Dakota Reference Manual  Version 6.4
Large-Scale Engineering Optimization and Uncertainty Analysis
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Response type suitable for calibration or least squares


Alias: least_squares_terms num_least_squares_terms

Argument(s): INTEGER

Required/Optional Description of Group Dakota Keyword Dakota Keyword Description
Optional scalar_calibration_terms Number of scalar calibration terms
Optional field_calibration_terms Number of field calibration terms
Optional primary_scale_types

Choose a scaling type for each response

Optional primary_scales

Supply a characteristic value to scale each reponse

Optional weights Apply different weights to each response
(Choose One)
Group 1 calibration_data

Supply calibration data in the case of field data or mixed data (both scalar and field data).


Specify a text file containing calibration data for scalar responses

Optional nonlinear_inequality_constraints

Group to specify nonlinear inequality constraints

Optional nonlinear_equality_constraints

Group to specify nonlinear equality constraints


Responses for a calibration study are specified using calibration_terms and optional keywords for weighting/scaling, data, and constraints. In general when calibrating, Dakota automatically tunes parameters $ \theta $ to minimize discrepancies or residuals between the model and the data:

\[ R_{i} = y^{Model}_i(\theta) - y^{Data}_{i}. \]

There are two use cases:

  • If calibration_data_file is NOT specified, then each of the calibration terms returned to Dakota through the interface is a residual $ R_{i} $ to be driven toward zero.
  • If calibration_data_file IS specified, then each of the calibration terms returned to Dakota must be a response $ y^{Model}_i(\theta) $, which Dakota will difference with the data in the specified data file.


The keywords nonlinear_inequality_constraints, and nonlinear_equality_constraints specify the number of nonlinear inequality constraints, and nonlinear equality constraints, respectively. When interfacing to external applications, the responses must be returned to Dakota in this order: calibration terms, nonlinear_inequality_constraints, then nonlinear_equality_constraints.

Any linear constraints present in an application need only be input to an optimizer at start up and do not need to be part of the data returned on every function evaluation. These are therefore specified in the method block.

Optional Keywords

The optional keywords relate to scaling responses (for better numerical results), dealing with multiple residuals, and importing data.

See the scaling keyword in the method section for more details on scaling. If scaling is specified, then it is applied to each residual prior to squaring:

\[f = \sum_{i=1}^{n} w_i (\frac{y^{Model}_i - y^{Data}_i}{s_i})^2\]

In the case where experimental data uncertainties are supplied, then the weights are automatically defined to be the inverse of the experimental variance:

\[f = \sum_{i=1}^{n} \frac{1}{\sigma^2_i} (\frac{y^{Model}_i - y^{Data}_i}{s_i})^2\]


Dakota calibration terms are typically used to solve problems of parameter estimation, system identification, and model calibration/inversion. Local least squares calibration problems are most efficiently solved using special-purpose least squares solvers such as Gauss-Newton or Levenberg-Marquardt; however, they may also be solved using any general-purpose optimization algorithm in Dakota. While Dakota can solve these problems with either least squares or optimization algorithms, the response data sets to be returned from the simulator are different when using objective_functions versus calibration_terms.

Least squares calibration involves a set of residual functions, whereas optimization involves a single objective function (sum of the squares of the residuals), i.e.,

\[f = \sum_{i=1}^{n} R_i^2 = \sum_{i=1}^{n} \left(y^{Model}_i(\theta) - y^{Data}_{i} \right)^2 \]

where f is the objective function and the set of $R_i$ are the residual functions, most commonly defined as the difference between a model response and data. Therefore, function values and derivative data in the least squares case involve the values and derivatives of the residual functions, whereas the optimization case involves values and derivatives of the sum of squares objective function. This means that in the least squares calibration case, the user must return each of n residuals separately as a separate calibration term. Switching between the two approaches sometimes requires different simulation interfaces capable of returning the different granularity of response data required, although Dakota supports automatic recasting of residuals into a sum of squares for presentation to an optimization method. Typically, the user must compute the difference between the model results and the observations when computing the residuals. However, the user has the option of specifying the observational data (e.g. from physical experiments or other sources) in a file.

See Also

These keywords may also be of interest: