 Research article
 Open Access
Large scale random fields generation using localized Karhunen–Loève expansion
 Alfonso M. Panunzio^{1, 2}Email author,
 Régis Cottereau^{2} and
 Guillaume Puel^{2}
https://doi.org/10.1186/s4032301801147
© The Author(s) 2018
 Received: 3 April 2018
 Accepted: 24 July 2018
 Published: 2 August 2018
Abstract
In this paper the generation of random fields when the domain is much larger than the characteristic correlation length is made using an adaptation of the Karhunen–Loève expansion (KLE). The KLE requires the computation of the eigenfunctions and the eigenvalues of the covariance operator for its modal representation. This step can be very expensive if the domain is much larger than the correlation length. To deal with this issue, the domain is split in subdomains where this modal decomposition can be comfortably computed. The random coefficients of the KLE are conditioned in order to guarantee the continuity of the field and a proper representation of the covariance function on the whole domain. This technique can also be parallelized and applied for nonstationary random fields. Some numerical studies, with different correlation functions and lengths, are presented.
Keywords
 Random fields generation
 Karhunen–Loève expansion
 Large scale random fields
Introduction
The representation of fluctuating parameters by means of random fields is very common in many scientific domains. Samples of stationary random fields can be generated through a sum of harmonic functions with random uniform phase and amplitude depending on the spectral density [1, 2]. This kind of representation can be performed in the spectral domain [3–5], leading to the spectral representation method that can be efficiently computed using the FFT [6]. In case of multidimensional random fields, the spectral representation has been combined with the turning bands methods [7] for a more efficient computation [8]. In a huge domain the numerical cost can be a major issue. In [9], to deal with this problem, the domain is split in several small subdomains in which the samples of the random fields are generated. Then, samples on the whole domain are obtained by using an overlapping technique.
Autoregressive models, in which a state only depends linearly on its own previous values, can also be employed to represent random fields [10–13]. The linear dependency coefficients can be computed by maximizing a likelihood function or by solving a linear system involving the inversion of a matrix representing the discretized covariance.
Other methods use the direct decomposition of the covariance to simulate generation of the field. The Cholesky decomposition of the discretized covariance can be used to correlate a set of random variables representing the discretization of the random field [14]. In some methods, when the random generation is required on a large size domain, polynomial approximations of the square root of the covariance are computed [15, 16]. The covariance decomposition can be combined with a ARMA representation to improve the computational efficiency [17].
Another way for generating random fields is the Karhunen–Loève expansion (KLE, [18, 19]), which is based on the covariance kernel modal decomposition on a finite domain [20, 21]. The KLE has been extensively used for representing random fluctuating properties in different engineering problems [22–24]. IWhile the spectral representation is optimal in meansquare sense on an infinitely large domain, the KLE is optimal on a finite domain. Moreover, one of the main advantages is that this expansion can also be directly applied for nonstationary processes. Different numerical methods exist for solving the covariance decomposition for the KLE [25, 26]. In the case of stationary covariances, the modal functions can be approximated by means of Fourier transforms [27–29] to reduce the computational complexity, but the application of the KLE in a very large domain compared to the correlation length still remains unaffordable. For a given meansquare error, The number of needed terms in the KLE grows as shown in [30], with the size of the domain. In some applications, for avoiding the KLE decomposition, known families of polynomials are used to parametrize the random field [31, 32], but they do not minimize the meansquare error as the KLE does.
The aim of this paper is to generate samples of a random field using the KLE, when the size of the domain is much larger than the correlation length and a direct KLE is not affordable because of the computational effort. The technique presented in [33], that deals with the representation of crosscorrelated random fields using the KLE, is here adapted to overcome this issue. At first the whole domain is split in small subdomains (with a size of few correlation lengths). The modal decomposition of the covariance operator is computed in a small subdomain where the computational effort is easily affordable and a reduced number of terms are needed for the KLE. Then, independent random samples are generated in each subdomain and, finally, the assembling is made by conditioning the KLE coefficients to obtain continuous samples of the random fields having the prescribed covariance function. In [33] the authors model a set of correlated random fields by imposing a correlation between the KLE coefficients of each random field. In this work, the same idea is used for correlating sets of KLE coefficients related to local regions of a large domain.
