Uncertainty quantification with stochastic collocation

Specification

Alias: nond_stoch_collocation

Argument(s): none

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Optional (Choose One)	Automated refinement type (Group 1)	p_refinement	Automatic polynomial order refinement
Optional (Choose One)	Automated refinement type (Group 1)	h_refinement	Employ h-refinement to refine the grid
Optional (Choose One)	Basis polynomial family (Group 2)	piecewise	Use piecewise local basis functions
		askey	Select the standardized random variables (and associated basis polynomials) from the Askey family that best match the user-specified random variables.
		wiener	Use standard normal random variables (along with Hermite orthogonal basis polynomials) when transforming to a standardized probability space.
Required (Choose One)	Interpolation grid type (Group 3)	quadrature_order	Cubature using tensor-products of Gaussian quadrature rules
Required (Choose One)	Interpolation grid type (Group 3)	sparse_grid_level	Set the sparse grid level to be used when peforming sparse grid integration or sparse grid interpolation
Optional		dimension_preference	A set of weights specifying the realtive importance of each uncertain variable (dimension)
Optional		use_derivatives	Use derivative data to construct surrogate models
Optional (Choose One)	Nesting of quadrature rules (Group 4)	nested	Enforce use of nested quadrature rules if available
Optional (Choose One)	Nesting of quadrature rules (Group 4)	non_nested	Enforce use of non-nested quadrature rules
Optional		variance_based_decomp	Activates global sensitivity analysis based on decomposition of response variance into main, interaction, and total effects
Optional (Choose One)	Covariance type (Group 5)	diagonal_covariance	Display only the diagonal terms of the covariance matrix
Optional (Choose One)	Covariance type (Group 5)	full_covariance	Display the full covariance matrix
Optional		sample_type	Selection of sampling strategy
Optional		probability_refinement	Allow refinement of probability and generalized reliability results using importance sampling
Optional		export_points_file	Output file for evaluations of a surrogate model
Optional		fixed_seed	Reuses the same seed value for multiple random sampling sets
Optional		reliability_levels	Specify reliability levels at which the response values will be estimated
Optional		response_levels	Values at which to estimate desired statistics for each response
Optional		distribution	Selection of cumulative or complementary cumulative functions
Optional		probability_levels	Specify probability levels at which to estimate the corresponding response value
Optional		gen_reliability_levels	Specify generalized relability levels at which to estimate the corresponding response value
Optional		rng	Selection of a random number generator
Optional		samples	Number of samples for sampling-based methods
Optional		seed	Seed of the random number generator
Optional		model_pointer	Identifier for model block to be used by a method

Description

Stochastic collocation is a general framework for approximate representation of random response functions in terms of finite-dimensional interpolation bases.

The stochastic collocation (SC) method is very similar to polynomial_chaos, with the key difference that the orthogonal polynomial basis functions are replaced with interpolation polynomial bases. The interpolation polynomials may be either local or global and either value-based or gradient-enhanced. In the local case, valued-based are piecewise linear splines and gradient-enhanced are piecewise cubic splines, and in the global case, valued-based are Lagrange interpolants and gradient-enhanced are Hermite interpolants. A value-based expansion takes the form

$R = \sum_{i=1}^{N_p} r_i L_i(\xi)$

where $N_p$ is the total number of collocation points, $r_i$ is a response value at the $i^{th}$ collocation point, $L_i$ is the $i^{th}$ multidimensional interpolation polynomial, and $\xi$ is a vector of standardized random variables.

Thus, in PCE, one forms coefficients for known orthogonal polynomial basis functions, whereas SC forms multidimensional interpolation functions for known coefficients.

