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KL-expansion (advantages (the method allows to capture the variability of…
KL-expansion
advantages
the method allows to capture the variability of uncertain quantities using a low number of random
variables. pg97-Felipe
It has been also verified that strongly correlated stochastic fields with smooth autocovariance functions exhibit fast eigenvalue decay and thus may be easier to simulate with the K–L expansion. Stefanou 2007
Due to non accumulation of eigenvalues around a non zero value, it is possible to order them in a descending series converging to zero. Sudret pg19
The KL approximate representation has the advantage that it can model both stationary and nonstationary processes. Grigoriu 2007
The covariance eigenfunction basis is optimal in the sense that the mean square error (integrated over domain) resulting from a truncation after the M-th term is minimized (with respect to the value it would take when any other complete basis hi(x) is chosen). Sudret pg 19
The use of Karhunen–Loeve (K–L) expansion with orthogonal deterministic basis functions and uncorrelated random coefficients has generated interest because of its bi-orthogonal property, that is, both the deterministic basis functions and the corresponding random coefficients are orthogonal. Phin 26
If the exact eigenvalues and eigenfunctions of the covariance function can be found, K–L expansion is the most efficient method for representing the random process, i.e. it requires the smallest number of random variables to represent the random process. Phon 26
The main advantage of the K–L expansion method is that it can be easily generalized to simulate non-stationary processes with little additional effort. Phon 26
disadvantage
the K-L approach, most of the methods used for the estimation of its parameters (i.e. the eigensolution) result in a considerable computational effort, since they often deal with the assembly of dense covariance matrices. pg97-Felipe
...method for Feredholm.eq.Most of these techniques improve the efficiency of the calculation by generating sparse covariance matrices. However, some numerical stability concerns may arise from rejecting elements of the full covariance matrix, e.g., the loss of its positive definiteness. In this regard, the construction of a proper random field discretization and/or the generation of adaptable covariance kernels, is necessary to achieve an efficient and accurate eigenvalue problem solution. pg97-Felipe
In addition, K–L expansion provides sample functions that are not ergodic (in mean and autocorrelation) in a sample-by-sample sense in contrast to the spectral representation. Stefanou 2007
An important fact is that the homogeneity of stochastic fields generated by the K–L series is questionable:
the ensemble variance of the produced sample functions fluctuates over the domain of definition of the field thus leading to non-homogeneous fields.
Generally, the implementation of the KL representation requires extensive calculations. This situation persists even when dealing with stationary processes if the analysis is based on current practice, which rarely accounts for the fact that the KL and SP representations coincide for weakly stationary processes. Grigoriu 2007
The KL representation is not adequate for modeling non-Gaussian processes. As for the KL representation.
The difficulty in solving an integral equation analytically involved in K–L expansion limits its application. Phon 26
Generally
The convergence and accuracy of K–L expansion depend on the terms in the truncated K–L expansion. The lower the ratio of the length of the process over correlation parameter, the lesser are the terms needed for a given accuracy. Smooth covariance models exhibit faster convergence than models with less smooth covariance function. It is further shown that numerical K–L expansion reduces the effectiveness of the K–L expansion. In using numerical K–L expansion rather than analytical K–L expansion, more terms are needed to represent the random process for a given accuracy. Phoon