In this paper Gaussian random fields with different correlation structure are considered. NonGaussian random fields can be obtained by using the Rosenblatt transform [34] that allows to modify a Gaussian random field according to a chosen marginal first order probability density function (memoryless transformation). This transform also changes the correlation structure although, in most of the cases, one can deal with this issue by modifying the original correlation function as done in [35–37], where stationary fields are transformed into nonstationary fields. In other methods, nonGaussian fields can be obtained through transformations with memory [38]. These aspects are not discussed in this paper, where only Gaussian fields are considered.
The proposed method is firstly presented for the case of 1dimensional (1D) random processes (“Karhunen–Loève expansion for large scale 1D random processes” section) with an example of the application of the method. Some considerations about the continuity of the generated samples are discussed in “Continuity of the generated samples” section. The method is then generalized to 2D and 3D random fields (“Generation of multidimensional random fields” section). Then, the generation of nonstationary fields is discussed in “Extension to nonstationary random fields” section. In this work only random fields with values in \(\mathbb {R}\) are considered. Some numerical applications are provided (“Numerical applications” section).
Karhunen–Loève expansion for large scale 1D random processes
In this section the Karhunen–Loève expansion is adapted to generate samples of a large scale 1D stationary random process. In “Standard 1D Karhunen–Loève expansion” section, the generalities of the standard KLE, applied on a domain of size equal to L, are presented.
For simplicity, the method is firstly illustrated for a domain composed of just two subdomains (“Principles of the generation method on a large domain” section). The continuity of the generated samples is investigated in “Continuity of the generated samples” section. Then the extension to a domain composed of an arbitrary number of subdomains is discussed in “Extension of the expansion on an arbitrary large domain” section.
Standard 1D Karhunen–Loève expansion
When \(L/l_c \rightarrow \infty \) (where \(l_c\) is the correlation length) the KLE is equivalent to the spectral representation of random fields [30]. Equation (2) can be analytically solved in a few cases, such as rational spectra processes as detailed in [45] or in case of Slepian processes where the eigenfunctions are finite trigonometric polynomial functions [46]. However, generally the problem has to be solved numerically. When the random field is discretized into \(n_s\) uniformly spaced points over a domain, Eq. 2 leads to a \(n_s \times n_s\) eigenvalue problem. This corresponds to the optimal linear estimation method [25], which is used in this paper. When the domain is huge and a fine discretization of the field is needed the eigenproblem is very heavy to solve, having \(\mathcal {O}(n_s^3)\) complexity.
Other methods, described and compared in [26] can be used to approximate the eigenfunctions more efficiently, such as collocation and Galerkin integration [26]. In this case, the eigenfunctions are approximated by a set of \(\tilde{n}_s<n_s\) basis functions, leading to \(\tilde{n}_s \times \tilde{n}_s\) matrix generalized eigenproblem whose complexity is \(\mathcal {O}(2\tilde{n}_s^3)\). However, since \(\tilde{n}_s\) increases with the size of the domain, solving the eigenproblem still remains an obstacle for large scale random fields. In the frame of this work, any numerical method can be used of solving the KLE but, for simplicity, the optimal linear estimation method is employed.
Principles of the generation method on a large domain
The general principles of the random fields generation method, which is the main object of this paper, are highlighted in this section through the explanation of a simple case.
In this section the eigenfunctions and eigenvalues calculated in Eq. (2), defined for \(s\in [0,L]\), are used to generate a sample of the random process in a domain with \(s\in [0,2L]\). The size of the domain is thus doubled. The method can be straightforwardly extended to an arbitrarysized domain by iterating the technique presented in this section. The general idea is to firstly generate two independent samples, each covering half of the domain, and then impose a correlation between the KLE coefficients of the two samples. At the end, continuous samples of the process on the whole domain with a respected correlation structure are obtained.
