Krylov method

Kolmogorov, A. N., 114,139,159 Konigs thorem applied to Bernoulli method, 81 Koopman, B., 307 Roster, G.F., 727,768 Kraft theorem, 201 Kronig-Penney problem, 726 antiferromagnetic, 747 Krylov-Bogoliubov method, 359 Krylov method, 73 Krylov, N., 322 Kuhn, W. H., 289,292,304 Kuratowski s theorem, 257... 

Banachiewicz method, 67 characteristic roots, 67 characteristic vectors, 67 Cholesky method, 67 Danilevskii method, 74 deflation, 71 derogatory form, 73 "equations of motion, 418 Givens method, 75 Hessenberg form, 73 Hessenberg method, 75 Householder method, 75 Jacobi method, 71 Krylov method, 73 Lanczos form, 78 method of modification, 67 method of relaxation, 62 method of successive displacements,... 

It should be noted that SLDM is related to other Krylov methods (Greenbaum, 1997) such as BiConjugate Gradient (BCG) method (or its modifications such as QMR), also used in induction logging simulations. But compared to the latter it (SLDM) is in real arithmetic and computes for multiple frequencies at no cost. 

Brown, R N., Hindmarsh, A. C., Petzold, L. R. (2006). Using Krylov methods in the solution of large-scale differential-algebraic systems. SIAM Journal Scientific Computing, 15(6), 1467-1488. 

Mukadi, L.S. Hayes, R.E. (2002) Modelling the three-way catalytic converter with mechanistic kinetics using the Newton-Krylov method on a parallel computer. Computers and Chemical Engineering 26,439-455. 

What might be called the Krylov method has been attempted and also compared. It is based on an article by Krylov [250], and is described more accessibly in the books of Saad[251, 252] and Wesseling [253]. Some electrochemists have tried it out [246, 247, 254-256] but generally in comparisons, SIP wins. As with SIP, it appears to have fallen out of favom. 

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 ... 

Large stepsizes result in a strong reduction of the number of force field evaluations per unit time (see left hand side of Fig. 4). This represents the major advantage of the adaptive schemes in comparison to structure conserving methods. On the right hand side of Fig. 4 we see the number of FFTs (i.e., matrix-vector multiplication) per unit time. As expected, we observe that the Chebyshev iteration requires about double as much FFTs than the Krylov techniques. This is due to the fact that only about half of the eigenstates of the Hamiltonian are essentially occupied during the process. This effect occurs even more drastically in cases with less states occupied. 

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... 

But then is the characteristic polynomial of A, and its coefficients are the elements of / and can be found by solving Eq. (2-11). This is essentially the method of Krylov, who chose, in particular, a vector et (usually ej) for vx. Several methods of reduction of the matrix A can be derived from applying particular methods of inverting or factoring V at the same time that the successive columns of V are being developed. Note first that if... 

The use of this theory in studies of nonlinear oscillations was suggested in 1929 (by Andronov). At a later date (1937) Krylov and Bogoliubov (K.B.) simplified somewhat the method of attack by a device resembling Lagrange s method of the variation of parameters, and in this form the method became useful for solving practical problems. Most of these early applications were to autonomous systems (mainly the self-excited oscillations), but later the method was extended to... 

The same system has been studied previously by Boguslavsky et al. [29], who also used the drop weight method. While qualitatively the same behavior was observed over the broad concentration range up to the solubility limit, the data were fitted to a Frumkin isotherm, i.e., the ions were supposed to be specifically adsorbed as the interfacial ion pair [29]. The equation of the Frumkin-type isotherm was derived by Krylov et al. [31], on assuming that the electrolyte concentration in each phase is high, so that the potential difference across the diffuse double layer can be neglected. 

Krylov AI (2006) Spin-flip equation-of-motion coupled-cluster electronic structure method for a description of excited states, bond breaking, diradicals, and triradicals. Acc Chem Res 39 83-91... 

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... 

Krylov, A. I., Slipchenko, L. V., and Levchenko, S. V. Breaking the Curse of the Non-dynamical Correlation Problem the Spin-Flip Method , ACS Symp. Ser., in press. 

We should note that this article by Ya.B. apparently remained little noticed in its time. In any case, we are unaware of any reference to it in the works of other authors. This is explained by the fact that its ideas were far ahead of their time. Only in recent years, due to the wide application of physical methods in studies of adsorption and catalysis, have the changes in the surface (and volume) structure of a solid body during adsorption and catalysis been proved. Critical phenomena have been discovered, phenomena of hysteresis and auto-oscillation related to the slowness of restructuring processes in a solid body compared to processes on its surface. Relaxation times of processes in adsorbents and catalysts and comparison with chemical process times on a surface were considered in papers by O. V. Krylov in 1981 and 1982 [1] (see references at end of Introduction). 

Krylov, D.M. and Hurley, J.B. (2001). Identification of proximate regions in a complex of retinal guanylyl cyclase 1 and guanylyl cyclase activating protein-1 by a novel mass spectrometry based method. J Biol. Chem. 276 30648-30654. 

The method has a long history. The name MOL seems to have become estabhshed around 1960. Before this, various authors either used the word line [254] or expressions like on certain lines [330] or a description of the idea. In the book by Kantorovich and Krylov [330], there is a reference to a 1934 paper [329]. It is also cited by Liskovets [366] as a source paper, along with Rothe (1930) [475], who might be the first. Hartree and Womersley [296] use, in their summary, the words approximating by use of finite intervals in one variable, and integrating exactly in the other variable . The book by Schiesser [497] is the standard work now (he calls the method NUMOL, for numerical method of hnes). Electrochemical use of MOL has been sparse. Lemos and coworkers [357, 359, 360] have investigated the method, using... 

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). 

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