Basis polynomial family (Group 2)

In addition to the askey and wiener basis types also supported by polynomial_chaos, SC supports the option of piecewise local basis functions. These are piecewise linear splines, or in the case of gradient-enhanced interpolation via the use_derivatives specification, piecewise cubic Hermite splines. Both of these basis options provide local support only over the range from the interpolated point to its nearest 1D neighbors (within a tensor grid or within each of the tensor grids underlying a sparse grid), which exchanges the fast convergence of global bases for smooth functions for robustness in the representation of nonsmooth response functions (that can induce Gibbs oscillations when using high-order global basis functions). When local basis functions are used, the usage of nonequidistant collocation points (e.g., the Gauss point selections described above) is not well motivated, so equidistant Newton-Cotes points are employed in this case, and all random variable types are transformed to standard uniform probability space. The global gradient-enhanced interpolants (Hermite interpolation polynomials) are also restricted to uniform or transformed uniform random variables (due to the need to compute collocation weights by integration of the basis polynomials) and share the variable support shown in variable_support for Piecewise SE. Due to numerical instability in these high-order basis polynomials, they are deactivated by default but can be activated by developers using a compile-time switch.

Interpolation grid type (Group 3)

To form the multidimensional interpolants $L_i$ of the expansion, two options are provided.

interpolation on a tensor-product of Gaussian quadrature points (specified with quadrature_order and, optionally, dimension_preference for anisotropic tensor grids). As for PCE, non-nested Gauss rules are employed by default, although the presence of p_refinement or h_refinement will result in default usage of nested rules for normal or uniform variables after any variable transformations have been applied (both defaults can be overridden using explicit nested or non_nested specifications).
interpolation on a Smolyak sparse grid (specified with sparse_grid_level and, optionally, dimension_preference for anisotropic sparse grids) defined from Gaussian rules. As for sparse PCE, nested rules are employed unless overridden with the non_nested option, and the growth rules are restricted unless overridden by the unrestricted keyword.

Another distinguishing characteristic of stochastic collocation relative to polynomial_chaos is the ability to reformulate the interpolation problem from a nodal interpolation approach into a hierarchical formulation in which each new level of interpolation defines a set of incremental refinements (known as hierarchical surpluses) layered on top of the interpolants from previous levels. This formulation lends itself naturally to uniform or adaptive refinement strategies, since the hierarchical surpluses can be interpreted as error estimates for the interpolant. Either global or local/piecewise interpolants in either value-based or gradient-enhanced approaches can be formulated using hierarchical interpolation. The primary restriction for the hierarchical case is that it currently requires a sparse grid approach using nested quadrature rules (Genz-Keister, Gauss-Patterson, or Newton-Cotes for standard normals and standard uniforms in a transformed space: Askey, Wiener, or Piecewise settings may be required), although this restriction can be relaxed in the future. A selection of hierarchical interpolation will provide greater precision in the increments to mean, standard deviation, covariance, and reliability-based level mappings induced by a grid change within uniform or goal-oriented adaptive refinement approaches (see following section).

It is important to note that, while quadrature_order and sparse_grid_level are array inputs, only one scalar from these arrays is active at a time for a particular expansion estimation. These scalars can be augmented with a dimension_preference to support anisotropy across the random dimension set. The array inputs are present to support advanced use cases such as multifidelity UQ, where multiple grid resolutions can be employed.

Automated refinement type (Group 1)

Automated expansion refinement can be selected as either p_refinement or h_refinement, and either refinement specification can be either uniform or dimension_adaptive. The dimension_adaptive case can be further specified as either sobol or generalized (decay not supported). Each of these automated refinement approaches makes use of the max_iterations and convergence_tolerance iteration controls. The h_refinement specification involves use of the same piecewise interpolants (linear or cubic Hermite splines) described above for the piecewise specification option (it is not necessary to redundantly specify piecewise in the case of h_refinement). In future releases, the hierarchical interpolation approach will enable local refinement in addition to the current uniform and dimension_adaptive options.

Covariance type (Group 5)

These two keywords are used to specify how this method computes, stores, and outputs the covariance of the responses. In particular, the diagonal covariance option is provided for reducing post-processing overhead and output volume in high dimensional applications.

Active Variables

The default behavior is to form expansions over aleatory uncertain continuous variables. To form expansions over a broader set of variables, one needs to specify active followed by state, epistemic, design, or all in the variables specification block.