The positive definiteness of the matrix \(\mathbf {I}\mathbf {K}^\mathrm {T}\mathbf {K}\) is demonstrated in B. Note that the numerical cost to perform this Cholesky decomposition is not related to the size of the whole domain: it is related to the size of the subdomain L and the KLE truncation error \(\epsilon ^2_{KL}\) (which determines the number of terms N).
Correlation structure across the coupled subdomains
In Eq. (14), the samples are piecewise generated in the two subdomains. The KLE coefficient set related to the second subdomain has been correlated to the first one in Eq. (13).
When \(N\rightarrow \infty \), because of the basis completeness (Eq. 4), the expression in Eq. (15) is exactly equal to \({C(st)}\). Otherwise, in the same way as the standard KLE, the crosscovariance is approximated. The crossspectrum of the two subsamples is equal to \(\mathfrak {S}(\omega ) \mathrm {e}^{{\imath }\omega L}\). The more similar the decays of \(\mathfrak {S}(\omega )\) and crossspectrum are, the fewer extra terms are needed for a good approximation of the correlation structure.
Continuity of the generated samples
Note that, because of the completeness of the KLE basis (Eq. 4), when \(N\rightarrow \infty \) the error tends to zero, making equal to one the probability that the left and the right limits take the same value (satisfying the continuity at the breaking location \(s=L\)).
Note that an overlapping method for representing large scale random fields is proposed in [9]. This overlapping strategy can be applied to any random fields generation methods (KLE included). By overlapping the subdomains the continuity issues are avoided, but an error on the correlation representation is introduced. In this paper, the correlation across the subdomains is imposed. Then the continuity error that can be reduced by adding more terms in the expansion. Since the modal decomposition is affordably performed in one subdomain, adding more terms in the expansion is not a numerically expensive task.
Extension of the expansion on an arbitrary large domain
In this section the generation in each subdomain is performed sequentially. Some aspects concerning the parallelisation are discussed in “Parallel computing of the random field generation” section.
Generation of multidimensional random fields
In this section, the random fields generation method presented in “Karhunen–Loève expansion for large scale 1D random processes” section for 1D processes is generalized to 2D and 3D random fields. Note that the term “multidimensional” refers to the dimension of the indexing variable of the fields. In this work only random fields with values in \(\mathbb {R}\) are considered. The principle of the method is essentially the same as what has been presented in “Standard 1D Karhunen–Loève expansion” section. The only difference is that, in case of multidimensional random fields, one subdomain must be conditioned with more subdomains along the different directions (and not only one direction as for 1D processes).
In this section the generation method is described for a general case. If the tensorization is possible, the computational cost for the modal decomposition of the covariance and the random field generation can be reduced. The case generation with tensorizable correlation is reported in A.
\(f({\mathbf {s}},\theta )\) is centred random field, indexed by the variable \({\mathbf {s}} \in \mathbb {R}^d\) (with d being the dimensionality), with covariance function equal to \(C({\mathbf {s}},{\mathbf {t}})\).
For the application of the KLE, over the domain \(\mathbf {D}=[0,L]^d\), the modal decomposition of the covariance operator is performed as in Eq. (2) with multidimensional eigenfunctions. Then, N eigenvalues and eigenfunctions are retained in the expansion to obtain realizations of the field on the domain \(\mathbf {D}\) as in Eq. (5).

Step 1: Domain subdivision
The first step consists in the domain subdivision into \(M^d\) subdomains \(\mathbf {D}_{k}\) (with \(k=1,\dots ,M^d\)), each one \(L^d\) sized. An example of the domain subdivision in a 2D case with \(M=3\) is shown in Fig. 2.

