stoch_collocation
Uncertainty quantification with stochastic collocation
Specification
Alias: nond_stoch_collocation
Arguments: None
Child Keywords:
Required/Optional 
Description of Group 
Dakota Keyword 
Dakota Keyword Description 

Optional (Choose One) 
Automated Refinement Type 
Automatic polynomial order refinement 

Employ hrefinement to refine the grid 

Optional 
Maximum number of expansion refinement iterations 

Optional 
Stopping criterion based on objective function or statistics convergence 

Optional 
define scaling of statistical metrics when adapting UQ surrogates 

Required (Choose One) 
Interpolation Grid Type 
Order for tensorproducts of Gaussian quadrature rules 

Level to use in sparse grid integration or interpolation 

Optional (Choose One) 
Basis Polynomial Family 
Use piecewise local basis functions 

Select the standardized random variables (and associated basis polynomials) from the Askey family that best match the userspecified random variables. 

Use standard normal random variables (along with Hermite orthogonal basis polynomials) when transforming to a standardized probability space. 

Optional 
Use derivative data to construct surrogate models 

Optional 
Number of samples at which to evaluate an emulator (surrogate) 

Optional 
Selection of sampling strategy 

Optional 
Selection of a random number generator 

Optional 
Allow refinement of probability and generalized reliability results using importance sampling 

Optional 
Output moments of the specified type and include them within the set of final statistics. 

Optional 
Values at which to estimate desired statistics for each response 

Optional 
Specify probability levels at which to estimate the corresponding response value 

Optional 
Specify reliability levels at which the response values will be estimated 

Optional 
Specify generalized relability levels at which to estimate the corresponding response value 

Optional 
Selection of cumulative or complementary cumulative functions 

Optional 
Activates global sensitivity analysis based on decomposition of response variance into main, interaction, and total effects 

Optional (Choose One) 
Covariance Type 
Display only the diagonal terms of the covariance matrix 

Display the full covariance matrix 

Optional 
Filename for points at which to evaluate the PCE/SC surrogate 

Optional 
Output file for surrogate model value evaluations 

Optional 
Seed of the random number generator 

Optional 
Reuses the same seed value for multiple random sampling sets 

Optional 
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 finitedimensional 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 valuebased or gradientenhanced. In the local case,
valuedbased are piecewise linear splines and gradientenhanced are
piecewise cubic splines, and in the global case, valuedbased are
Lagrange interpolants and gradientenhanced are Hermite interpolants.
A valuebased expansion takes the form
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
gradientenhanced 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 highorder 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 NewtonCotes
points are employed in this case, and all random variable types are
transformed to standard uniform probability space. The
global gradientenhanced 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 highorder basis polynomials, they are deactivated
by default but can be activated by developers using a compiletime switch.
Interpolation grid type (Group 3)
To form the multidimensional interpolants \(L_i\) of the expansion, two options are provided.
interpolation on a tensorproduct of Gaussian quadrature points (specified with
quadrature_order
and, optionally,dimension_preference
for anisotropic tensor grids). As for PCE, nonnested Gauss rules are employed by default, although the presence ofp_refinement
orh_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 explicitnested
ornon_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 thenon_nested
option, and the growth rules are restricted unless overridden by theunrestricted
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 valuebased or
gradientenhanced 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 (GenzKeister, GaussPatterson, or NewtonCotes 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
reliabilitybased level mappings induced by a grid change within
uniform or goaloriented 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 postprocessing 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 GaussLegendre rules if nonnested, GaussPatterson rules if nested, and NewtonCotes 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 approximationbased, so management of this expense is less
critical). This option allows for refinement of probability and
generalized reliability results using importance sampling.
Multifidelity 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 tensorproduct Gaussian quadrature is again the
preferred choice. For larger n, tensorproduct quadrature quickly
becomes too expensive and the sparse grid approach is preferred. For
selfconsistency 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 nonnested Gauss rules (integrand precision
\(i=4l+1\) for sparse grid level l and \(i=2m1\) for tensor
grid order m).
Additional Resources
Dakota provides access to SC methods through the NonDStochCollocation class. Refer to Uncertainty Quantification and Stochastic Expansion Methods for additional information on the SC algorithm.
Expected HDF5 Output
If Dakota was built with HDF5 support and run with the
hdf5
keyword, this method
writes the following results to HDF5:
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 valuebased expansion takes the form
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 gradientenhanced expansion (selected
via the use_derivatives
keyword) involves both type 1 and type 2
basis functions as follows:
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 illconditioning that can occur in the gradientenhanced PCE approach based on regression. However, the calculation of highorder 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.