Several examples follow. The first example shows a minimal specification for a single model.
model, \ single
The next example displays a surrogate model specification which selects a quadratic polynomial from among the global approximation methods. It uses a pointer to a design of experiments method for generating the data needed for building the global approximation, reuses any old data available for the current approximation region, and employs the first-order multiplicative approach to correcting the approximation at the center of the current approximation region.
model, \ id_model = 'M1' \ variables_pointer = 'V1' \ responses_pointer = 'R1' \ surrogate global \ quadratic polynomial \ dace_method_pointer = 'DACE' \ reuse_samples region \ correction multiplicative first_order
Finally, an advanced nested model example would be
model, \ id_model = 'M1' \ variables_pointer = 'V1' \ responses_pointer = 'R1' \ nested \ optional_interface_pointer = 'OI1' \ optional_interface_responses_pointer = 'OIR1' \ sub_method_pointer = 'SM1' \ primary_variable_mapping = '' '' 'X' 'Y' \ secondary_variable_mapping = '' '' 'mean' 'mean' \ primary_response_mapping = 1. 0. 0. 0. 0. 0. 0. 0. 0. \ secondary_response_mapping = 0. 0. 0. 1. 3. 0. 0. 0. 0. \ 0. 0. 0. 0. 0. 0. 1. 3. 0.
model, \ <model independent controls> \ <model selection> \ <model dependent controls>
<model independent controls> are those controls which are valid for all models. Referring to dakota.input.spec, these controls are defined externally from and prior to the model selection blocks. The model selection blocks are all required group specifications separated by logical OR's, where the model selection must be
nested. If a
surrogate model is specified, a secondary selection must be made for its type:
<model dependent controls> are those controls which are only meaningful for a specific model. These controls are defined within each model selection block. Defaults for model independent and model dependent controls are defined in DataModel. The following sections provide additional detail on the model independent controls followed by the model selections and their corresponding model dependent controls.
id_modeland is used to provide a unique identifier string for use within method specifications (refer to
model_pointerin Method Independent Controls).
The type of model can be
surrogate. Each of these model specifications supports
responses_pointer strings for identifying the variables and responses specifications used in constructing the model (by cross-referencing with
id_responses strings from particular variables and responses keyword specifications). These pointers are valid for each model type since each model contains a set of variables that is mapped into a set of responses -- only the specifics of the mapping differ. Additional pointers are used for each model type for constructing the components of the variable to response mapping. As a strategy specification identifies one or more methods and a method specification identifies a model, a model specification identifies variables, interface, and responses specifications. This top-down flow specifies all of the object interrelationships.
For each of these pointer specifications, if a pointer string is specified and no corresponding id string is available, DAKOTA will exit with an error message. If the pointer is optional and no pointer string is specified, then the last specification parsed will be used. It is appropriate to omit optional cross-referencing whenever the relationships are unambiguous due to the presence of only one specification.
Table 6.1 provides the specification detail for the model independent controls involving identifiers, model type controls, and pointers.
|Model set identifier||string||Optional||method use of last model parsed|
|Model type||none||Required group||N/A (|
|Variables set pointer||string||Optional||model use of last variables parsed|
|Responses set pointer||string||Optional||model use of last responses parsed|
singlemodel case, a single interface is used to map the variables into responses. The optional
interface_pointerspecification identifies this interface by cross-referencing with the
id_interfacestring input from a particular interface keyword specification.
Table 6.2 provides the specification detail for single models.
|Interface set pointer||string||Optional||model use of last interface parsed|
surrogatemodel case, the specification first allows a mixture of surrogate and actual response mappings through the use of the optional
id_surrogatesspecification. This identifies the subset of the response functions by number that are to be approximated (the default is all functions). The valid response function identifiers range from 1 through the total number of response functions (see Function Specification). Next, the specification selects a global, multipoint, local, or hierarchical approximation. Table 6.3 provides the specification detail for surrogate models.
|Surrogate response ids||list of integers||Optional||All response functions are approximated|
|Surrogate type selection||none||Required group||N/A|
Each of these surrogate types provides an approximate representation of a "truth" model which is used to perform the parameter to response mappings. This approximation is built and updated using data from the truth model. This data is generated in some cases using a design of experiments iterator applied to the truth model (global approximations with a
dace_method_pointer). In other cases, truth model data from a single point (local, hierarchical approximations), from a few previously evaluated points (multipoint approximations), or from the restart database (global approximations with
reuse_samples) can be used. Surrogate models are used extensively in the surrogate-based optimization strategy (see SurrBasedOptStrategy and Surrogate-based Optimization (SBO) Commands), in which the goals are to reduce expense by minimizing the number of truth function evaluations and to smooth out noisy data with a global data fit. However, the use of surrogate models is not restricted in any way to optimization techniques, and in fact, the uncertainty quantification methods and optimization under uncertainty strategy are other primary users.
