Simplex stochastic collocation with ENO-type stencil selection for robust uncertainty quantification
Introduction
Resolving discontinuities in stochastic problems is important, since they can lead to high sensitivities with respect to input uncertainties. They can also result in oscillatory approximations and, consequently, in the prediction of non-zero probabilities for unphysical realizations such as negative static pressures. In order to avoid these problems, the polynomial interpolation degree can locally be reduced to a linear approximation to avoid overshoots at the discontinuity in a multi-element uncertainty quantification (UQ) approach. In this context, two points are essential to maintain a highly accurate solution despite the locally first degree approximation. Firstly, the region in which the interpolation is reduced to a piecewise linear function should be as small as possible. This means that the samples need to be concentrated around the discontinuity to pinpoint its location. Secondly, the higher degree interpolation in the smooth regions should be extended as close as possible up to the discontinuity to maintain high order accuracy near the singularity. These two objectives are achieved here by introducing an Essentially Non-Oscillatory (ENO) type stencil selection into the Simplex Stochastic Collocation (SSC) method.
The ENO scheme has been developed by Harten and Osher [13] as a robust spatial discretization in the finite volume method (FVM) for deterministic Computational Fluid Dynamics (CFD) [14]. In that field, the robust approximation of discontinuities is critical for resolving shock waves and contact surfaces in the flow field. Therefore, it was proposed by Abgrall [1] and Barth [6] to use shock-capturing FVM to discretize also the parametric space to obtain robust approximations for stochastic CFD. These FVM discretizations of the combined physical and parametric space use the ENO and Weighted Essentially Non-Oscillatory (WENO) [16] schemes. Since FVM has originally been developed for the three-dimensional physical space, the direct extension of FVM to these high dimensional parametric spaces can, however, be inefficient due to the curse-of-dimensionality. Because of the different nature of the parametric space, there are also no physical fluxes between the cells in the stochastic directions, which form the basis of FVM. Therefore, we follow a different approach to extend the robustness of FVM to parametric space. We reformulate the robustness principles of FVM in terms of the parametric space and develop new UQ methods that satisfy these concepts in the stochastic dimensions. This approach has the advantages that it maintains the FVM robustness in combination with the effectiveness of specifically designed methods for UQ, which can also be used with other spatial discretizations than FVM. In this way, we have previously introduced, for instance, the Total Variation Diminishing (TVD) [12], [31], Extremum Diminishing (ED) [15], [33], and Local Extremum Diminishing (LED) [15], [35] principles into UQ and proposed the Essentially Extremum Diminishing (EED) [34] concept.
The ENO spatial discretization [13] achieves an essentially non-oscillatory approximation of the solution of hyperbolic conservation laws. Non-oscillatory means, in this context, that the number of local extrema in the solution does not increase with time. The ENO scheme obtains this property using an adaptive-stencil approach with a uniform polynomial degree for reconstructing the spatial fluxes. Each spatial cell is assigned r stencils of degree p, all of which include the cell itself. Out of this set of candidate stencils , the stencil is selected for cell that results in the interpolation which is smoothest in some sense based on an indicator of smoothness . In this way, a cell next to a discontinuity is adaptively given a stencil consisting of the smooth part of the solution, which avoids Gibbs-like oscillations in physical space. Attention has been paid to the efficient implementation of ENO schemes by Shu and Osher [24], [25]. Fig. 1 shows an example of the ENO stencil selection in a FVM discretization of a discontinuity in one spatial dimension using piecewise quadratic polynomials.
ENO-type stencil selection is here used in the SSC multi-element UQ method to obtain an accurate approximation of discontinuities in parametric space. Multi-element UQ methods discretize the stochastic dimensions using multiple subdomains comparable to spatial discretizations in physical space. These local methods [3], [19], [28] can be based on Stochastic Galerkin (SG) projections of Polynomial Chaos (PC) expansions [10], [36] in each of the subdomains. Other methods [2], [8], [17] use a Stochastic Collocation (SC) approach [4], [37] to construct the local polynomial approximations based on sampling at quadrature points in the elements. These methods commonly use sparse grids of Gauss quadrature rules in hypercube subdomains combined with solution-based refinement measures for resolving nonlinearities. Because of the hypercube elements, these methods are most effective in capturing discontinuities that are aligned with one of the stochastic coordinates.
