1Blurb:: Uncertainty quantification with stochastic collocation 2 3Description:: 4Stochastic collocation is a general framework for approximate 5representation of random response functions in terms of 6finite-dimensional interpolation bases. 7 8The stochastic collocation (SC) method is very similar to 9\ref method-polynomial_chaos, with the key difference that the orthogonal 10polynomial basis functions are replaced with interpolation polynomial 11bases. The interpolation polynomials may be either local or global 12and either value-based or gradient-enhanced. In the local case, 13valued-based are piecewise linear splines and gradient-enhanced are 14piecewise cubic splines, and in the global case, valued-based are 15Lagrange interpolants and gradient-enhanced are Hermite interpolants. 16A value-based expansion takes the form 17 18\f[R = \sum_{i=1}^{N_p} r_i L_i(\xi) \f] 19 20where \f$N_p\f$ is the total number of collocation points, \f$r_i\f$ 21is a response value at the \f$i^{th}\f$ collocation point, \f$L_i\f$ 22is the \f$i^{th}\f$ multidimensional interpolation polynomial, and 23\f$\xi\f$ is a vector of standardized random variables. 24 25Thus, in PCE, one forms coefficients for known orthogonal polynomial 26basis functions, whereas SC forms multidimensional interpolation 27functions for known coefficients. 28 29<!-- 30The following provides details on the various stochastic collocation 31method options in Dakota. 32 33The groups and optional keywords relating to method specification are: 34\li Group 1 35\li Group 2 36\li Group 3 37\li Group 4 38\li \c dimension_preference 39\li \c use_derivatives 40\li \c fixed_seed 41 42In addition, this method treats variables that are not 43aleatoric-uncertain different, despite the \ref variables-active keyword. 44 45Group 5 and the remainder of the optional keywords relate to the output 46of the method. 47--> 48 49<b> Basis polynomial family (Group 2) </b> 50 51In addition to the \ref method-stoch_collocation-askey and \ref 52method-stoch_collocation-wiener basis types also supported by \ref 53method-polynomial_chaos, SC supports the option of \c piecewise local 54basis functions. These are piecewise linear splines, or in the case of 55gradient-enhanced interpolation via the \c use_derivatives 56specification, piecewise cubic Hermite splines. Both of these basis 57options provide local support only over the range from the 58interpolated point to its nearest 1D neighbors (within a tensor grid 59or within each of the tensor grids underlying a sparse grid), which 60exchanges the fast convergence of global bases for smooth functions 61for robustness in the representation of nonsmooth response functions 62(that can induce Gibbs oscillations when using high-order global basis 63functions). When local basis functions are used, the usage of 64nonequidistant collocation points (e.g., the Gauss point selections 65described above) is not well motivated, so equidistant Newton-Cotes 66points are employed in this case, and all random variable types are 67transformed to standard uniform probability space. The 68global gradient-enhanced interpolants (Hermite interpolation 69polynomials) are also restricted to uniform or transformed uniform 70random variables (due to the need to compute collocation weights by 71integration of the basis polynomials) and share the variable support 72shown in \ref topic-variable_support for Piecewise SE. Due to numerical 73instability in these high-order basis polynomials, they are deactivated 74by default but can be activated by developers using a compile-time switch. 75 76<b> Interpolation grid type (Group 3) </b> 77 78To form the multidimensional interpolants \f$L_i\f$ of the expansion, 79two options are provided. 80 81<ol> 82<li> interpolation on a tensor-product of Gaussian quadrature points 83 (specified with \c quadrature_order and, optionally, \c 84 dimension_preference for anisotropic tensor grids). As for PCE, 85 non-nested Gauss rules are employed by default, although the 86 presence of \c p_refinement or \c h_refinement will result in 87 default usage of nested rules for normal or uniform variables 88 after any variable transformations have been applied (both 89 defaults can be overridden using explicit \c nested or \c 90 non_nested specifications). 91<li> interpolation on a Smolyak sparse grid (specified with \c 92 sparse_grid_level and, optionally, \c dimension_preference for 93 anisotropic sparse grids) defined from Gaussian rules. As for 94 sparse PCE, nested rules are employed unless overridden with the 95 \c non_nested option, and the growth rules are restricted unless 96 overridden by the \c unrestricted keyword. 97</ol> 98 99Another distinguishing characteristic of stochastic collocation 100relative to \ref method-polynomial_chaos is the ability to reformulate the 101interpolation problem from a \c nodal interpolation approach into a \c 102hierarchical formulation in which each new level of interpolation 103defines a set of incremental refinements (known as hierarchical 104surpluses) layered on top of the interpolants from previous levels. 