13.13. Calculation algorithm “ParticleSwarmOptimization”¶

13.13.1. Description¶

This algorithm estimates the state of a system, by minimizing a cost function $J$ , using an evolutionary particle swarm method. It is a meta-heuristic that does not require any particular information about the functional, and does not require derivatives (except in its hybrid “VLS” version).

It falls in the same category than the Calculation algorithm “DerivativeFreeOptimization”, Calculation algorithm “DifferentialEvolution”, Calculation algorithm “SimulatedAnnealing”, Calculation algorithm “TabuSearch”.

This is a mono-objective optimization method, allowing for global minimum search of a general error function $J$ of type $L^1$ , $L^2$ or $L^{\infty}$ , with or without weights, as described in the section for Going further in the state estimation by optimization methods. As it is an optimization meta-heuristic, reaching a global or local optimal result is nevertheless not guaranteed (except in its hybrid “VLS” version). The default error function is the augmented weighted least squares function, classically used in data assimilation.

It is based on the evolution of a population (called a “swarm”) of states (each state is called a “particle” or an “insect”). There exists various variants of this algorithm. The following stable and robust formulations are proposed here:

“CanonicalPSO” (Canonical Particle Swarm Optimization, see [ZambranoBigiarini13]), classical algorithm called “canonical” of particle swarm, robust and defining a reference for particle swarm algorithms,
“OGCR” (Simple Particle Swarm Optimization), simplified algorithm of particle swarm with no bounds on insects or velocities, not recommended because less robust, but sometimes a lot more efficient,
“SPSO-2011” (Standard Particle Swarm Optimization 2011, voir [ZambranoBigiarini13]), 2011 reference algorithm of particle swarm, robust, efficient and defined as a reference for particle swarm algorithms. This algorithm is sometimes called “ $\omega$ -PSO” or “Inertia PSO” because it incorporates a so-called inertia contribution, or also called “AIS” (for “Asynchronous Iteration Strategy”) or “APSO” (for “Advanced Particle Swarm Optimization”) because it incorporates evolutionary updating of the best elements, leading to intrinsically improved convergence of the algorithm.
“SPSO-2011-SIS” (Standard Particle Swarm Optimisation 2011 with Synchronous Iteration Strategy), very similar to the 2011 reference algorithm, and with a synchronous particle update, called “SIS”,
“SPSO-2011-PSIS” (Standard Particle Swarm Optimisation 2011 with Parallel Synchronous Iteration Strategy), similar to the “SPSO-2011-SIS” algorithm with synchronous updating and parallelization, known as “PSIS”, of the particles.

Each of the above methods can have the local aspect of its search accelerated by a “VLS” (Variational Local Search) method concerning the best insects in the swarm. The global aspect of the search remains preserved by the particle swarm method, but great care must be taken experimentally to avoid premature stagnation of the best particles in the swarm. Simply suffix the name of the algorithm variant with “-VLS” to activate the acceleration, e.g. “CanonicalPSO-VLS”.

The following are a few practical suggestions for the effective use of these algorithms:

The recommended variant of this algorithm is the “SPSO-2011” even if the “CanonicalPSO” algorithm remains by default the more robust one. If the state evaluation can be carried out in parallel, the “SPSO-2011-PSIS” algorithm can be used, even if its convergence is sometimes a little less efficient.
The number of particles or insects usually recommended varies between 40 and 100 depending on the algorithm, more or less independently of the dimension of the state space. Usually, the best performances are obtained for populations of 70 to 500 particles. Even if the default value for this elementary parameter comes from extended knowledge on these algorithms, it is recommended to adapt it to the difficulty of the given problems.
The recommended number of generations for population evolution is often around 50, but it can easily vary between 25 and 500.
The maximum number of evaluations of the simulation function should usually be limited to between a few thousand and a few tens of thousands of times the dimension of the state space.
The error functional usually decreases by levels (thus with a zero progression of the value of the functional at each generation when we stay in the level), making it not recommended to stop on the criterion of decrease of the cost function. It is normally wiser to adapt the number of iterations or generations to accelerate the convergence of the algorithms.
If the problem is constrained, it is necessary to define the bounds of the variables (by the variable “Bounds”). If the problem is totally unconstrained, it is essential to define increment bounds (by the variable “BoxBounds”) to delimit the optimal search in a useful way. Similarly, if the problem is partially constrained, it is recommended (but not required) to define increment bounds. In case these increment bounds are not defined, the variable bounds will be used as increment bounds.
The use of hybrid “VLS”-type acceleration variants will require the availability of the operator gradient (either given explicitly or requested by finite differences at definition), but normally speeds up the local search. It is therefore a good idea to reduce the number of generations (which can sometimes be reduced to as few as 5 or 10), while specifying a reasonable number of insects (of the order of 1 to 5, for example) to be the subject of this local search. However, the parameters of the hybrid search have to be finely tuned for each case, in the form of a compromise, in order to have both the global and local capacity of this optimization. Indeed, too fast local convergence or too limited global exploration can reduce the algorithm’s ability to explore the solution space.