In contrast, the SSC method [35], [34] is based on a simplex tessellation of the parametric space with sampling points at the vertexes of the simplex elements. The polynomial approximation in the simplexes is built using higher degree interpolation stencils , with local polynomial degree , consisting of samples in the vertexes of surrounding simplexes. The degree is controlled by a Local Extremum Conserving (LEC) limiter, which reduces and the stencil size to avoid overshoots in the interpolation of the samples where necessary. The limiter, therefore, leads to a non-uniform polynomial degree that reduces to a linear interpolation in simplexes which contain a discontinuity and that increases away from singularities. SSC employs adaptive refinement measures based on the hierarchical surplus and the geometrical properties of the simplexes to identify the location of discontinuities. However, the limiter can result in an excessive reduction of the polynomial degree also at significant distances away from a discontinuity. Since the polynomial degree affects the refinement criteria, this can also deteriorate the effectiveness of the refinement to sharply resolve singularities.
In order to obtain a more accurate solution of nonlinear response surfaces, ENO-type stencil selection is introduced into the SSC method. For each simplex stencils are constructed that contain , and the stencil is selected that results in the smoothest interpolation . The polynomial degree of the candidate stencils that are accepted by the LEC limiter, is used as the indicator of smoothness . A simplex near a discontinuity, therefore, achieves a higher order approximation by assigning a higher degree interpolation stencil that does not contain the discontinuity. The higher polynomial degree leads also to lower values of the refinement measures in , which restricts the refinement more to the simplexes that contain the discontinuity. The stencil selection does not affect the linear approximation in the latter simplexes, since the LEC limiter rejects the higher degree stencils that contain these elements.
The developed SSC–ENO method is applied to a chemical model problem and a shock tube problem. The chemical model problem is used to analyze the impact of uncertainty in initial conditions on the formation of a discontinuity in time. The initial-value problem with a normal input distribution was proposed by Reagan et al. [23] as a model involving nonlinearity and bifurcation behavior, which are both characteristic for chemical systems. They used it to determine the effectiveness of global PC formulations for problems resulting in a large coefficient of variation (CoV). The model was also considered by Le Maître et al. [20] to compare the global PC expansion with their multi-resolution scheme. They both conclude that the problem is too challenging for global representations. A system with similar behavior describing the movement of a particle subject to a potential field and a friction force with an uncertain initial position was also studied by Le Maître et al. [18]. A different class of unsteady models that involves the long time integration problem is treated, for instance, in [9], [21], [30].
The shock tube problem is considered to study the solution of a system of hyperbolic conservation laws with uncertain discontinuity locations in space. Uncertainty analysis of hyperbolic systems has received relatively little attention [11]. The example involves Sod’s Riemann problem [26] for the Euler equations of inviscid compressible gas dynamics. This Riemann problem was used by Poëtte et al. [22] to illustrate their PC method based on an entropic variable in an example with steep fronts and shocks. The problem was also studied by Tryoen et al. [27] using upwinding in a multi-resolution approach based on local SG projections of the PC expansion. Abgrall [1] used shock tube like test cases to demonstrate the application of his method to the Euler equations.
The efficient implementation of ENO-type stencil selection in the SSC method is discussed in Section 2. In Section 3, the effectiveness of the resulting SSC–ENO approach is analyzed for a discontinuous test function. The chemical model problem and the shock tube problem are considered in Sections 4 Chemical model problem, 5 Shock tube problem, respectively. The conclusions are summarized in Section 6.
Section snippets
Simplex stochastic collocation with stencil selection
An efficient algorithm for the SSC–ENO interpolation stencil selection is developed below. The implementation and the effect on the adaptive refinement are also discussed.
Multidimensional discontinuous test function
In order to isolate the effect of the introduced stencil selection, results of the SSC–ENO method are compared to those of SSC without stencil selection. A test function with discontinuities that are not aligned with the grid is considered up to five dimensions. The impact of the refinement measures on the comparison is reported in Appendix A.
Chemical model problem
The SSC–ENO method is applied to a chemical model problem with a discontinuity developing from two different smooth initial conditions.
Shock tube problem
A shock tube problem is used here to study the impact of uncertain discontinuity locations on a spatial flow field compared to the deterministic solution.
Conclusions
Essentially Non-Oscillatory (ENO) type stencil selection is introduced into the Simplex Stochastic Collocation (SSC) method to achieve an accurate approximation of discontinuities in parametric space. The stencil selection for simplex chooses the stencil with the highest polynomial degree that is accepted by the Local Extremum Conserving (LEC) limiter. This results in an increase of the local polynomial degree in the smooth regions and a concentration of the refinement in the simplexes
Acknowledgments
This work was supported by the Netherlands Organization for Scientific Research (NWO) and the European Union Marie Curie Cofund Action under Rubicon grant 680-50-1002.
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