The following sections present the
hierarchical specification groups in further detail.
kriging. These specifications invoke a layered perceptron artificial neural network approximation, a polynomial regression approximation, a multivariate adaptive regression spline approximation, a hermite polynomial approximation, a gaussian process approximation, or a kriging interpolation approximation, respectively. In the polynomial case, the order of the polynomial (linear, quadratic, or cubic) must be specified, and in the kriging case, a vector of correlations can be optionally specified in order to bypass the internal kriging calculations of correlation coefficients. Note that the gaussian process approximation is new, and currently always invokes an internal optimization procedure to determine the correlation coefficients.
For each of the global surrogates,
use_gradients can be optionally specified. The
dace_method_pointer specification points to a design of experiments iterator which can be used to generate truth model data for building a global data fit. The
reuse_samples specification can be used to employ old data (either from previous function evaluations performed in the run or from function evaluations read from a restart database or text file) in the building of new global approximations. The default is no reuse of old data (since this can induce directional bias), and the settings of
samples_file result in reuse of all available data, reuse of all data available in the current trust region, and reuse of all data from a specified text file, respectively. The combination of new build data from
dace_method_pointer and old build data from
reuse_samples must be sufficient for building the global approximation. If not enough data is available, the system will abort with an error message. Both
reuse_samples are optional specifications, which gives the user maximum flexibility in using design of experiments data, restart/text file data, or both.
correction specification specifies that the approximation will be corrected to match truth data, either matching truth values in the case of
zeroth_order matching, matching truth values and gradients in the case of
first_order matching, or matching truth values, gradients, and Hessians in the case of
second_order matching. For
multiplicative corrections, the correction is local in that the truth data is matched at a single point, typically the center of the approximation region. The
additive correction adds a scalar offset (
zeroth_order), a linear function (
first_order), or a quadratic function (
second_order) to the approximation to match the truth data at the point, and the
multiplicative correction multiplies the approximation by a scalar (
zeroth_order), a linear function (
first_order), or a quadratic function (
second_order) to match the truth data at the point. The
first_order case is due to [Lewis and Nash, 2000] and the
first_order case is commonly known as beta correction [Haftka, 1991]. For the
combined correction, the use of both additive and multiplicative corrections allows the satisfaction of an additional matching condition, typically the truth function values at the previous correction point (e.g., the center of the previous trust region). The
combined correction is then a multipoint correction, as opposed to the local
multiplicative corrections. Each of these correction capabilities is described in detail in [Eldred et al., 2004a].
use_gradients flag specifies a future capability for the use of gradient data in the global approximation builds. This capability is currently supported in SurrBasedOptStrategy, SurrogateDataPoint, and Approximation::build(), but is not yet supported in any global approximation derived class redefinitions of Approximation::find_coefficients(). Tables 6.4 and 6.5 summarizes the global approximation specifications.
|Global approximations||none||Required group (1 of 4 selections)||N/A|
|Artificial neural network||none||Required (1 of 6 selections)||N/A|
|Polynomial||Required group (1 of 6 selections)||N/A|
|Multivariate adaptive regression splines||none||Required (1 of 6 selections)||N/A|
|Hermite polynomial||none||Required (1 of 6 selections)||N/A|
|Gaussian process||none||Required (1 of 6 selections)||N/A|
|Kriging interpolation||none||Required group (1 of 6 selections)||N/A|
|Kriging correlations||list of reals||Optional||internally computed correlations|
|Design of experiments method pointer||string||Optional||no design of experiments data|
|Sample reuse in global approximation builds||Optional group||no sample reuse|
|Surrogate correction approach||Optional group||no surrogate correction|
|Use of gradient data in global approximation builds||none||Optional||gradient data not used in global approximation builds|
actual_model_pointerspecification. Table 6.6 summarizes the multipoint approximation specifications.
|Multipoint approximation||none||Required group (1 of 4 selections)||N/A|
|Two-point adaptive nonlinear approximation||none||Required||N/A|
|Pointer to the truth model specification||string||Required||N/A|
taylor_seriesselection. The order of the Taylor series may be either first-order or second-order, which is automatically determined from the gradient and Hessian specifications in the responses specification (see Gradient Specification and Hessian Specification) for the truth model.
