Minimal residual method

We confine ourselves here to the minimal residual method and the method of steepest descent relating to two-layer schemes. As usual, the explicit scheme is considered first ... 

The only difference between the methods we have mentioned above lies in the selection rules for the parameter In the minimal residual... 

Tj, 2/(7i -b 72), thereby justifying estimate (17) and the convergence of the minimal residual method with the same rate as occurred before for the simple iteration method with the exact values 71 and... 

The implicit minimal residual method can be designed in line with established practice ... 

Applying here the explicit minimal residual method yields... 

No doubt, several conclusions can be drawn from such reasoning. First, the method being employed above converges in the space Ha with the same rate as the simple iteration method although it occurs in one of the subordinate norms. Second, the minimal residual method converges in the space Ha, that is, in a more stronger norm. 

The minimal residual method can be employed for the equation Au = / with a non-self-adjoint operator A, the convergence rate of which coincides with that of scheme (30) for r = f. 

This is certainly true for the minimal residual method (13)-(14) under conditions (34). Here is a solution of problem (13) and p is specified by... 

Parameter estimation. Integral reactor behavior was used for the interpretation of the experimental data, using N2O conversion levels up to 70%. The temperature dependency of the rate parameters was expressed in the Arrhenius form. The apparent rate parameters have been estimated by nonlinear least-squares methods, minimizing the sum of squares of the residual N2O conversion. Transport limitations could be neglected. 

Within the above scheme, we implemented the generalized minimal residual (GM-RES) method [52], which is a robust linear solver that ensures convergence of the iterative solution. 

Since Ap is the Fourier transform of AF, Eq. (5.12) implies that minimization of J (Fobs - Pcaic )2 dr and of J (Fobs - Fcalc)2 dS are equivalent. Thus, the structure factor least-squares method also minimizes the features in the residual density. Since the least-squares method minimizes the sum of the squares of the discrepancies in reciprocal space, it also minimizes the features in the difference density. The flatness of residual maps, which in the past was erroneously interpreted as the insensitivity of X-ray scattering to bonding effects, is an intrinsic result of the least-squares technique. If an inadequate model is used, the resulting parameters will be biased such as to produce a flat Ap(r). 

Montillo M, Schinkoethe T, Elter T. Eradication of minimal residual disease with alemtuzumab in B-cell chronic lymphocytic leukemia (B-CLL) patients the need for a standard method of detection and the potential impact of bone marrow clearance on disease outcome. Cawcer/wvext 2005 23 488-496. 

Durand J. P., Bre A., Beboulene J. J., Ducrozet A., and Carbonneaux S. (1998) Simulated distillation methods for petroleum fractions with minimal residue in the boiling range of 35-700 °C. J. Chromatogr. Sci. 36, 431-434. 

Based on formulae (4.22) and (4.23), we sec that the minimal residual method converges if L is a positively determined (PD) liii( ar continuous operator, acting in a real Hilbert s[)ac( A/. Actually, t.he following imi)ortant theorem holds. 

Thus, we conclude that the sequence of elements m , generated by the minimal residual method, is a Cauchy sequence, because the distance between any two elements goes to zero, m — m —> 0, as f,n oo (see Appendix A, section A.2). Since the Hilbert space M is a complete linear space, the Cauchy sequence m converges to the element m G M m — in, if n —+ oo. 

Theorem 19 Let L be an absolutely positively determined (APD) linear continuous operator, acting in a complex Hilbert space M. Then the solution of the linear operator equation (4-6) exists and is unique in M, and the minimal residual method, based on the recursive formulae (4-V and (4-13), converges to this solution for any initial approximation mo... 

Note, in conclusion, that one can estimate also the convergence rate of the minimal residual method based on formula (4.22) for the residuals ... 

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