.. _method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio: """"""""""""""""" collocation_ratio """"""""""""""""" Set the number of points used to build a PCE via regression to be proportional to the number of terms in the expansion. .. toctree:: :hidden: :maxdepth: 1 method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-collocation_points_sequence method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-least_squares method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-orthogonal_matching_pursuit method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-basis_pursuit method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-basis_pursuit_denoising method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-least_angle_regression method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-least_absolute_shrinkage method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-cross_validation method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-ratio_order method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-response_scaling method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-use_derivatives method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-tensor_grid method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-reuse_points method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-max_solver_iterations **Specification** - *Alias:* None - *Arguments:* REAL **Child Keywords:** +-------------------------+--------------------+---------------------------------+---------------------------------------------+ | Required/Optional | Description of | Dakota Keyword | Dakota Keyword Description | | | Group | | | +=========================+====================+=================================+=============================================+ | Optional | `collocation_points_sequence`__ | Sequence of collocation point counts used | | | | in a multi-stage expansion | +-------------------------+--------------------+---------------------------------+---------------------------------------------+ | Optional (Choose One) | Regression | `least_squares`__ | Compute the coefficients of a polynomial | | | Algorithm | | expansion using least squares | | | +---------------------------------+---------------------------------------------+ | | | `orthogonal_matching_pursuit`__ | Compute the coefficients of a polynomial | | | | | expansion using orthogonal matching pursuit | | | | | (OMP) | | | +---------------------------------+---------------------------------------------+ | | | `basis_pursuit`__ | Compute the coefficients of a polynomial | | | | | expansion by solving the Basis Pursuit | | | | | :math:`\ell_1` -minimization problem using | | | | | linear programming. | | | +---------------------------------+---------------------------------------------+ | | | `basis_pursuit_denoising`__ | Compute the coefficients of a polynomial | | | | | expansion by solving the Basis Pursuit | | | | | Denoising :math:`\ell_1` -minimization | | | | | problem using second order cone | | | | | optimization. | | | +---------------------------------+---------------------------------------------+ | | | `least_angle_regression`__ | Compute the coefficients of a polynomial | | | | | expansion by using the greedy least angle | | | | | regression (LAR) method. | | | +---------------------------------+---------------------------------------------+ | | | `least_absolute_shrinkage`__ | Compute the coefficients of a polynomial | | | | | expansion by using the LASSO problem. | +-------------------------+--------------------+---------------------------------+---------------------------------------------+ | Optional | `cross_validation`__ | Use cross validation to choose the 'best' | | | | polynomial order of a polynomial chaos | | | | expansion. | +----------------------------------------------+---------------------------------+---------------------------------------------+ | Optional | `ratio_order`__ | Specify a non-linear the relationship | | | | between the expansion order of a polynomial | | | | chaos expansion and the number of samples | | | | that will be used to compute the PCE | | | | coefficients. | +----------------------------------------------+---------------------------------+---------------------------------------------+ | Optional | `response_scaling`__ | Perform bounds-scaling on response values | | | | prior to surrogate emulation | +----------------------------------------------+---------------------------------+---------------------------------------------+ | Optional | `use_derivatives`__ | Use derivative data to construct surrogate | | | | models | +----------------------------------------------+---------------------------------+---------------------------------------------+ | Optional | `tensor_grid`__ | Use sub-sampled tensor-product quadrature | | | | points to build a polynomial chaos | | | | expansion. | +----------------------------------------------+---------------------------------+---------------------------------------------+ | Optional | `reuse_points`__ | This describes the behavior of reuse of | | | | points in constructing polynomial chaos | | | | expansion models. | +----------------------------------------------+---------------------------------+---------------------------------------------+ | Optional | `max_solver_iterations`__ | Maximum iterations in determining | | | | polynomial coefficients | +----------------------------------------------+---------------------------------+---------------------------------------------+ .. __: method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-collocation_points_sequence.html __ method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-least_squares.html __ method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-orthogonal_matching_pursuit.html __ method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-basis_pursuit.html __ method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-basis_pursuit_denoising.html __ method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-least_angle_regression.html __ method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-least_absolute_shrinkage.html __ method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-cross_validation.html __ method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-ratio_order.html __ method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-response_scaling.html __ method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-use_derivatives.html __ method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-tensor_grid.html __ method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-reuse_points.html __ method-multilevel_polynomial_chaos-expansion_order_sequence-collocation_ratio-max_solver_iterations.html **Description** Set the number of points used to build a PCE via regression to be proportional to the number of terms in the expansion. To avoid requiring the user to calculate N from n and p, the collocation_ratio allows for specification of a constant factor applied to N (e.g., collocation_ratio = 2. produces samples = 2N). In addition, the default linear relationship with N can be overridden using a real-valued exponent specified using ratio_order. In this case, the number of samples becomes :math:`cN^o` where :math:`c` is the collocation_ratio and :math:`o` is the ratio_order. The use_derivatives flag informs the regression approach to include derivative matching equations (limited to gradients at present) in the least squares solutions, enabling the use of fewer collocation points for a given expansion order and dimension (number of points required becomes :math:`\frac{cN^o}{n+1}` ).