Approximation error estimates

In applications, one is often interested in approximating time averages over a time interval [0, T] via associated mean values of a , k = 1. ..Tfr. For T (or r) small enough, the above backward analysis may lead to much better error estimates than the worst case estimates of forward analysis. 

Unfortunately, this local error Cr cannot be calculated, since we do not know the exact solution to the QCMD equations. The clue to this problem is given by the introduction of an approximation to Let us consider another discrete evolution with an order q > p and define an error estimation via er t + z i) - z t). 

The error estimate approximates the error of the propagation with the less accurate method p. Nonetheless, the next step is started with the more precise result of... 

In [13], an efficient residual error estimation scheme has been introduced for controlling the quality of the approximation. This gives us a stopping criterion for the iteration guaranteeing that the quality of the approximation fits to the accuracy requirements of the stepsize control. 

We consider penalized operator equations approximating variational inequalities. For equations with strongly monotonous operators we construct an iterative method, prove convergence of solutions, and obtain error estimates. 

Any truncation is an approximation. WAMCUT estimates the error of truncation. WAMBAM does this also but does not display the cutsets. The GO code contains Icatincs toi csiimaiing ihc eftects of truncation on the probability but not the ri.sk. 

In the estimation of the approximation error the well-founded choice of the norm depends on the structure of an operator and needs investigation in every particular case. A precise relationship between an operator and a norm in the process of searching the error of approximation will be established in the general case in Section 4. Its concretization for the example of interest leads naturally to the appearance of the negative norm ... 

These assertions follow from the representation of the approximation error in the form (6) (8) and a priori estimate (12). On the basis of the estimates for T/j and obtained in Section 3.2 we find that... 

The proof of convergence of scheme (19) reduces to the estimation of a solution of problem (21) in terms of the approximation error. In the sequel we obtain such estimates using the maximum principle for domains of arbitrary shape and dimension. In an attempt to fill that gap, a non-equidistant grid... 

Upon substituting the estimates for the approximation error obtained in Section 3 into (34) we find that... 

The accuracy of the error equations (Eqs. (22) and (23)] also depends on the selected wavelet. A short and compactly supported wavelet such as the Haar wavelet provides the most accurate satisfaction of the error estimate. For longer wavelets, numerical inaccuracies are introduced in the error equations due to end effects. For wavelets that are not compactly supported, such as the Battle-Lemarie family of wavelets, the truncation of the filters contributes to the error of approximation in the reconstructed signal, resulting in a lower compression ratio for the same approximation error. 

Figure 3 The collapse of the peptide Ace-Nle30-Nme under deeply quenched poor solvent conditions monitored by both radius of gyration (Panel A) and energy relaxation (Panel B). MC simulations were performed in dihedral space 81% of moves attempted to change angles, 9% sampled the w angles, and 10% the side chains. For the randomized case (solid line), all angles were uniformly sampled from the interval —180° to 180° each time. For the stepwise case (dashed line), dihedral angles were perturbed uniformly by a maximum of 10° for 4>/ / moves, 2° for w moves, and 30° for side-chain moves. In the mixed case (dash-dotted line), the stepwise protocol was modified to include nonlocal moves with fractions of 20% for 4>/ J/ moves, 10% for to moves, and 30% for side-chain moves. For each of the three cases, data from 20 independent runs were combined to yield the traces shown. CPU times are approximate, since stochastic variations in runtime were observed for the independent runs. Each run comprised of 3 x 107 steps. Error estimates are not shown in the interest of clarity, but indicated the results to be robust.

Metal-vacuum-metal tunneling 49—50 Method of Harris and Liebsch 110, 123 form of corrugation function 111 leading-Bloch-waves approximation 123 Microphone effect 256 Modified Bardeen approach 65—72 derivation 65 error estimation 69 modified Helmholtz equation 348 Modulus of elasticity in shear 367 deflection 367 Mo(lOO) 101, 118 Na-atom-tip model 157—159 and STM experiments 157 NaCl 322 NbSej 332 NionAu(lll) 331 Nucleation 331... 

