14.4. Checking algorithm “GradientTest”¶

14.4.1. Description¶

This algorithm allows to check the quality of an adjoint operator, by calculating a residue with known theoretical properties. Different residue formula are available. The test is applicable to any operator, of evolution $\mathcal{D}$ or observation $\mathcal{H}$ .

In any cases, one take $\mathbf{dx}_0=Normal(0,\mathbf{x})$ and $\mathbf{dx}=\alpha*\mathbf{dx}_0$ with $\alpha_0$ a user scaling of the initial perturbation, with default to 1. $F$ is the calculation code (given here by the user by using the observation operator command “ObservationOperator”).

14.4.1.1. “Taylor” residue¶

One observe the residue coming from the Taylor development of the $F$ function, normalized by the value at the nominal point:

$R(\alpha) = \frac{|| F(\mathbf{x}+\alpha*\mathbf{dx}) - F(\mathbf{x}) - \alpha * \nabla_xF(\mathbf{dx}) ||}{|| F(\mathbf{x}) ||}$

If the residue is decreasing and the decrease change in $\alpha^2$ with respect to $\alpha$ , it signifies that the gradient is well calculated until the stopping precision of the quadratic decrease, and that $F$ is not linear.

If the residue is decreasing and the decrease change in $\alpha$ with respect to $\alpha$ , until a certain level after which the residue remains small and constant, it signifies that the $F$ is linear and that the residue is decreasing due to the error coming from $\nabla_xF$ term calculation.

14.4.1.2. “TaylorOnNorm” residue¶

One observe the residue coming from the Taylor development of the $F$ function, with respect to the $\alpha$ parameter to the square:

$R(\alpha) = \frac{|| F(\mathbf{x}+\alpha*\mathbf{dx}) - F(\mathbf{x}) - \alpha * \nabla_xF(\mathbf{dx}) ||}{\alpha^2}$

This is a residue essentially similar to the classical Taylor criterion previously described, but its behavior can differ depending on the numerical properties of the calculation.

If the residue is constant until a certain level after which the residue will growth, it signifies that the gradient is well calculated until this stopping precision, and that $F$ is not linear.

If the residue is systematically growing from a very small value with respect to $||F(\mathbf{x})||$ , it signifies that $F$ is (quasi-)linear and that the gradient calculation is correct until the precision for which the residue reaches the numerical order of $||F(\mathbf{x})||$ .

14.4.1.3. “Norm” residue¶

One observe the residue based on the gradient approximation:

$R(\alpha) = \frac{|| F(\mathbf{x}+\alpha*\mathbf{dx}) - F(\mathbf{x}) ||}{\alpha}$

which has to remain stable until the calculation precision is reached.

14.4.2. 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 require the derivation of the objective function or of one of the operators. It requires that at least one or both of the observation or evolution operators be differentiable, and this implies an additional cost in the case where the derivatives are calculated numerically by multiple evaluations.

The methods proposed by this algorithm have no internal parallelism, but use the numerical derivation of operator(s), which can be parallelized. The potential interaction, between the parallelism of the numerical derivation, and the parallelism that may be present in the observation or evolution operators embedding user codes, must therefore be carefully tuned.

14.4.3. Optional and required commands¶

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

CheckingPoint: Vector. The variable indicates the vector used as the state around which to perform the required check, noted $\mathbf{x}$ and similar to the background $\mathbf{x}^b$ . It is defined as a “Vector” or “VectorSerie” type object. Its availability in output is conditioned by the boolean “Stored” associated with input.

ObservationOperator: Operator. The variable indicates the observation operator, usually noted as $H$ , which transforms the input parameters $\mathbf{x}$ to results $\mathbf{y}$ to be compared to observations $\mathbf{y}^o$ . Its value is defined as a “Function” type object or a “Matrix” type one. In the case of “Function” type, different functional forms can be used, as described in the section Requirements for functions describing an operator. If there is some control $U$ included in the observation, the operator has to be applied to a pair $(X,U)$ .