Step 2: Determination of the coupling matrices
The next step is the computation of the coupling matrices. Each subdomain is connected with the surrounding subdomains. \(\mathbf {K}^{(pq)}\) indicates the coupling matrix concerning the subdomains \(\mathbf {D}_p\) and \(\mathbf {D}_q\). Its elements are calculated as:with \({\mathbf {o}}_{k}=[\min s_1,\dots ,\min s_d] \mid {\mathbf {s}} \in \mathbf {D}_{k} \). Note that, when the field is stationary, \(\mathbf {K}^{(pq)}=\mathbf {K}^{(qp){\text {T}}}\). Moreover, two coupling matrices are equal if the relative position between their respective subdomains is the same in stationary conditions. For instance, with respect to Fig. 2, \(\mathbf {K}^{(12)}=\mathbf {K}^{(23)}\), \(\mathbf {K}^{(14)}=\mathbf {K}^{(47)}\), \(\mathbf {K}^{(15)}=\mathbf {K}^{(59)}\), and so on. In this way the number of coupling matrices, and the consecutive Cholesky decompositions, is reduced.$$\begin{aligned} {K}_{ij}^{(pq)} =\frac{1}{\sqrt{\lambda _i \lambda _j}}\int _{{\mathbf {s}} \in \mathbf {D}_p} \int _{ {\mathbf {t}} \in \mathbf {D}_q} C({\mathbf {s}},{\mathbf {t}}) \varphi _i({\mathbf {s}}{\mathbf {o}}_p)\varphi _j({\mathbf {t}}{\mathbf {o}}_q) \mathrm {d}{\mathbf {s}} \mathrm {d} {\mathbf {t}} \end{aligned}$$(23) 
Step 3: KLE coefficients conditioning
The third step is the conditioning of the KLE coefficients sets in each subdomain with respect to its neighbour subdomain. At first, the order in which the KLE coefficients sets are conditioned is chosen. Different strategies can be adopted. For example, referring to Fig. 2, the order can be chosen by simply using the domain numbering. Then, each set is generated and conditioned with the sets of all its neighbour subdomains that have been already generated.
The structure of the linear system ensures that the crosscorrelation between the neighbour sets is taken into account: by using the expectation of Eq. (24) and the definition of the system in Eq. (25) it can be proven that: \({{\mathbb {E}}[\tilde{{\mathbf {H}}}^{(k)}({\varvec{\theta }} _k) \tilde{{\mathbf {H}}}^{(q)^\text {T}}({\varvec{\theta }} _q)]=\mathbf {K}^{(kq)}}\). This ensures the respect of the correlation structure in the same way as for the 1D case (Eq. 15).

Step 4: Random field generation
Parallel computing of the random field generation
In this section the strategies to adopt for parallelizing the generation method are discussed. The sequential conditioning presented above is simply applicable but, since each part is conditioned by the previous ones iteratively, this technique is not parallelizable. When one wants to use several processors to generate a very large sample of the field another strategy is more advisable.
The first part of this section discusses the parallelization of the 1D processes generation technique. Then, the parallelization strategy for general multidimensional fields generation is described.
In this section the parallelization is performed with respect to the indexing variable of the random field (computation of several subdomains at the same time). However, it is always possible to run distributed computations along the statistical axis, if one needs to sample several realizations of the random fields.
1D processes generation parallel computing
Note that when M is even, the only difference is that the last subdomain (\(m=M\)) is only conditioned from the left side (as done in “Principles of the generation method on a large domain” section).
Multidimensional fields generation parallel computing
For parallelizing the multidimensional fields generation method presented in “Generation of multidimensional random fields” section, the only difference with the sequential conditioning technique is the subdomains ordering. Indeed, if two subdomains do not interact, i.e. they are enough far to consider their correlation equal to zero, they can be processed at the same time. Then the equations for the conditioning are the same as in “Generation of multidimensional random fields” section.
Extension to nonstationary random fields
All the methods cited above concern stationary random fields. The representation of nonstationary random fields can be achieved by modifying a stationary field (already generated) using one of the methods described in the introduction or the method proposed in this paper. A stationary process can be multiplied by a deterministic slowlyvarying function for reproducing a nonstationary effect as in [48, 49]. In [3, 50] the spectral representation is extended to nonstationary processes with evolutionary power spectrum [51], i.e. when the power spectral density can be modulated by a deterministic function depending on the frequency and the support variable, with the possibility to improve the computation by using the FFT [52]. Concerning the ARMA method, it has been extended to nonstationary processes by introducing timedependent coefficients [53].
