Matrix ill-conditioned

The norm is useful when doing numerical calculations. If the computer s floating-point precision is 10" , then K = 10 indicates an ill-conditioned matrix. If the floating-point precision is I0" (double precision), then a matrix with K = I0 may be ill-conditioned. Two other measures are useful and are more easily calculated ... 

It should be noted that the generalized inverse of is determined with PCA. The inversion step, which is risky in case of badly resolved peaks because of the probability of an ill-conditioned matrix [C], is circumvented. 

Riley, J.D., Solving systems of linear equations with a positive definite symmetric but possibly ill-conditioned matrix, Math. Table Aids Comput., 9, 96-101, 1955. 

On the other hand, if the correlation matrix has variables that are 100% redundant, then the inverse of the matrix cannot be computed it is the so-called ill conditioning matrix. This happens when there are high intercorrelated variables (e.g. a variable that is the sum of two other variables). The statistical... 

Repeated convergence test failures occurred on the last attempted step. An inaccurate or ill-conditioned matrix 0 t) may be the cause. If you are absolutely sure you want to continue, MAIN should restart the solution with lnfo(l)=0. 

Comparing (3.130) with (3.11G), wc can see now that vc have to invei t tlie well-conditioned matrix F -f al rather than tlic possibly ill-conditioned matrix F. Substituting (3.130) into (3.125), wc write... 

A characteristic of A for ill-posed problems is that it has a very large condition number. In other words, the ill-conditioned matrix A is very near to being singular. Briefly, the condition number is defined as k A) = A IIA II or the ratio of maximum to minimum singular values measured in the l2 norm. The ideal problem conditioning occurs for orthogonal matrices which have k (A) 1, while an ill-conditioned... 

The inverse of a matrix cannot always be determined accurately. There are many matrices that are ill-conditioned. An ill-conditioned matrix can be identified using the following criterion When the ratio of the absolute values of the largest and smallest eigenvalues of the matrix is very large, the matrix is ill-conditioned. 

The finite-element technique is based on dividing the cell domain into polygonal sections. The potential within each of the elements is assumed to be a linear combination of the value at the vertices. However, unlike the finite-dilference method, which solves the finite-difference approximation of the Laplace equation, the finite-elements method seeks a solution for the potential distribution within the cell, which best fits the Laplace equation and the boundary conditions. The degree of accuracy is similar to that of the finite-difference method however, curved boundaries and narrow corners can be described with more precision and ease. On the other hand, the presence of electrochemical nonlinear boundary conditions leads to ill-conditioned matrix equations which are more difficult to solve than the finite-difference system. 

A matrix with a large condition number is commonly referred to as ill-conditioned and particularly vulnerable to round-off errors. Special techniques. 

This set is said to be ill-conditioned because the second equation is almost an exact multiple of the first. The matrix of coefficients is almost singular. 

The higher the Condition Number, the more ill-conditioned the % matrix is... 

If a matrix is ill-conditioned, its inverse may be inaccurate or the solution vector for its set of equations may be inaccurate. Two of the many ways to recognize possible ill-conditioning are... 

In practice, the solution of Equation 3.16 for the estimation of the parameters is not done by computing the inverse of matrix A. Instead, any good linear equation solver should be employed. Our preference is to perform first an eigenvalue decomposition of the real symmetric matrix A which provides significant additional information about potential ill-conditioning of the parameter estimation problem (see Chapter 8). 

If two or more of the unknown parameters are highly correlated, or one of the parameters does not have a measurable effect on the response variables, matrix A may become singular or near-singular. In such a case we have a so called ill-posed problem and matrix A is ill-conditioned. 

A measure of the degree of ill-conditioning of a nonsingular square matrix is through the condition number which is defined as... 

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... 

If however, matrix A is reasonably well-conditioned at the optimum, A could easily be ill-conditioned when the parameters are away from their optimal values. This is quite often the case in parameter estimation and it is particularly true for highly nonlinear systems. In such cases, we would like to have the means to move the parameters estimates from the initial guess to the optimum even if the condition number of matrix A is excessively high for these initial iterations. 

If matrix A is well-conditioned, the above equation should be used. If however, A is ill-conditioned, we have the option without any additional computation effort, to use instead the pseudoinverse of A. Essentially, instead of A 1 in Equation 8.31, we use the pseudoinverse of A, A. ... 

When the parameters differ by more than one order of magnitude, matrix A may appear to be ill-conditioned even if the parameter estimation problem is well-posed. The best way to overcome this problem is by introducing the reduced sensitivity coefficients, defined as... 

With this modification the conditioning of matrix A is significantly improved and cond AR) gives a more reliable measure of the ill-conditioning of the parameter estimation problem. This modification has been implemented in all computer programs provided with this book. 

The remedies to increase the region of convergence include the use of a pseudoinverse or Marquardt s modification that overcome the problem of ill-conditioning of matrix A. However, if the basic sensitivity information is not there, the estimated direction Ak +I) cannot be obtained reliably. 

In this problem it is very difficult to obtain convergence to the global optimum as the condition number of matrix A at the above local optimum is 3xl018. Even if this was the global optimum, a small change in the data would result in widely different parameter estimates since this parameter estimation problem appears to be fairly ill-conditioned. 

The LS objective function was found to be 0.7604x10"9. This value is almost three orders of magnitude smaller than the one found earlier at a local optimum. The estimated parameter values were At=22.672, A2=132.4, A3=585320, Ej=l3899, E2=2439.6 and E3=13506 where parameters A, and E were estimated back from Ai and E. With this reparameterization we were able to lessen the ill-conditioning of the problem since the condition number of matrix A was now 5.6x108. 

Furthermore, the implementation of the Gauss-Newton method also incorporated the use of the pseudo-inverse method to avoid instabilities caused by the ill-conditioning of matrix A as discussed in Chapter 8. In reservoir simulation this may occur for example when a parameter zone is outside the drainage radius of a well and is therefore not observable from the well data. Most importantly, in order to realize substantial savings in computation time, the sequential computation of the sensitivity coefficients discussed in detail in Section 10.3.1 was implemented. Finally, the numerical integration procedure that was used was a fully implicit one to ensure stability and convergence over a wide range of parameter estimates. 

In the 3rd run the porosity of the ten zones was estimated by using an initial guess of 0.1. Finally, in the 4,h run the porosity of all fifteen zones was estimated by using the same initial guess (0.1) as above. In this case, matrix A was found to be extremely ill-conditioned and the pseudo-inverse option had to be used. 

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