These suggestions are to be used as experimental indications, not as requirements, because they are to be appreciated or adapted according to the physics of each problem that is treated.

The count of the number of evaluations of the function to be simulated during this algorithm is deterministic, namely the “number of iterations or generations” multiplied by the “number of individuals in the population”. With the default values of 40 to 100 individuals on 50 generations, it takes between 40x50=2000 and 100*50=5000 evaluations, which is significantly more than in variational optimization. It is for this reason that this algorithm is usually interesting when the dimension of the state space is large, or when the non-linearity of the simulation make the evaluation of the gradient of the functional by numerical approximation complicated or invalid. But it is also necessary that the calculation of the function to be simulated is not too costly to avoid a prohibitive optimization time length.

To reduce the number of generations, we can use the ASAPSO adaptive method [Wang09] to modify the localization of the optimal swarm search. This consists in linearly reducing the algorithm’s global search behavior, thus favoring swarm localization. When the problem is favorable, it’s possible to reduce the number of generations by a factor of 2 to 5, while still achieving virtually the same quality for the best particle. Two keywords are available to control the use of this adaptive method.

As with other algorithms, it is possible to compute a quantity called the a posteriori covariance of the particle swarm. However, the meaning of the indicator calculated here differs in nature from a covariance established in filtering or variational analysis, for example. The indicator is calculated, in the standard way, as the dispersion of all the particles. But its value is to be compared exclusively with the evaluation of the same quantity for the initial swarm, and used in a relative sense of variation between the beginning and the end of the algorithm’s optimal search. This is because the initial dispersion of the particles is controlled by the initialization parameters of the swarm, whose aim is to cover the search space, and the final dispersion of the particles reflects the localization performance of the optimal search. Unlike the covariance of the background errors, which is classically involved in calculating the objective to be optimized and is related to the variance of the optimal state, the covariance calculated a posteriori reflects only the dispersion of the particles. For the correct use of this algorithm, when the a posteriori covariance calculation is requested, it is performed for the initial swarm and for the final swarm.

13.13.2. 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 optimization methods proposed by this algorithm perform a non-local search for the minimum, without however ensuring a global search. This is the case when optimization methods have the ability to avoid being trapped by the first local minimum found. These capabilities are sometimes heuristic.

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.

The methods proposed by this algorithm have internal parallelism, and can therefore take advantage of computational distribution resources. 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.

The methods proposed by this algorithm achieve their convergence on one or more number criteria. In practice, there may be simultaneously several convergence criteria.

The number is frequently a significant value for the algorithm, such as a number of iterations or a number of evaluations, but it can also be, for example, a number of generations for an evolutionary algorithm.

Convergence thresholds need to be carefully adjusted, to reduce the gobal calculation cost, or to ensure that convergence is adapted to the physical case encountered.

13.13.3. Optional and required commands¶

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

Background: Vector. The variable indicates the background or initial vector used, previously noted as $\mathbf{x}^b$ . Its value is defined as a “Vector” or “VectorSerie” type object. Its availability in output is conditioned by the boolean “Stored” associated with input.

BackgroundError: Matrix. This indicates the background error covariance matrix, previously noted as $\mathbf{B}$ . Its value is defined as a “Matrix” type object, a “ScalarSparseMatrix” type object, or a “DiagonalSparseMatrix” type object, as described in detail in the section Requirements to describe covariance matrices. Its availability in output is conditioned by the boolean “Stored” associated with input.

EvolutionError: Matrix. The variable indicates the evolution error covariance matrix, usually noted as $\mathbf{Q}$ . It is defined as a “Matrix” type object, a “ScalarSparseMatrix” type object, or a “DiagonalSparseMatrix” type object, as described in detail in the section Requirements to describe covariance matrices. Its availability in output is conditioned by the boolean “Stored” associated with input.

EvolutionModel: Operator. The variable indicates the evolution model operator, usually noted $M$ , which describes an elementary step of evolution. 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 evolution model, the operator has to be applied to a pair $(X,U)$ .

