Saibaba, Arvind K.; Bakhos, Tania; Kitanidis, Peter K. A flexible Krylov solver for shifted systems with application to oscillatory hydraulic tomography. (English) Zbl 1288.65041 SIAM J. Sci. Comput. 35, No. 6, A3001-A3023 (2013). 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. Reviewer: Adhemar Bultheel (Leuven) Cited in 19 Documents 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 Keywords:Krylov solvers; shifted systems; inverse problems; oscillatory hydraulic tomography; Arnoldi method; preconditioner; algorithm; full orthogonal method (FOM); generalized minimal residual (GMRES); generalized eigenvalue problem; numerical experiment Citations:Zbl 1141.65356; Zbl 1048.65032 Software:PyAMG; DOLFIN PDFBibTeX XMLCite \textit{A. K. Saibaba} et al., SIAM J. Sci. Comput. 35, No. 6, A3001--A3023 (2013; Zbl 1288.65041) Full Text: DOI arXiv