Sparse polynomial chaos expansion based on D-MORPH regression

Active learning polynomial chaos expansion for reliability analysis by maximizing expected indicator function prediction error

“…The selection criterions for retaining the basis functions are based on the determination coefficient R 2 and least angle regression technique . Later, many other attempts also have been made to develop sparse PCE model in the field of UQ, the common idea hold in these methods is that the PCE coefficients are sparse (ie, having only several dominant coefficients). Given the training sample { X , Y }, where X = { x 1 , …, x N } T is the input data, Y = { Y 1 , …, Y N } T is the corresponding model response and N is the size of sample, the dominant PCE coefficients can be recovered by solving the following optimization problem where ‖ ω α ‖ 1 is the l 1 norm of PCE coefficients, and ϵ is a tolerance parameter necessitated by the truncation error and Φ ( i , j ) = ψ j ( x i )( i = 1, …, N ; j = 1, …, P + 1) is the measure matrix.…”

“…Given the training sample { X , Y }, where X = { x 1 , …, x N } T is the input data, Y = { Y 1 , …, Y N } T is the corresponding model response and N is the size of sample, the dominant PCE coefficients can be recovered by solving the following optimization problem where ‖ ω α ‖ 1 is the l 1 norm of PCE coefficients, and ϵ is a tolerance parameter necessitated by the truncation error and Φ ( i , j ) = ψ j ( x i )( i = 1, …, N ; j = 1, …, P + 1) is the measure matrix. To solve above optimization problem, a large number of powerful algorithms have been proposed, such as some adaptive methods and l 1 minimization methods. The adaptive methods aim at selecting the significant basis functions from full PCE sequentially using only few samples based on the well‐defined selection criterion, such as the correlation criterion in References and the variance contribution criterion in Reference .…”

“…For PCE model, the key point is to determine the expansion coefficients. Existing approaches for obtaining PCE coefficients include least square regression, least angle regression, compressive sensing, support vector regression, orthogonal match pursuit, and sparse grid approximation . Unfortunately, all these methods cannot provide the local prediction error information for PCE model, which hinders the application of PCE for structural reliability analysis adaptively.…”

A new surrogate modeling method combining polynomial chaos expansion and Gaussian kernel in a sparse Bayesian learning framework

“…This algorithm is available from UQLab software introduced by Marelli and Sudret . On the sparsity assumption, many other techniques are established in the context of compressed sensing . PCE is more suitable for capturing the global trend.…”

“…15 On the sparsity assumption, many other techniques are established in the context of compressed sensing. [16][17][18][19] PCE is more suitable for capturing the global trend. However, PCE is often not adequate for capturing local accuracy in the close neighborhood of the sample points.…”