105This formulation lends itself naturally to uniform or adaptive 106refinement strategies, since the hierarchical surpluses can be 107interpreted as error estimates for the interpolant. Either global or 108local/piecewise interpolants in either value-based or 109gradient-enhanced approaches can be formulated using \c hierarchical 110interpolation. The primary restriction for the hierarchical case is 111that it currently requires a sparse grid approach using nested 112quadrature rules (Genz-Keister, Gauss-Patterson, or Newton-Cotes for 113standard normals and standard uniforms in a transformed space: Askey, 114Wiener, or Piecewise settings may be required), although this 115restriction can be relaxed in the future. A selection of \c 116hierarchical interpolation will provide greater precision in the 117increments to mean, standard deviation, covariance, and 118reliability-based level mappings induced by a grid change within 119uniform or goal-oriented adaptive refinement approaches (see following 120section). 121 122It is important to note that, while \c quadrature_order and \c 123sparse_grid_level are array inputs, only one scalar from these arrays 124is active at a time for a particular expansion estimation. These 125scalars can be augmented with a \c dimension_preference to support 126anisotropy across the random dimension set. The array inputs are 127present to support advanced use cases such as multifidelity UQ, where 128multiple grid resolutions can be employed. 129 130<b> Automated refinement type (Group 1) </b> 131 132Automated expansion refinement can be selected as either \c 133p_refinement or \c h_refinement, and either refinement specification 134can be either \c uniform or \c dimension_adaptive. The \c 135dimension_adaptive case can be further specified as either \c sobol or 136\c generalized (\c decay not supported). Each of these automated 137refinement approaches makes use of the \c max_iterations and \c 138convergence_tolerance iteration controls. 139The \c h_refinement specification involves use of the same piecewise 140interpolants (linear or cubic Hermite splines) described above for the 141\c piecewise specification option (it is not necessary to redundantly 142specify \c piecewise in the case of \c h_refinement). In future 143releases, the \c hierarchical interpolation approach will enable local 144refinement in addition to the current \c uniform and \c 145dimension_adaptive options. 146 147<b> Covariance type (Group 5) </b> 148 149These two keywords are used to specify how this method computes, stores, 150and outputs the covariance of the responses. In particular, the diagonal 151covariance option is provided for reducing post-processing overhead and 152output volume in high dimensional applications. 153 154<b> Active Variables </b> 155 156The default behavior is to form expansions over aleatory 157uncertain continuous variables. To form expansions 158over a broader set of variables, one needs to specify 159\c active followed by \c state, \c epistemic, \c design, or \c all 160in the variables specification block. 161 162For continuous design, continuous state, and continuous epistemic 163uncertain variables included in the expansion, interpolation points 164for these dimensions are based on Gauss-Legendre rules if non-nested, 165Gauss-Patterson rules if nested, and Newton-Cotes points in the case 166of piecewise bases. Again, when probability integrals are evaluated, 167only the aleatory random variable domain is integrated, leaving behind 168a polynomial relationship between the statistics and the remaining 169design/state/epistemic variables. 170 171<b> Optional Keywords regarding method outputs </b> 172 173Each of these sampling specifications refer to sampling on the SC 174approximation for the purposes of generating approximate statistics. 175\li \c sample_type 176\li \c samples 177\li \c seed 178\li \c fixed_seed 179\li \c rng 180\li \c probability_refinement 181\li \c distribution 182\li \c reliability_levels 183\li \c response_levels 184\li \c probability_levels 185\li \c gen_reliability_levels 186 187Since SC approximations are formed on structured grids, there should 188be no ambiguity with simulation sampling for generating the SC expansion. 189 190When using the \c probability_refinement control, the number of 191refinement samples is not under the user's control (these evaluations 192are approximation-based, so management of this expense is less 193critical). This option allows for refinement of probability and 194generalized reliability results using importance sampling. 195 196<b> Multi-fidelity UQ </b> 197 198When using multifidelity UQ, the high fidelity expansion generated 199from combining the low fidelity and discrepancy expansions retains the 200polynomial form of the low fidelity expansion (only the coefficients 201are updated). Refer to \ref method-polynomial_chaos for information 202on the multifidelity interpretation of array inputs for \c 203quadrature_order and \c sparse_grid_level. 