calibration_terms

Response type suitable for calibration or least squares

Specification

• Alias: least_squares_terms num_least_squares_terms

• Arguments: INTEGER

Child Keywords:

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_scales	Characteristic values to scale each calibration term
Optional		weights	Specify weights for each objective function
Optional (Choose One)	Calibration Data	calibration_data	Supply field or mixed field/scalar calibration data
		calibration_data_file	Supply scalar calibration data only
Optional		simulation_variance	Variance applied to simulation responses
Optional		nonlinear_inequality_constraints	Group to specify nonlinear inequality constraints
Optional		nonlinear_equality_constraints	Group to specify nonlinear equality constraints

Description

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:

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

Note that the problem specification affects what must be returned to Dakota in the results_file :

• If calibration data 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 is specified, then each of the calibration terms returned to Dakota must be a response :math:` y^{Model}_i(theta)` , which Dakota will difference with the data in the specified data file.

Constraints

(See general problem formulation at objective_functions.) The keywords nonlinear_inequality_constraints and nonlinear_equality_constraints specify the number of nonlinear inequality constraints em g, and nonlinear equality constraints em h, respectively. When interfacing to external applications, the responses must be returned to Dakota in this order in the results_file : 0. calibration terms nonlinear inequality constraints nonlinear equality constraints

An optimization problem’s linear constraints are provided to the solver at startup only and do not need to be included in the data returned on every function evaluation. Linear constraints are therefore specified in the variables block through the linear_inequality_constraint_matrix \(A_i\) and linear_equality_constraint_matrix \(A_e\) .

Lower and upper bounds on the design variables em x are also specified in the variables block.

Problem Transformations

Weighting or scaling calibration terms is often appropriate to account for measurement error or to condition the problem for easier solution. Weighting or scaling transformations are applied in the following order:

1. When present, observation error variance \(\sigma_i\) or full covariance \(\Sigma\) , optionally specified through experiment_variance_type, is applied to residuals first: .. math:: R^{(1)}_i = frac{R_{i}}{sigma_{i}} = frac{y^{Model}_i(theta) - y^{Data}_{i}}{sigma_{i}} textrm{, or} .. math:: R^{(1)} = Sigma^{-1/2} R = Sigma^{-1/2} left(y^{Model}(theta) - y^{Data}right), resulting in the typical variance-weighted least squares formulation .. math:: textrm{min}_theta ; R(theta)^T Sigma^{-1} R(theta) 2. Any active scaling transformations are applied next, e.g., for characteristic value scaling: .. math:: R^{(2)}_i = frac{R^{(1)}_i}{s_i} 3. Finally the optional weights are applied in a way that preserves backward compatibility: .. math:: R^{(3)}_i = sqrt{w_i}{R^{(2)}_i} so the ultimate least squares formulation, e.g., in a scaled and weighted case would be .. math:: f = sum_{i=1}^{n} w_i left( frac{y^{Model}_i - y^{Data}_i}{s_i} right)^2 Note that specifying observation error variance and weights are mutually exclusive in a calibration problem.

Theory

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., .. math:: 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.