The center manifold approach of Mercer and Roberts (see the article Mercer and Roberts, 1990 and the subsequent article by Rosencrans, 1997) allowed to calculate approximations at any order for the original Taylor s model. Even if the error estimate was not obtained, it gives a very plausible argument for the validity of the effective model. This approach was applied to reactive flows in the article by Balakotaiah and Chang (1995). A number of effective models for different Damkohler numbers were obtained. Some generalizations to reactive flows through porous media are in Mauri (1991) and the preliminary results on their mathematical justification are in Allaire and Raphael (2007). 

Recent approach using the anisotropic singular perturbation is the article by Mikelic et al. (2006). This approach gives the error estimate for the approximation and, consequently, the rigorous justification of the proposed effective models. It uses the strategy introduced by Rubinstein and Mauri (1986) for obtaining the effective models. 

Once more, for the non-negligible local Peclet and Damkohler numbers, taking the simple mean over the section does not lead to a good approximation and our numerical simulations, presented in the last section, will confirm these theoretical results. For an error estimate analogous to Theorem 1, we refer to the articles Mikelic and Rosier (2007) and Choquet and Mikelic (2008). 

Here ip(x) = u -(- f(x) is the approximation error of scheme (44). For all sufficiently smooth functions u(x) it is well-known that tp(x) is a quantity of order 0(/i2), thus causing the same type of the problem for the function z(x) as occured for the function y(x). Because of this fact, estimate (48) is still valid for z(x) ... 

It is a truism that the SchrSdinger equation very seldom may be solved exactly. It is then desirable to obtain error estimates, namely, upper and lower bounds to the eigenvalues and accuracy criteria for the approximate eigenfunctions. 

Points 1 and 12 were assumed to correspond at their observed activities to the upper curve of run II, which had been evaluated as described in Improvements. The compositions so determined were used with the observed balance readings to establish the two calculation constants used in the evaluation of all other data in run I. Point 128 was confirmed within analytical error by a chemical analysis. The compositions associated with run III (Figure 2) could not be determined directly because of the unknown amount of oxide present. However, since the absolute composition-activity relationships for CeCd 6(4) are known approximately, an estimate of the composition was possible. The resulting calculated values for CeCd 4 5 were increased by 0.05%, in order to be consistent with the position of the upper structure in Figure 1. The resulting composition values should not be assumed to correspond precisely to those in Figure 1 however, it is not essential that the composition be known with greater precision. 

A typical result of numerical analysis is an estimate of the error U — Uh between the solution U of the continuous problem (. e., the solution of the initial boundary value problem) and the solution Uh of the discrete problem (also called approximate problem). In what follows the error estimates are obtained with the assumption that U is sufficiently regular. In many realistic situations the geometry of the flow has singularities (corners for example), the solution U is not regular, and these results do not apply. (As a matter of fact existence of a solution has not been shown yet in those singular situations.)... 

Theorem 6.1 (Existence of continuous and approximate flows. Error estimate)... 

Because of the combinatorial nature of systematic search, one is often faced with large numbers of conformers that have to be analyzed. For some problems, energetic considerations are appropriate and conformers can be clustered with the closest local minimum, providing to a first approximation an estimate of the entropy associated with each minima by the number of conformers associated, in that they can come from a grid search that approximates the volume of the potential well. A single conformer, perhaps the one of lowest en-Qgy, can be used with appropriately adjusted error limits in further analyses as representative of the family. 

The error estimation of this approximation can be based on the equality... 

The performance of the FO approach for the analysis of observational and experimental data have been evaluated by Sheiner and Beal with the Michaelis-Menten pharmacokinetic modek and the one- and two-compartment models. In all instances, a comparison was made with the NPD and STS approaches for the analysis of the two types of data. The FO approach outperformed the NPD and the STS approaches on both data types. Despite the approximation, the FO approach provides good parameter estimates. If the residual error increases, the STS approach quickly deteriorates, especially with respect to variance parameters. However, the STS still performs reasonably well but the bias and imprecision of the estimates tend to increase with increasing residual error. Estimates of residual variability have been shown to deteriorate with the FO approach when residual error increases. " ... 

In addition to allowing fine-tuning of the fitted parameters, the final step of simplex searching offers a convenient means of estimating the error associated with each parameter. This process has been described by Phillips and Eyring [30]. Briefly, one determines a quadratic approximation to the error surface, from which an error matrix is developed. This matrix can then be used to calculate standard deviations of the fitted parameters. These standard deviations are reported as error estimates of the parameters in Table 2. 

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