Note that, because of covariance nonstationarity, the sets of eigenfunctions (and also their number) \(\varphi ^{(m)}_i(s)\) and \(\varphi ^{(m+1)}_i(s)\) related to two subdomains can be different. For this reason, the matrices \(\mathbf {K}^{(m)}\) and \(\mathbf {L}^{(m)}\) are not necessarily square.
Numerical applications
Example of generation on a large domain
The field is discretized into \((n_sM)^2\) steps, with \(n_s=100\). The KLE truncation error \(\epsilon ^2_{KL}\) is set to 0.001, giving a number of retained terms in the expansion N equal to 9600. Using directly the standard KLE on such a large domain as the one here considered is unaffordable: using the optimal linear estimation method [25] would require an eigendecomposition of a \({(n_sM)^2\times (n_sM)^2}\), i.e. \(4{,}410{,}000\times 4{,}410{,}000\), sized matrix.
The method described in “Generation of multidimensional random fields” section is employed in this section to generate the sample. No parallel computing is performed: this field is generated sequentially in each subdomain.
Computational time for generating a 2D random field with correlation function as in Eq. (37) on an extended domain \([0,ML]^2\), with \(M=21\) using the conditioned KLE
Operation  Elapsed time (s) 

Kernel modal decomposition  56.31 
Conditioning matrices computation  2220.17 
Random field sampling  1351.95 
Computational complexity
In this section the computation time of the standard KLE is compared with the generation method proposed in this work in the case of tensorizable and nontensorizable correlation functions. A 2D random field defined on the domain \({\mathbf {s}} \in [0,ML]^d\), with d being the dimension. The modal decomposition is solved using the optimal linear estimation method [25] in which the domain is uniformly discretized. Let us indicate as \(n_s\) the number of discretization steps of a segment of length L along one of the directions. The domain is thus discretized in \((n_sM)^d\) parts.
In the first part of this section, an example concerning a tensorizable 2D random field is presented. In the second part, the numerical complexity of the standard and the conditioned KLE are compared.
Numerical complexity of the standard KLE and the conditioned KLE
Operation  Standard KLE  Conditioned KLE 

Kernel modal decomposition  \(\mathcal {O}((n_sM)^{3d})\)  \(\mathcal {O}(n_s^{3d})\) 
Conditioning matrices computation  –  \(\mathcal {O}(N^3(3^d1)^3)\) 
Random field sampling  \(\mathcal {O}(n_s^d M^{2d} N)\)  \(\mathcal {O}((n_sM)^dN)\) 
Numerical complexity of the standard KLE and the conditioned KLE
Operation  Standard KLE  Conditioned KLE 

Kernel modal decomposition  \(\mathcal {O}((n_sM)^3d)\)  \(\mathcal {O}(n_s^3d)\) 
Conditioning matrices computation  –  \(\mathcal {O}(\bar{N}^3d)\) 
Random field sampling  \(\mathcal {O}(n_s^dM^{d+1}\bar{N})\)  \(\mathcal {O}((n_sM)^d\bar{N})\) 
Kernel modal decomposition
The modal decomposition complexity does not depend on M when the conditioned KLE is employed. For the standard KLE, this complexity grows with \(\mathcal {O}(M^3)\), when the covariance kernel is tensorizable, and \(\mathcal {O}(M^{3d})\) when not. This is the main limit for directly using the KLE on the whole domain.
Conditioning matrices computation
The computation of the matrices used for conditioning the KLE coefficients, described in “Generation of multidimensional random fields” section, requires some Cholesky decomposition operations. Its complexity does not depend on M, but on the number of KLE terms (depending on the truncation error) and the sequential prolongation strategy. In fact, the size of the linear system in Eq. (25), that is solved with Cholesky decomposition of the coefficient matrix, depends on the number of connections of the considered subdomain. The complexity, indicated in Table 2, for the nontensorizable kernel case is referred to the resolution of the largest linear system (the one that is associated with the subdomain having the largest number of neighbours).