204 205<b> Usage Tips </b> 206 207If \e n is small, then tensor-product Gaussian quadrature is again the 208preferred choice. For larger \e n, tensor-product quadrature quickly 209becomes too expensive and the sparse grid approach is preferred. For 210self-consistency in growth rates, nested rules employ restricted 211exponential growth (with the exception of the \c dimension_adaptive \c 212p_refinement \c generalized case) for consistency with the linear 213growth used for non-nested Gauss rules (integrand precision 214\f$i=4l+1\f$ for sparse grid level \e l and \f$i=2m-1\f$ for tensor 215grid order \e m). 216 217<b> Additional Resources </b> 218 219%Dakota provides access to SC methods through the NonDStochCollocation 220class. Refer to the Uncertainty Quantification Capabilities chapter of 221the Users Manual \cite UsersMan and the Stochastic Expansion Methods 222chapter of the Theory Manual \cite TheoMan for additional information 223on the SC algorithm. 224 225<b> Expected HDF5 Output </b> 226 227If Dakota was built with HDF5 support and run with the 228\ref environment-results_output-hdf5 keyword, this method 229writes the following results to HDF5: 230 231- \ref hdf5_results-se_moments 232- \ref hdf5_results-pdf 233- \ref hdf5_results-level_mappings 234- \ref hdf5_results-vbd 235 236 237Topics:: 238 239Examples:: 240\verbatim 241method, 242 stoch_collocation 243 sparse_grid_level = 2 244 samples = 10000 seed = 12347 rng rnum2 245 response_levels = .1 1. 50. 100. 500. 1000. 246 variance_based_decomp 247\endverbatim 248 249Theory:: 250As mentioned above, a value-based expansion takes the form 251 252\f[R = \sum_{i=1}^{N_p} r_i L_i(\xi) \f] 253 254The \f$i^{th}\f$ interpolation polynomial assumes the value of 1 at 255the \f$i^{th}\f$ collocation point and 0 at all other collocation 256points, involving either a global Lagrange polynomial basis or local 257piecewise splines. It is easy to see that the approximation reproduces 258the response values at the collocation points and interpolates between 259these values at other points. A gradient-enhanced expansion (selected 260via the \c use_derivatives keyword) involves both type 1 and type 2 261basis functions as follows: 262 263\f[R = \sum_{i=1}^{N_p} [ r_i H^{(1)}_i(\xi) 264 + \sum_{j=1}^n \frac{dr_i}{d\xi_j} H^{(2)}_{ij}(\xi) ] \f] 265 266where the \f$i^{th}\f$ type 1 interpolant produces 1 for the value at 267the \f$i^{th}\f$ collocation point, 0 for values at all other 268collocation points, and 0 for derivatives (when differentiated) at all 269collocation points, and the \f$ij^{th}\f$ type 2 interpolant produces 2700 for values at all collocation points, 1 for the \f$j^{th}\f$ 271derivative component at the \f$i^{th}\f$ collocation point, and 0 for 272the \f$j^{th}\f$ derivative component at all other collocation points. 273Again, this expansion reproduces the response values at each of the 274collocation points, and when differentiated, also reproduces each 275component of the gradient at each of the collocation points. Since 276this technique includes the derivative interpolation explicitly, it 277eliminates issues with matrix ill-conditioning that can occur in the 278gradient-enhanced PCE approach based on regression. However, the 279calculation of high-order global polynomials with the desired 280interpolation properties can be similarly numerically challenging such 281that the use of local cubic splines is recommended due to numerical 282stability. 283 284<!-- Rhe orthogonal polynomials used in defining 285the Gauss points that make up the interpolation grid are governed by 286the Wiener, Askey, or Extended options. The Wiener option uses 287interpolation points from Gauss-Hermite (non-nested) or Genz-Keister 288(nested) integration rules for all random variables and employs the 289same nonlinear variable transformation as the local and global 290reliability methods (and therefore has the same variable support). 291The Askey option, however, employs interpolation points from 292Gauss-Hermite (Genz-Keister if nested), Gauss-Legendre 293(Gauss-Patterson if nested), Gauss-Laguerre, Gauss-Jacobi, and 294generalized Gauss-Laguerre quadrature. The Extended option avoids the 295use of any nonlinear variable transformations by augmenting the Askey 296approach with Gauss points from numerically-generated orthogonal 297polynomials for non-Askey probability density functions. As for PCE, 298the Wiener/Askey/Extended selection defaults to Extended, can be 299overridden by the user using the keywords \c askey or \c wiener, and 300automatically falls back from Extended/Askey to Wiener on a per 301variable basis as needed to support prescribed correlations.--> 302 303Faq:: 304See_Also:: method-adaptive_sampling, method-gpais, method-local_reliability, method-global_reliability, method-sampling, method-importance_sampling, method-polynomial_chaos 305