nl2sol

Trust-region method for nonlinear least squares

Topics

nonlinear_least_squares

Specification

• Alias: None

• Arguments: None

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Optional		function_precision	Specify the maximum precision of the analysis code responses
Optional		absolute_conv_tol	Absolute convergence tolerance
Optional		x_conv_tol	X-convergence tolerance
Optional		singular_conv_tol	Singular convergence tolerance
Optional		singular_radius	Singular radius
Optional		false_conv_tol	False convergence tolerance
Optional		initial_trust_radius	Initial trust region radius
Optional		covariance	Determine how the final covariance matrix is computed
Optional		regression_diagnostics	Turn on regression diagnostics
Optional		convergence_tolerance	Stopping criterion based on objective function or statistics convergence
Optional		max_iterations	Number of iterations allowed for optimizers and adaptive UQ methods
Optional		speculative	Compute speculative gradients
Optional		max_function_evaluations	Number of function evaluations allowed for optimizers
Optional		scaling	Turn on scaling for variables, responses, and constraints
Optional		model_pointer	Identifier for model block to be used by a method

Description

NL2SOL is available as nl2sol and addresses unconstrained and bound-constrained least squares problems. It uses a trust-region method (and thus can be viewed as a generalization of the Levenberg-Marquardt algorithm) and adaptively chooses between two Hessian approximations, the Gauss-Newton approximation alone and the Gauss-Newton approximation plus a quasi-Newton approximation to the rest of the Hessian. Even on small-residual problems, the latter Hessian approximation can be useful when the starting guess is far from the solution. On problems that are not over-parameterized (i.e., that do not involve more optimization variables than the data support), NL2SOL usually exhibits fast convergence.

Several internal NL2SOL convergence tolerances are adjusted in response to function_precision, which gives the relative precision to which responses are computed.

These tolerances may also be specified explicitly using:

• convergence_tolerance (NL2SOL’s rfctol)

• x_conv_tol (NL2SOL’s xctol)

• absolute_conv_tol (NL2SOL’s afctol)

• singular_conv_tol (NL2SOL’s sctol)

• false_conv_tol (NL2SOL’s xftol)

• initial_trust_radius (NL2SOL’s lmax0)

The internal NL2SOL defaults can be obtained for many of these controls by specifying the value -1. The internal defaults are often functions of machine epsilon (as limited by function_precision).

Expected HDF5 Output

If Dakota was built with HDF5 support and run with the environment-results_output-hdf5 keyword, this method writes the following results to HDF5:

• hdf5_results-best_params

• hdf5_results-best_constraints

• hdf5_results-calibration (when responses-calibration_terms are specified)

• hdf5_results-lsq_confidence_intervals (when responses-calibration_terms-calibration_data-num_experiments equals 1)

Examples

An example of nl2sol is given below, and is discussed in the User’s Manual.

Note that in this usage of responses-calibration_terms, the driver script rosenbrock, is returning “residuals”, which the nl2sol method is attempting to minimze. Another use case is to provide a data file, which Dakota will attempt to match the model responses to. See responses-calibration_terms-calibration_data_file. Finally, as of Dakota 6.2, the field data capability may be used with nl2sol. That is, the user can specify field simulation data and field experiment data, and Dakota will interpolate and provide the proper residuals for the calibration.

# Dakota Input File: rosen_opt_nls.in
environment
  tabular_data
    tabular_data_file = 'rosen_opt_nls.dat'

method
  max_iterations = 100
  convergence_tolerance = 1e-4
  nl2sol

model
  single

variables
  continuous_design = 2
    initial_point    -1.2      1.0
    lower_bounds     -2.0     -2.0
    upper_bounds      2.0      2.0
    descriptors       'x1'     "x2"

interface
  analysis_driver = 'rosenbrock'
    direct

responses
  calibration_terms = 2
  analytic_gradients
  no_hessians

Theory

NL2SOL has a variety of internal controls as described in AT&T Bell Labs CS TR 153 (http://cm.bell-labs.com/cm/cs/cstr/153.ps.gz). A number of existing Dakota controls (method independent controls and responses controls) are mapped into these NL2SOL internal controls. In particular, Dakota’s convergence_tolerance, max_iterations, max_function_evaluations, and fd_gradient_step_size are mapped directly into NL2SOL’s rfctol, mxiter, mxfcal, and dltfdj controls, respectively. In addition, Dakota’s fd_hessian_step_size is mapped into both delta0 and dltfdc, and Dakota’s output verbosity is mapped into NL2SOL’s auxprt and outlev (for normal/ verbose/ debug output, NL2SOL prints initial guess, final solution, solution statistics, nondefault values, and changes to the active bound constraint set on every iteration; for quiet output, NL2SOL prints only the initial guess and final solution; and for silent output, NL2SOL output is suppressed).