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:

AmplitudeOfInitialDirection

Real value. This key indicates the scaling of the initial perturbation build as a vector used for the directional derivative around the nominal checking point. The default is 1, that means no scaling. It’s useful to modify this value, and in particular to decrease it when the biggest perturbations are going out of the allowed domain for the function.

Example: {"AmplitudeOfInitialDirection":0.5}

AmplitudeOfTangentPerturbation

Real value. This key indicates the relative numerical magnitude of the perturbation used to estimate the tangent value of the operator at the evaluation point, i.e. its directional derivative. The conservative default is 1.e-2 i.e. 1%, and it is strongly recommended to adapt it to the needs of real problems, by decreasing its value by several orders of magnitude.

Example : {"AmplitudeOfTangentPerturbation":1.e-2}

EpsilonMinimumExponent

Integer value. This key indicates the minimal exponent value of the power of 10 coefficient to be used to decrease the increment multiplier. The default is -8, and it has to be negative between 0 and -20. For example, its default value leads to calculate the residue of the scalar product formula with a fixed increment multiplied from 1.e0 to 1.e-8.

Example: {"EpsilonMinimumExponent":-12}

InitialDirection

Vector. This key indicates the vector direction used for the directional derivative around the nominal checking point. It has to be a vector of the same vector size than the checking point. If not specified, this direction defaults to a random perturbation around zero of the same vector size than the checking point.

Example: {"InitialDirection":[0.1,0.1,100.,3} for a state space of dimension 4

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}

ResiduFormula

Predefined name. This key indicates the residue formula that has to be used for the test. The default choice is “Taylor”, and the possible ones are “Taylor” (normalized residue of the Taylor development of the operator, which has to decrease with the square power of the perturbation), “TaylorOnNorm” (residue of the Taylor development of the operator with respect to the perturbation to the square, which has to remain constant) and “Norm” (residue obtained by taking the norm of the Taylor development at zero order approximation, which approximate the gradient, and which has to remain constant).

Example : {"ResiduFormula":"Taylor"}

SetSeed

Integer value. This key allow to give an integer in order to fix the seed of the random generator used in the algorithm. By default, the seed is left uninitialized, and so use the default initialization from the computer, which then change at each study. To ensure the reproducibility of results involving random samples, it is strongly advised to initialize the seed. A simple convenient value is for example 123456789. It is recommended to put an integer with more than 6 or 7 digits to properly initialize the random generator.

Example: {"SetSeed":123456789}

StoreSupplementaryCalculations

List of names. This list indicates the names of the supplementary variables, that can be available during or at the end of the algorithm, if they are initially required by the user. Their availability involves, potentially, costly calculations or memory consumptions. The default is then a void list, none of these variables being calculated and stored by default (excepted the unconditional variables). The possible names are in the following list (the detailed description of each named variable is given in the following part of this specific algorithmic documentation, in the sub-section “Information and variables available at the end of the algorithm”): [ “CurrentState”, “Residu”, “SimulatedObservationAtCurrentState”, ].

Example : {"StoreSupplementaryCalculations":["CurrentState", "Residu"]}

14.4.4. 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 the Inventory of potentially available information at the output.

Permanent outputs (non conditional)

The unconditional outputs of the algorithm are the following:

Residu

List of values. Each element is the value of the particular residue checked during the running of the algorithm, in the order of the tests.

Example: r = ADD.get("Residu")[:]

Set of on-demand outputs (conditional or not)

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

CurrentState

List of vectors. Each element is a usual state vector used during the iterative algorithm procedure.

Example: xs = ADD.get("CurrentState")[:]

Residu

List of values. Each element is the value of the particular residue checked during the running of the algorithm, in the order of the tests.

Example: r = ADD.get("Residu")[:]

SimulatedObservationAtCurrentState

List of vectors. Each element is an observed vector simulated by the observation operator from the current state, that is, in the observation space.

Example: hxs = ADD.get("SimulatedObservationAtCurrentState")[-1]

14.4.5. See also¶

References to other sections: