.. _method-richardson_extrap:

"""""""""""""""""
richardson_extrap
"""""""""""""""""


Estimate order of convergence of a response as model fidelity increases


.. toctree::
   :hidden:
   :maxdepth: 1

   method-richardson_extrap-estimate_order
   method-richardson_extrap-converge_order
   method-richardson_extrap-converge_qoi
   method-richardson_extrap-refinement_rate
   method-richardson_extrap-convergence_tolerance
   method-richardson_extrap-max_iterations
   method-richardson_extrap-model_pointer


**Specification**

- *Alias:* None

- *Arguments:* None


**Child Keywords:**

+-------------------------+--------------------+---------------------------+---------------------------------------------+
| Required/Optional       | Description of     | Dakota Keyword            | Dakota Keyword Description                  |
|                         | Group              |                           |                                             |
+=========================+====================+===========================+=============================================+
| Required (Choose One)   | Verification Mode  | `estimate_order`__        | Compute the best estimate of the            |
|                         |                    |                           | convergence order from three points         |
|                         |                    +---------------------------+---------------------------------------------+
|                         |                    | `converge_order`__        | Refine until the estimated covergence order |
|                         |                    |                           | converges                                   |
|                         |                    +---------------------------+---------------------------------------------+
|                         |                    | `converge_qoi`__          | Refine until the response converges         |
+-------------------------+--------------------+---------------------------+---------------------------------------------+
| Optional                                     | `refinement_rate`__       | Rate at which the state variables are       |
|                                              |                           | refined                                     |
+----------------------------------------------+---------------------------+---------------------------------------------+
| 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                                     | `model_pointer`__         | Identifier for model block to be used by a  |
|                                              |                           | method                                      |
+----------------------------------------------+---------------------------+---------------------------------------------+

.. __: method-richardson_extrap-estimate_order.html
__ method-richardson_extrap-converge_order.html
__ method-richardson_extrap-converge_qoi.html
__ method-richardson_extrap-refinement_rate.html
__ method-richardson_extrap-convergence_tolerance.html
__ method-richardson_extrap-max_iterations.html
__ method-richardson_extrap-model_pointer.html



**Description**


Solution verification procedures estimate the order of convergence of
the simulation response data during the course of a refinement study.
This branch of methods is new and currently only contains one algorithm:
Richardson extrapolation.

*Refinement of the model*

The model fidelity must be parameterized by one or more continuous state variable(s).

The refinement path is
determined from the ``initial_state`` of the ``continuous_state``
variables specification in combination with the
``refinement_rate``, where each of the state variables is treated as an
independent refinement factor and each of the initial state values is
repeatedly divided by the refinement rate value to define new
discretization states.

*Results*

Three algorithm options are currently provided:


- ``estimate_order``
- ``converge_order``
- ``converge_qoi``

*Stopping Criteria*

The method employs the
``max_iterations`` and ``convergence_tolerance`` method independent
controls as stopping criteria.



**Theory**


In each of these cases, convergence order for a response quantity of
interest (QoI) is estimated from

.. math:: p = ln\left(\frac{QoI_3 - QoI_2}{QoI_2 - QoI_1}\right)/ln(r)

where :math:`r`  is the uniform refinement rate specified by
``refinement_rate``.