Sequential Simulation with Iterative Methods

Abstract

The sequential Gaussian algorithm is widespread to simulate Gaussian random fields. In practice, the determination of the successive conditional distributions only accounts for the information available in a moving neighborhood centered on the target location, which provokes a loss of accuracy with respect to a unique neighborhood implementation. In order to reduce this loss of accuracy, iterative methods for solving large kriging systems of equations are used to improve the determination of the conditional distributions, taking the results obtained in a moving neighborhood as a first approximation. Numerical experiments are presented to show the proposed strategies and the improvements in the reproduction of the correlation structure of the simulated random field.

Appendix: Analysis of Computational Cost

In this appendix we assess the computational cost of sequential simulation using a moving neighborhood of fixed size k, and the additional cost of solving the kriging system with an iterative method in each step of sequential simulation.

1.1 A.1 Solving Kriging Systems with a Moving Neighborhood

If we sequentially simulate a Gaussian random vector with n components using a moving neighborhood containing k variables, one has to solve one kriging system of size 1×1, one system of size 2×2,… , one system of size (k−1)×(k−1) and (n−k) systems of size k×k. To solve a linear system of equations of size p×p by means of the Gaussian elimination (direct method), the number of floating point operations is 2p ³/3 [3]. Accordingly, the computational cost in floating point operations to solve all the kriging systems in the sequential simulation with a neighborhood of k variables is O(nk ³):

(9)

If we consider the ideal case of a unique neighborhood implementation (k=n−1), the number of floating point operations is O(n ⁴):

(10)

1.2 A.2 Solving Kriging Systems with Iterative Methods

The computational cost to calculate the kriging weights and that are used to construct initial approximations (to apply the iterative methods), using a moving neighborhood of size k and Gaussian elimination, is given by (9). To this cost, one has to add the cost of the iterative method from component (k+2) to component n of the simulated vector.

1.2.1 A.2.1 Gauss-Seidel Method

To solve a linear system of equations of size p×p with the Gauss-Seidel iterative method, the number of floating point operations is 2sp ², where s is the number of iterations performed [3]. Therefore, the number of floating point operations for the (n−k−1) systems where we apply the Gauss-Seidel method is:

(11)

With respect to the classical implementation (9), the additional cost is O(sn ³).

1.2.2 A.2.2 Conjugate Gradient Method

The cost to solve a linear system of equations of size p×p with Conjugate Gradient method is s(6p+3+2u _p ), where u _p is the number of nonzero elements of the left-hand side matrix of the system of equations [9]. Accordingly, the number of floating point operations for solving the last (n−k−1) kriging systems with the Conjugate Gradient method is:

The computational order of this additional is O(sn ²) when the matrix of the system has many zero entries.

1.3 A.3 CPU Time on a Numerical Example

To illustrate the gain in CPU time when using an iterative method (here, the Conjugate Gradient) rather than the direct method (Gaussian elimination) to solve the successive kriging systems, we have run an experiment on a regular 2D grid with 100×100 nodes, which are visited through a random sequence using multiple grids [15]. Table 5 shows the standardized Frobenius norm of $C_{\tilde{Y}}-C_{Y}$ and the CPU time for direct sequential Gaussian simulation (SGS) and for the proposed algorithm (SGS + CG), using the same set of conditioning variables to construct the successive conditional distributions (specifically, the variables located within a circle of radius 30 centered on the target node). The initial approximation for the Conjugate Gradient method is a vector of zeros. It is seen that the accuracy is comparable for both approaches (SGS and SGS+CG), insofar as the standardized Frobenius norms are practically the same, but there are considerable differences in CPU times. These differences increase for larger simulation grids and for larger sizes of the neighborhood used to select the conditioning variables.

Table 5 Standardized Frobenius norms of $C_{\tilde{Y}}-C_{Y}$ (η) and CPU times (in seconds) for sequential Gaussian simulation (SGS) and sequential Gaussian simulation with Conjugate Gradient method (SGS + CG) when the first and second neighborhoods are equal (a circle of radius 30). Smallest CPU times are indicated in bold type, largest CPU times in italic type