct oscillatory hydraulic tomography (OHT) one has to solve a number of systems \((K+\sigma_j M)x_j=b\), \(j=1,\dots,n_f\). It is proposed to solve this by an Arnoldi method with right preconditioner of the form \((K+\tau M)^{-1}\). Flexible choices \(\tau_k\) for \(\tau\) are implemented as in the work by G. D. Gu et al. [J. Comput. Math. 25, No. 5, 522–530 (2007; Zbl 1141.65356)]. An important element that makes the algorithms efficient is the fact that Krylov spaces are shift invariant, i.e., that \(\mathcal{K}_m(A,b)=\mathcal{K}_m(A+\sigma I,b)\). The convergence is analysed for both a full orthogonal method (FOM) and a generalized minimal residual (GMRES) version of the algorithm by studying the generalized eigenvalue problem \(Kx=\lambda Mz\). Also, the error introduced when the preconditioner is approximately inverted by an iterative method is given, following V. Simoncini and D. B. Szyld [SIAM J. Sci. Comput. 25, No. 2, 454–477 (2003; Zbl 1048.65032)]. Finally, this is applied in the OHT context, where also the inverse problem is considered. The latter, called geostatistical approach, is a stochastic inverse modeling problem that requires the solution of a nonlinear least squares problem. Numerical experiments illustrate the algorithms.

MSC:

65F10 Iterative numerical methods for linear systems
86-08 Computational methods for problems pertaining to geophysics
65F08 Preconditioners for iterative methods
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
86A05 Hydrology, hydrography, oceanography
86A22 Inverse problems in geophysics

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