Ill-conditioned problem

Thus, we obtain = 0.159 for a step size of Aj = 0.03125. The ill-conditioning problem has been solved, but the solution remains inaccurate due to the simple integration scheme and the large step size. 

Thus, the error in the solution vector is expected to be large for an ill-conditioned problem and small for a well-conditioned one. In parameter estimation, vector b is comprised of a linear combination of the response variables (measurements) which contain the error terms. Matrix A does not depend explicitly on the response variables, it depends only on the parameter sensitivity coefficients which depend only on the independent variables (assumed to be known precisely) and on the estimated parameter vector k which incorporates the uncertainty in the data. As a result, we expect most of the uncertainty in Equation 8.29 to be present in Ab. 

If matrix A is ill-conditioned at the optimum (i.e., at k=k ), there is not much we can do. We are faced with a truly ill-conditioned problem and the estimated parameters will have highly questionable values with unacceptably large estimated variances. Probably, the most productive thing to do is to reexamine the structure and dependencies of the mathematical model and try to reformulate a better posed problem. Sequential experimental design techniques can also aid us in... 

In certain occasions the volume criterion is not appropriate. Fn particular when we have an ill-conditioned problem, use of the volume criterion results in an elongated ellipsoid (like a cucumber) for the joint confidence region that has a small volume however, the variance of the individual parameters can be very high. We can determine the shape of the joint confidence region by examining the cond( ) which is equal to and represents the ratio of the principal axes of... 

The Laplace inversion (LI) is the key mathematical tool of the DDIF experiment. The ability to convert the measured multi-exponential decay into a distribution of decay times is crucial to the DDIF pore size distribution application. However, unlike other mathematical operations, the Laplace inversion is an ill-conditioned problem in that its solution is not unique, and is fairly sensitive to the noise in the input data. In this light, significant research effort has been devoted to optimizing the transform and understanding its boundaries [17, 53, 54],... 

Firstly, it has been found that the estimation of all of the amplitudes of the LI spectrum cannot be made with a standard least-squares based fitting scheme for this ill-conditioned problem. One of the solutions to this problem is a numerical procedure called regularization [55]. In this method, the optimization criterion includes the misfit plus an extra term. Specifically in our implementation, the quantity to be minimized can be expressed as follows [53] ... 

The recovery of /(C) is an ill-conditioned problem and it is reasonable to take a Gaussian distribution. Moreover, when probes experience two distinct environments (as expected for instance for gel/fluid heterogeneity), a sum of two Gaussian curves should be adequate for data analysis3 ... 

As a numerical analyst you may have to solve inherently ill-conditioned problems, but in scientific computing there are further opportunities. Neglecting or coupling unimportant variables, seeking further constraints or... 

A.N. Tihonov and V. Ya. Arsenin, Solution Methods of Ill-Conditioned Problems, Nauka, Moscow, 1979. (in Russian)... 

Particularly note that the error check is sizable for this example and not nearly zero as it should be. We will comment on this numerically ill-conditioned problem on p. 398 after we have displayed and discussed the graphics output of this particular call and following the on-screen output of several other similar calls. 

It is well known that the determination of abscissas and weights of the Gaussian quadrature from power moments is an exponentially ill-conditioned problem due to the presence of rounding errors. 

The modified moments are useful whenever it is possible to guess an auxiliary distribution n E) closely simulating the actual one. However, in the construction of the N matrix, the principal ingredients are just the modified moments, whose determination from the power moment is still a notoriously ill-conditioned problem. 

Referring to Eqn (7.9), we see that in any treatment of surface heterogeneity, we have to deal with three functions, any two of which, if known, assumed or determined can be used in theory to obtain the third. Equation (7.9) represents a Fredholm s integral of the first kind. The solution of equations of this type is well known to present an iU-posed or ill-conditioned problem. For our purposes, this means that the data, Q(p), can be well represented by many function pairs in the integrand hence, simply fitting the data does not guarantee that the kernel function or the distribution are individually correct. In addition, the mathematical difBculties of handling Eqn (7.9) analytically have severely restricted the number of possible variations that have been pubHshed and these are now only of historical interest. 

This procedure transforms the ill-conditioned problem of finding the roots of a polynomial into the well-conditioned problem of finding the eigenvalues and eigenvectors of a tridiagonal symmetric matrix. As shown by Wilf (1962), the N weights can then be calculated as Wa = OToV ai where tpai is the first component of the ath eigenvector (pa of the Jacobi matrix. 

Initially the model was compiled using both experimentally measured and theoretically calculated kinetic parameters. Then, the results of simulations were compared with the data of multiple experiments and sensitivity analysis was employed to select the parameters, which should be corrected for the better agreement between experimentally observed and simulated kinetic behavior. The computation routine can perform the modification of each kinetic parameter within the range of its initial uncertainty. Such an approach gives a serious cause for criticism, since the discrepancies with experimental data are eliminated (or minimized) by changing the values of multiple parameters. First, this makes all of them correlated. Next, an independent correction of just one parameter in the model, or just a slight modification of the micro-chemical scheme leads to the readjustment of the whole system of kinetic parameters. This is in a certain sense equal to the solution of the inverse kinetic task, which, as we mentioned above, is an ill-conditioned problem. 

Second, it is evident that the selection of different basic experiments should lead to different sets of optimized kinetic parameters. Since, as it was already mentioned, the inverse kinetic task is in general an ill-conditioned problem, such sets can differ significantly and it is difficult to define the criteria for the selection of the one (only one ) that should be used for modeling. 

Equation 13 indicates that when TG(r, q) is plotted as the ordinate and log(T Uf) is plotted as the abscissa, the area under each peak corresponds to the weight of each peak. It should be added that the Laplace inversion is a delicate process due to ill-conditioned problems involved, so that care must be exercised in extracting information [88],... 

Over-parameterization. This is one of the two aspects of what is defined in regression analysis as an ill-conditioned problem. When a problem is ill-conditioned the parameter vector which minimizes the residual sum of squares function S may be difficult to obtain computationally. Ill-conditioning could indicate a model that is over-parameterized, that is, a model that has more parameters than are needed, or could reveal the existence of inadequate data (e.g. too few data), which will not allow us to estimate the parameters postulated. Because these are two sides of the same coin, the choice of whether one or the other is the culprit depends on a priori knowledge about the practical problem and one s point of view (Seber and Wild, 1989 Salthammer, 1996). 

Among the implicit methods are the Gaussian elimination and methods such as the modified strongly implicit (MSI) procedure, the LU-SSOR, and the implicit Runge-Kutta. The parallelization of implicit methods is more elaborate than for explicit methods. Implicit methods are frequently employed for solving ill-conditioned problems, such as those that arise in reactive flows. Thus, in physical terms, implicit methods are best suited to address ill-conditioned systems, while in computational terms these methods are preferred to resolve small matrix systems. 

Liu YS, Ware WR (1993) Photophysics of polycyclic aromatic hydrocarbons adsorbed on silica gel surfaces. 1. Fluorescence lifetime distribution analysis an ill-conditioned problem. J Phys Chem 97 5980-5986... 

Recently, maximum entropy image enhancement techniques have been successfully applied to a variety of ill-conditioned problems, including the inversion of imaginary... 

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