The truth model to be used to generate the value/gradient/Hessian data used in the series expansion is identified through the required
actual_model_pointer specification. The use of a model pointer (as opposed to an interface pointer) allows additional flexibility in defining the approximation. In particular, the derivative specification for the truth model may differ from the derivative specification for the approximation , and the truth model results being approximated may involve a model recursion (e.g., the values/gradients from a nested model). Table 6.7 summarizes the local approximation interface specifications.
|Local approximation||none||Required group (1 of 4 selections)||N/A|
|Taylor series local approximation||none||Required||N/A|
|Pointer to the truth model specification||string||Required||N/A|
low_fidelity_model_pointerspecification points to the low fidelity model specification. This model is used to generate low fidelity responses which are then corrected and returned to an iterator. The required
high_fidelity_model_pointerspecification points to the specification for the high fidelity truth model. This model is used only for verifying low fidelity results and updating low fidelity corrections. The
correctionspecification specifies which correction technique will be applied to the low fidelity results in order to match the high fidelity results at one or more points. In the hierarchical case (as compared to the global case), the
correctionspecification is required, since the omission of a correction technique would effectively eliminate the purpose of the high fidelity model. If it is desired to use a low fidelity model without corrections, then a hierarchical approximation is not needed and a
singlemodel should be used. Refer to Global approximations for additional information on available correction approaches. Table 6.8 summarizes the hierarchical approximation specifications.
|Hierarchical approximation||none||Required group (1 of 4 selections)||N/A|
|Pointer to the low fidelity model specification||string||Required||N/A|
|Pointer to the high fidelity model specification||string||Required||N/A|
|Surrogate correction approach||Required group||N/A|
nestedmodel case, a
sub_method_pointermust be provided in order to specify the nested iterator, and
optional_interface_responses_pointerprovide an optional group specification for the optional interface portion of nested models (where
optional_interface_pointerpoints to the interface specification and
optional_interface_responses_pointerpoints to a responses specification describing the data to be returned by this interface). This interface is used to provide non-nested data, which is then combined with data from the nested iterator using the
secondary_response_mappinginputs (see mapping discussion below).
Table 6.9 provides the specification detail for nested model pointers.
|Interface set pointer||string||Optional group||no optional interface|
|Responses pointer for nested model optional interfaces||string||Optional||reuse of top-level responses specification|
|Sub-method pointer for nested models||string||Required||N/A|
Nested models may employ mappings for both the variable inputs to the sub-model and the response outputs from the sub-model. In the former case, the
secondary_variable_mapping specifications are used to map from the top-level variables into the sub-model variables, and in the latter case, the
secondary_response_mapping specifications are used to map from the sub-model responses back to the top-level responses. For the variable mappings, the primary and secondary specifications provide lists of strings which are used to target active sub-model variables and their distribution parameters, respectively. The primary strings are matched to variable labels such as
'cdv_1' (either user-supplied or default labels), and the secondary strings are matched to distribution parameters such as
'std_deviation' (the singular form of the uncertain distribution parameter keywords, lacking the prepended distribution type identifier). Both specifications are optional, which is designed to support three possibilities:
These different variable mapping possibilities may be used in any combination by employing empty strings (
'') for particular omitted mappings (the number of strings in user-supplied primary and secondary variable mapping specifications must equal the number of active top-level variables).
For the response mappings, the primary and secondary specifications provide real-valued multipliers to be applied to sub-iterator response results. The sub-iterator response results are defined as follows for different sub-iterator types:
reliability_levels, in that order.
The primary values map sub-iterator response results into top-level objective functions, least squares terms, or generic response functions, depending on the declared top-level response set. The secondary values map sub-iterator response results into top-level nonlinear inequality and equality constraints. Refer to NestedModel::response_mapping() for additional details.
An example of variable and response mappings is provided below:
primary_variable_mapping = '' '' 'X' 'Y' \ secondary_variable_mapping = '' '' 'mean' 'mean' \ primary_response_mapping = 1. 0. 0. 0. 0. 0. 0. 0. 0. \ secondary_response_mapping = 0. 0. 0. 1. 3. 0. 0. 0. 0. \ 0. 0. 0. 0. 0. 0. 1. 3. 0. \
'Y'(option 1 above). The response mappings correspond to 9 sub-iterator response functions (e.g., a set of UQ final statistics for 3 response functions, each with a mean, a standard deviation, and a reliability level). The primary response mapping maps the first sub-iterator response function (mean) into a single objective function, least squares term, or generic response function (as dictated by the top-level response specification), and the secondary response mapping maps the fourth sub-iterator response function plus 3 times the fifth sub-iterator response function (mean plus 3 standard deviations) into one top-level nonlinear constraint and the seventh sub-iterator response function plus 3 times the eighth sub-iterator response function (mean plus 3 standard deviations) into another top-level nonlinear constraint (these top-level nonlinear constraints may be inequality or equality, as dictated by the top-level response specification).
Table 6.10 provides the specification detail for the model independent controls involving nested model mappings.
|Primary variable mappings for nested models||list of strings||Optional||augmentation of sub-model variables (no insertion)|
|Secondary variable mappings for nested models||list of strings||Optional||primary mappings into sub-model variables are value-based|
|Primary response mappings for nested models||list of reals||Optional||no sub-iterator contribution to primary functions|
|Secondary response mappings for nested models||list of reals||Optional||no sub-iterator contribution to secondary functions|