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Application of Numerical Methods to Selected Model Equations
Published in Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar, Computational Fluid Mechanics and Heat Transfer, 2020
Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar
There is another useful family of iterative methods that has been widely used in solving large systems of algebraic equations. These are loosely related to the conjugate gradient method, which, in its original form, was a direct method. The methods are also known in some circles as Krylov methods or Krylov subspace methods after the Russian scientist A. N. Krylov. The methods solve a system of the form [A]u = C by repeatedly performing matrix–vector multiplications involving [A]. The conjugate gradient method (Hestenes and Stiefel, 1952) is the “original” Krylov subspace iterative method. The conjugate gradient method requires that [A] be symmetric and positive definite, which limits its use for many equations arising in CFD. However, a modification known as the generalized minimal residual method (GMRES) (Saad and Schultz, 1986) is often used for the more general systems of equations arising in CFD. To provide a background for the development of GMRES, the conjugate gradient method will be outlined first.
Numerical Methods and Computational Tools
Published in Raj P. Chhabra, CRC Handbook of Thermal Engineering Second Edition, 2017
Atul Sharma, Salil S. Kulkarni, K. Hrisheekesh, Amit Agrawal, Shyamprasad Karagadde, Amitabh Bhattacharya, Rajneesh Bhardwaj
Iterative solvers. In these solvers, one starts with an initial guess x0 to the solution of Equation 5.4.21 and generates a sequence of approximations xi, which converges to the exact solution of Equation 5.4.21. The main advantage of these method is that they do not need any decompositions and primarily involve matrix vector multiplications. For symmetric positive definite matrices, one can use the conjugate gradient method (see Hestenes and Stiefel, 1952), whereas for a general matrix one can use the Generalized minimal residual method (GMRES) method (see Saad and Schultz, 1986). The rate of convergence of an iterative method is heavily influenced by the choice of the preconditioners. For symmetric positive definite matrices, some of the preconditioners that can be used include incomplete Cholesky and diagonal scaling. For a general matrix, the preconditioners that are commonly used are based on the incomplete LU factorizations. More details of the iterative methods are found in Saad (2003). The detailed survey on the types of preconditioners for large systems is given in Benzi (2002).
Matrix-Free Iterative Solution Procedure for Finite Element Problems
Published in João Pedro A. Bastos, Nelson Sadowski, Magnetic Materials and 3D Finite Element Modeling, 2017
João Pedro A. Bastos, Nelson Sadowski
A slight modification of the procedure enables its application in non-stationary solvers such as the conjugate gradient method (CG), the biconjugate gradient method (BiCG) and the stabilized version (BiCGstab), the quasiminimal residual method (QMR), or the generalized minimal residual method (GMRES). Algorithmic descriptions of these solvers are provided in [13, 23, 24]. In all the Krylov subspace methods mentioned, the system matrix A is only referenced in the context of a matrix-vector product of the form q(i)=Ap(i) $$ q^{(i)}=Ap^{(i)} $$
Active space approaches combining coupled-cluster and perturbation theory for ground states and excited states
Published in Molecular Physics, 2020
Malte F. Lange, Timothy C. Berkelbach
Although the composite method gives results that are comparable to the interacting method for the ground state and first excited states, it has a number of disadvantages. First, the composite method does not yield an accurate wavefunction; instead, it relies upon error cancellation in the energies of various inaccurate wavefunctions. Therefore, it does not provide a natural route towards the calculation of accurate observables other than the energy (however, see Ref. [44] for a recent discussion). Second, not all applications of EOM-CCSD require the explicit enumeration of eigenvalues and eigenvectors. In particular, the calculation of frequency-dependent quantities, such as the Green's function [49] or other spectra [50], only requires the solution of a system of linear equations, , where is a known state. The solution of this sytem of linear equations is naturally obtained using iterative algorithms, such as the generalised minimal residual method method, which only require matrix-vector multiplications. While a composite approach does not provide a natural solution to these calculations (again see Ref. [44]), the interacting method yields a single, accurate, reduced-cost matrix-vector multiplication. The development and testing of such an approach for the the calculation of spectra, especially for solid-state systems, is now ongoing in our research group.