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| GMRES | |
|---|---|
| Name | GMRES |
| Full name | Generalized Minimal Residual method |
| Introduced | 1986 |
| Authors | Youcef Saad; Martin H. Schultz |
| Type | Krylov subspace method |
| Applications | Computational fluid dynamics, Structural engineering, Electromagnetics, Reservoir simulation |
GMRES provides an iterative solver for large, sparse, nonsymmetric linear systems arising in numerical simulations and scientific computing. Developed in the 1980s, GMRES is widely used alongside direct solvers and other Krylov methods in projects at institutions such as Los Alamos National Laboratory, Lawrence Livermore National Laboratory, and universities including Stanford University and Massachusetts Institute of Technology. It frequently appears in software libraries maintained by organizations like Netlib and incorporated into codes from vendors such as Intel and NVIDIA.
GMRES targets linear systems A x = b where A is typically nonsymmetric and sparse, encountered in problems studied at Princeton University, Caltech, and research centers such as CERN. The method builds on earlier Krylov ideas used by researchers at Cornell University and in algorithms developed by teams led by Gene Golub and Lanczos-related work. Implementations in packages from IBM and the National Center for Atmospheric Research make GMRES a standard option for solvers used in projects funded by agencies like the National Science Foundation and the U.S. Department of Energy.
GMRES minimizes the residual norm over a Krylov subspace K_m(A,r_0) = span{r_0, A r_0, ..., A^{m-1} r_0}. The approach employs orthonormal bases produced by the Arnoldi process, an algorithm with roots in work by Walter Kohn and whose matrix factorizations relate to developments by John von Neumann and Eugene Wigner. At iteration m the method seeks x_m in x_0 + K_m that minimizes ||b - A x_m||_2; this reduced minimization uses a small (m+1)×m Hessenberg matrix whose properties were studied by theorists at Princeton Plasma Physics Laboratory and groups collaborating with SIAM authors.
The core routine alternates Arnoldi orthonormalization and least-squares solves. Practical implementations balance orthogonalization strategies (classical or modified Gram–Schmidt) with numerical stability concerns analyzed in studies at Harvard University and University of Cambridge. Libraries such as PETSc, Trilinos, and ScaLAPACK provide parallel GMRES kernels optimized for architectures from Cray to modern clusters at Oak Ridge National Laboratory. Implementation choices include restarting, double precision arithmetic per IEEE standards endorsed by ISO, and communication-avoiding variants developed in collaborations involving Argonne National Laboratory.
Convergence depends on the spectrum of A and its field of values, topics investigated by researchers affiliated with Princeton University, University of California, Berkeley, and the Max Planck Society. For normal matrices, convergence relates to eigenvalue distribution studied in contexts like the Navier–Stokes equations and eigenproblems tackled by Courant Institute groups. Nonnormality and pseudospectra effects, topics explored by teams at Caltech and Imperial College London, can produce slow convergence despite favorable eigenvalues; such behavior connects to analyses by Lloyd N. Trefethen and collaborators. Rounding error accumulation and loss of orthogonality are practical issues documented in benchmarks from Sandia National Laboratories.
Effective preconditioning transforms A to improve spectral properties; choices include incomplete LU (ILU), algebraic multigrid (AMG), and block preconditioners. ILU schemes trace to work at IBM Research and Bell Labs, while AMG was advanced by researchers at Lawrence Berkeley National Laboratory and École Polytechnique Fédérale de Lausanne. Block and field-split preconditioners are used in multiphysics codes at Argonne National Laboratory and NASA for coupled problems. Left, right, and flexible preconditioning variants have been compared in studies published with contributions from SIAM authors and implemented in toolchains supported by Microsoft Research and national laboratories.
Variants such as restarted GMRES(m), Flexible GMRES (FGMRES), and shifted GMRES address memory, nonlinear, and parameter-dependent systems; these extensions were proposed and analyzed by researchers at University of Illinois Urbana–Champaign and Duke University. Communication-avoiding GMRES and pipelined GMRES target high-performance computing environments at facilities like Oak Ridge National Laboratory and Argonne National Laboratory. Hybrid approaches combining GMRES with direct solvers or deflation techniques have been developed in collaborations involving ETH Zurich and University of Oxford. Software projects from Sandia National Laboratories and CWI incorporate many of these variants.
GMRES is applied to discretized partial differential equations in Computational Fluid Dynamics codes used by Boeing and Rolls-Royce, to electromagnetic simulations in projects at Siemens and Thales, and to reservoir simulation workflows in companies such as Schlumberger. Performance benchmarks reported by conglomerates and research teams compare GMRES against BiCGSTAB, QMR, and direct solvers on matrices from collections maintained by NAFEMS and SuiteSparse Matrix Collection contributors at University of Florida. Scalability studies on leadership-class machines at Oak Ridge National Laboratory and Lawrence Berkeley National Laboratory quantify trade-offs between iteration counts, preconditioner costs, and communication overhead, guiding selection in industrial and academic deployments.