14.6. Checking algorithm “InterpolationByReducedModelTest”

14.6.1. Description

This algorithm provides a simple way of analyzing the quality of the empirical interpolation obtained by a reduced basis for complete states, using measurements at precise points.

The results displayed by default are simple statistics related to the normalized errors between the interpolation with a reduced basis and the complete states we seek to represent on the reduced basis. The test requires a reduced basis and a set of optimal measurement positions, and uses pseudo-measurements from each complete state (“snapshot”) included in the given test set.

Please note: to be consistent, this test must use the same mathematical norm as that used to construct the reduced basis. As the norm is chosen by the user when defining the test, the norm used to construct the reduced basis must be verified.

14.6.2. Some noteworthy properties of the implemented methods

To complete the description, we summarize here a few notable properties of the algorithm methods or of their implementations. These properties may have an influence on how it is used or on its computational performance. For further information, please refer to the more comprehensive references given at the end of this algorithm description.

  • The methods proposed by this algorithm do not require derivation of the objective function or of one of the operators, thus avoiding this additional calculation time when derivatives are calculated numerically by multiple evaluations.

14.6.3. Optional and required commands

The general required commands, available in the editing user graphical or textual interface, are the following:

EnsembleOfSnapshots

List of vectors or matrix. This key contains an ordered collection of physical state vectors \mathbf{y}, called “snapshots” in reduced basis terminology. At each step index, there is 1 state per column if this list is in matrix form, or 1 state per element if it’s actually a list. Caution: the numbering of the support or points, on which or to which a state value is given in each vector, is implicitly that of the natural order of numbering of the state vector, from 0 to the “size minus 1” of this vector.

Example : {"EnsembleOfSnapshots":[y1, y2, y3...]}

MeasurementLocations

Integer series. This series contains the indices of the positions at which a measurement is required to support a reduced basis interpolation, according to the order of the variables of a state vector arbitrarily considered in one-dimensional form, and in the same order as the information contained in the vectors of the reduced basis.

This is similar to the set of optimal positions or points resulting from a Task algorithm “MeasurementsOptimalPositioningTask”, or the set of positions required to run a Task algorithm “InterpolationByReducedModelTask”.

ReducedBasis

List of matrix. Each element is a matrix, containing in each column a vector of the reduced basis obtained by the optimal search, ordered by decreasing preference and in the same order as the ideal points found iteratively.

When it is an input, it is identical to a single output of a Task algorithm “MeasurementsOptimalPositioningTask”.

Example : rb = ADD.get("ReducedBasis")[-1]

The general optional commands, available in the editing user graphical or textual interface, are indicated in List of commands and keywords for an ADAO checking case. Moreover, the parameters of the command “AlgorithmParameters” allow to choose the specific options, described hereafter, of the algorithm. See Description of options of an algorithm by “AlgorithmParameters” for the good use of this command.

The options are the following:

ErrorNorm

Predefined name. This key indicates the norm used for the residue that controls the optimal search. The default is the “L2” norm. The possible criteria are in the following list: [“L2”, “Linf”].

Example : {"ErrorNorm":"L2"}

NumberOfPrintedDigits

Integer value. This key indicates the number of digits of precision for floating point printed output. The default is 5, with a minimum of 0.

Example: {"NumberOfPrintedDigits":5}

ShowElementarySummary

Boolean value. This variable leads to the activation, or not, of the calculation and display of a summary at each elementary evaluation of the test. The default value is “True”, the choices are “True” or “False”.

Example : {"ShowElementarySummary":False}

14.6.4. Information and variables available at the end of the algorithm

At the output, after executing the algorithm, there are information and variables originating from the calculation. The description of Variables and information available at the output show the way to obtain them by the method named get, of the variable “ADD” of the post-processing in graphical interface, or of the case in textual interface. The input variables, available to the user at the output in order to facilitate the writing of post-processing procedures, are described in an Inventory of potentially available information at the output.

Permanent outputs (non conditional)

The unconditional outputs of the algorithm are the following:

None

Set of on-demand outputs (conditional or not)

The whole set of algorithm outputs (conditional or not), sorted by alphabetical order, is the following:

None (messages or statistics are nevertheless displayed)