Random field sampling
The sample generation is the only operation growing with M when the conditioned KLE is used: in this case the complexity grows with \(\mathcal {O}(M^d)\). The use of the standard KLE requires a complexity, for this stage, of \(\mathcal {O}(M^{2d})\), in case of nontensorizable kernel and \(\mathcal {O}(M^{d+1})\), in case of a tensorizable kernel. Another advantage of using the method proposed in this work, is the possibility, due to the domain splitting, of storing separately each part of the sample corresponding to a subdomain. This can prevent memory issues when the total size of the domain is huge.
Since the modal decomposition is the most expensive stage, it is convenient to partition the domain in smaller subdomains, even tough, in this case, the random field sampling stage will be more expensive. However, the correlation structure should be well represented in one subdomain, i.e. the correlation should tend to zero at a lag equal to L. This is because, with the method proposed in this paper, the correlation across two subdomains which are not neighbour is neglected.
Influence of the correlation kernel
Correlation functions used in “Inuence of the correlation kernel” section and relative truncation errors, number of retained KLE terms and continuity errors, with \(l_c=0.15L\) in all the cases and \(\tau =st\)
Kernel name  Function  Truncation error \(\varvec{\epsilon _{KL}^2}\)  Retained terms \(\varvec{N}\)  Continuity error \(\varvec{\epsilon _c}\) 

Exponential  \(\exp \left( \dfrac{\tau }{l_c} \right) \)  0.001  98  \(1.7\times 10^{6}\) 
Triangular  \(\max \left( 1\dfrac{\tau }{l_c},0 \right) \)  0.001  86  \(1.8\times 10^{3}\) 
Damped sine  \(\dfrac{l_c}{10\tau }\sin \left( \dfrac{10\tau }{l_c} \right) \)  0.001  24  \(1.9\times 10^{2}\) 
Gaussian  \(\exp \left( \dfrac{\tau }{l_c},\right) ^2\)  0.001  12  \(9.2\times 10^{3}\) 
The KLE truncation error \(\epsilon _{KL}^2\) (Eq. 8) is set to 0.001. The choice of this truncation error determines the number of retained KLE terms N and the continuity error (defined in Eq. 19), that are indicated in Table 4. Note how the number of terms increases as the spectral density decay (Fig. 11) is slower. For instance, the exponential correlated processed requires more than 8 times more terms than the Gaussian correlated process.
Note that the samplepath continuity of the field is ensured only if the number of terms in the KLE tends to infinity (use of a complete orthonormal basis). In the other cases, the samplepath continuity is not recovered in the breaking points locations, but the discontinuity jump can be reduced in order to be enough small for the numerical applications.
Conclusion
Solving the KLE modal decomposition, when the domain is much larger than the correlation length and a small discretization step is needed, represents a computational issue that can become unaffordable. To deal with this issue, a “conditioned” Karhunen–Loève expansion is proposed. The domain is subdivided in subdomains where the modal decomposition can be comfortably computed. Then, parts of the field are generated in each subdomain and conditioned with their neighbours in order to ensure the continuity and the correlation structure.
This generation method is applicable to multidimensional fields with a strong simplification in case of tensorizable covariance kernels. The capability of the KLE for generating nonstationary random fields is also preserved with the proposed generation method. In every case the computational cost is largely reduced. Another important advantage is that the proposed technique can be easily parallelized to further reduce the computational time.
Moreover, another advantage of using the method proposed in this work is the possibility, due to the domain splitting, of working locally on each part of the sample corresponding to a subdomain. This can prevent memory issues when the total size of the domain is huge.
The method presented in this paper concerns Gaussian centred random fields. NonGaussians fields can be obtained by transforming the generated Gaussian fields by the application of the Rosenblatt transformation to obtain the prescribed first order marginal probability function.
Declarations
Author's contributions
AMP, RC and GP conceived the presented technique and carried out the mathematical description. AMP wrote this manuscript with support from RC and GP. All authors read and approved the final manuscript.
Acknowlegements
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
Data will be available on demand.
Consent for publication
All authors consent for pubblication.
Ethics approval and consent to participate
Not applicable.
Funding
Not applicable.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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