For continuous design, continuous state, and continuous epistemic uncertain variables included in the expansion, interpolation points for these dimensions are based on Gauss-Legendre rules if non-nested, Gauss-Patterson rules if nested, and Newton-Cotes points in the case of piecewise bases. Again, when probability integrals are evaluated, only the aleatory random variable domain is integrated, leaving behind a polynomial relationship between the statistics and the remaining design/state/epistemic variables.

Optional Keywords regarding method outputs

Each of these sampling specifications refer to sampling on the SC approximation for the purposes of generating approximate statistics.

sample_type
samples
seed
fixed_seed
rng
probability_refinement
distribution
reliability_levels
response_levels
probability_levels
gen_reliability_levels

Since SC approximations are formed on structured grids, there should be no ambiguity with simulation sampling for generating the SC expansion.

When using the probability_refinement control, the number of refinement samples is not under the user's control (these evaluations are approximation-based, so management of this expense is less critical). This option allows for refinement of probability and generalized reliability results using importance sampling.

Multi-fidelity UQ

When using multifidelity UQ, the high fidelity expansion generated from combining the low fidelity and discrepancy expansions retains the polynomial form of the low fidelity expansion (only the coefficients are updated). Refer to polynomial_chaos for information on the multifidelity interpretation of array inputs for quadrature_order and sparse_grid_level.

Usage Tips

If n is small, then tensor-product Gaussian quadrature is again the preferred choice. For larger n, tensor-product quadrature quickly becomes too expensive and the sparse grid approach is preferred. For self-consistency in growth rates, nested rules employ restricted exponential growth (with the exception of the dimension_adaptive p_refinement generalized case) for consistency with the linear growth used for non-nested Gauss rules (integrand precision $i=4l+1$ for sparse grid level l and $i=2m-1$ for tensor grid order m).

Additional Resources

Dakota provides access to SC methods through the NonDStochCollocation class. Refer to the Uncertainty Quantification Capabilities chapter of the Users Manual[5] and the Stochastic Expansion Methods chapter of the Theory Manual[4] for additional information on the SC algorithm.

Examples

method,
    stoch_collocation
      sparse_grid_level = 2 
      samples = 10000 seed = 12347 rng rnum2    
      response_levels = .1 1. 50. 100. 500. 1000.   
      variance_based_decomp

Theory

As mentioned above, a value-based expansion takes the form

$R = \sum_{i=1}^{N_p} r_i L_i(\xi)$

The $i^{th}$ interpolation polynomial assumes the value of 1 at the $i^{th}$ collocation point and 0 at all other collocation points, involving either a global Lagrange polynomial basis or local piecewise splines. It is easy to see that the approximation reproduces the response values at the collocation points and interpolates between these values at other points. A gradient-enhanced expansion (selected via the use_derivatives keyword) involves both type 1 and type 2 basis functions as follows:

$R = \sum_{i=1}^{N_p} [ r_i H^{(1)}_i(\xi) + \sum_{j=1}^n \frac{dr_i}{d\xi_j} H^{(2)}_{ij}(\xi) ]$

where the $i^{th}$ type 1 interpolant produces 1 for the value at the $i^{th}$ collocation point, 0 for values at all other collocation points, and 0 for derivatives (when differentiated) at all collocation points, and the $ij^{th}$ type 2 interpolant produces 0 for values at all collocation points, 1 for the $j^{th}$ derivative component at the $i^{th}$ collocation point, and 0 for the $j^{th}$ derivative component at all other collocation points. Again, this expansion reproduces the response values at each of the collocation points, and when differentiated, also reproduces each component of the gradient at each of the collocation points. Since this technique includes the derivative interpolation explicitly, it eliminates issues with matrix ill-conditioning that can occur in the gradient-enhanced PCE approach based on regression. However, the calculation of high-order global polynomials with the desired interpolation properties can be similarly numerically challenging such that the use of local cubic splines is recommended due to numerical stability.

Specification

Description

Examples

Theory

See Also