The Krylov-subspace method

An effective method to acetderato the convttrgerice of the MRM algorithm is based on the Krylov-subspace method (Kleinman and van den Berg, 1993). VVe introduced the Krylov subspace in Chapter 2 as the finite dirncnsiorial subsjtace A, of the Hilbert space M, spanned by the vec.tors r , Lr . lAr,.. 

Note that the dimension, s, of the Krylov subspace is always less than or 

The Krylov-subspace method is based on approximating the iteration step, Am , in the recursive formula (4.7) by an element of the Krylov subspace 

The difference between the iteration procedure (4.58)-(4.59) and (4.7) is that now we move from one iteration to another not only in the direction of one residual vector r , but along a multidimensional Krylov subspace spanned by the vectors Vn,Lrn,.L Tn]  

It is obvious that the coefficients kni should be selected based on the minimum of the norm of the residual rn+i  

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Krylov subspace methods (such as Conjugate Gradient CG, the improved BiCGSTAB, and GMRES) in combination with preconditioners for matrix manipulations aimed at enhanced convergence, and... 

The power method uses only the last vector in the recursive sequence in Eq. [21], discarding all information provided by preceding vectors. It is not difficult to imagine that significantly more information may be extracted from the space spanned by these vectors, which is often called the Krylov subspace- 0,14... 

For the RDE, the operating range of rotation frequency is between approximately 1 and 50 Hz and a typical radius is 0.25 cm. Dimensionless rate constants were interpolated from working curves generated from a fully implicit simulation using preconditioned Krylov subspace methods (Alden, unpublished work). 

We note here in passing that Chebyshev propagation is related to several other recursive methods based on the Krylov subspace = span i//(, Hy/(, ...,H Wq) ... 

The best known Krylov subspace method is the method of Conjugate Gradients (CG) by Hestenes and Stiefel [70]. If A is S5mimetric positive definite, the solution of the problem Ax = b corresponds to determining a local minimum of the quadratic function ... 

Preconditioning is a technique which improves the condition number of a matrix and thereby increases the convergence rate of Krylov subspace methods. Thus, if the preconditioner A4 is a symmetric, positive definite matrix, the original problem Ax = b can be solved indirectly by solving M Ax = M h. The the residual can then be written as ... 

The SLDM algorithm can be summarized as follows. The numerical approximation of eq. 12.7 is performed by applying the Lanczos method generating the eigenvectors of matrix A. Then the solution is represented as a projection into the Krylov subspace K" A,(f>) ... 

A solution is to use Krylov subspace methods, such as the conjugate gradient (CG) method, the biconjugate gradient (BiCG)... 

An important consequence of this analysis is that the spectrum of A, and thus convergence should not depend significantly on the discretization. This fact was also confirmed empirically [56, 67, 68], Budko et al. [69] derived optimal value of y when using the general overrelaxation iterative method. This estimate of y is similar but always greater than Eq. [2.36], since this stationary iterative method also constructs the solution in the Krylov subspace but not in the most optimal way. 

The generalized minimum residual (GMRES) Krylov subspace method... 

Krylov Approximation of the Matrix Exponential The iterative approximation of the matrix exponential based on Krylov subspaces (via the Lanczos method) has been studied in different contexts [12, 19, 7]. After the iterative construction of the Krylov basis ui,..., Vn j the matrix exponential is approximated by using the representation A oi H(g) in this basis ... 

We here describe the alternative of approximating <,c(S)b via Lanczos method. The Lanczos process [18, 22] recursively generates an orthonormal basis Qm = [qi,.., qm] of the mth Krylov subspace... 

There is a range of iterative diagonalization routines to choose between, including classical orthogonal polynomial expansion methods [48], Davidson iteration[58] and Krylov subspace iteration methods. Here the popular Lanezos method[59] will be discussed in the context of finding the eigenstates of the surface Hamiltonian appearing in the hyperspherical coordinate method. 

In the work of Lindborg et al [119], the resulting linear equation systems were solved with preconditioned Krylov subspace projection methods [166]. The Poisson equation was solved by a conjugate gradient (CG)-solver, while the other transport equations were solved using a bi-conjugate gradient (BCG)-solver which can handle also non-symmetric equations systems. The solvers were preconditioned with a Jacobi preconditioner. 

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