Observation: List of vectors. The variable indicates the observation vector used for data assimilation or optimization, and usually noted $\mathbf{y}^o$ . Its value is defined as an object of type “Vector” if it is a single observation (temporal or not) or “VectorSeries” if it is a succession of observations. Its availability in output is conditioned by the boolean “Stored” associated in input.

ObservationError: Matrix. The variable indicates the observation error covariance matrix, usually noted as $\mathbf{R}$ . It is defined as a “Matrix” type object, a “ScalarSparseMatrix” type object, or a “DiagonalSparseMatrix” type object, as described in detail in the section Requirements to describe covariance matrices. 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 data assimilation or optimization case. Moreover, the parameters of the command “AlgorithmParameters” allows 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:

Bounds

List of pairs of real values. This key allows to define pairs of upper and lower bounds for every state variable being optimized. Bounds have to be given by a list of list of pairs of lower/upper bounds for each variable, with a value of None each time there is no bound. The bounds can always be specified, but they are taken into account only by the constrained optimizers. If the list is empty, there are no bounds.

Example: {"Bounds":[[2.,5.],[1.e-2,10.],[-30.,None],[None,None]]}

BoxBounds

List of pairs of real values. This key allows to define pairs of upper and lower bounds for increments on every state variable being optimized (and not on state variables themselves, whose bounds can be indicated by the “Bounds” variable). Increment bounds have to be given by a list of list of pairs of lower/upper bounds for each increment on variable, with a value of None each time there is no bound. This key is only required if there are no variable bounds, and there are no default values.

Example : {"BoxBounds":[[-0.5,0.5], [0.01,2.], [0.,None], [None,None]]}

CognitiveAcceleration

Real value. This key indicates the recall rate at the best previously known value of the current insect’s history. It is a floating point positive value. The default value is about $1/2+ln(2)=1.19315$ , and it is recommended to adapt it, rather by reducing it, to the physical case that is being treated.

In the standard (non-adaptive) case, this rate is constant and is equal to the indicated value. In the ASAPSO adaptive case [Wang09], the value of this key indicates the initial recall rate, which then decreases linearly with the number of generations and the associated “CognitiveAccelerationControl” factor.

Example : {"CognitiveAcceleration":1.19315}

CognitiveAccelerationControl

Real value. This key indicates the factor of change in the recall rate towards the best previously known value of the current insect’s history. It is a positive real value whose default is 0, that is, by default, there is no change in the recall rate.

In the ASAPSO adaptive case [Wang09], the value of this key indicates the linear decrease factor of the recall rate with the number of generations (with respect to the requested total number of generations), given that the initial value of the rate is indicated by the associated “CognitiveAcceleration” factor. There is no recommended value, but you could, for example, use the initial value $1.19315$ of the associated factor if you want to cancel any recall towards the best known value of the history at the end of the iterations.

Example : {"CognitiveAccelerationControl":0.}

DistributionByComponents

List of predefined names. This keyword only needs to be defined when the particle swarm initialization for the “ParticleSwarmOptimization” algorithm is set to “DistributionByComponents”. In this case, for each state component, the chosen distribution must be indicated in the form of a predefined name. Possible names are “uniform”, “loguniform”, “logarithmic”, “[‘normal’,σ]”, “[‘lognormal’,σ]” and “[‘logarithmicnormal’,σ]”. All “normal” distributions must be accompanied by an indication of the standard deviation σ, given that they are centered in the definition domain. The values “uniform”, “loguniform”, “normal”, “lognormal” only affect position initialization by applying the indicated distribution, while the values “logarithmic” and “logarithmicnormal” affect both position and motion initialization. There must be the same number of values indicated as the size of an individual state. Distributions conform to the limits specified for the “*ParticleSwarmOptimization” algorithm. Distributions can be different for each axis. When an identical distribution is chosen for all components, this is equivalent to choosing the global value of the previous keyword instead, if it exists.

Example : {"DistributionByComponents" : ['uniform', 'loguniform', 'logarithmic', ['normal', 1]]} for a state space of dimension 4.

GlobalCostReductionTolerance

Real value. This key indicates the limit reduction factor, leading to the iterative optimization process being stopped when the cost function decreases by at least this tolerance over the entire optimal search. The default value is 1.e-16 (equivalent to no effect), and it is recommended to adapt it to the needs of real problems.

Example: {"GlobalCostReductionTolerance":1.e-16}

HybridCostDecrementTolerance

Real value. This key indicates a limit value, leading to stop successfully the optimization process for the variational part in the coupling, when the cost function decreases less than this tolerance at the last step. The default is 1.e-7, and it is recommended to adapt it to the needs on real problems. One can refer to the section describing ways for Convergence control for calculation cases and iterative algorithms for more detailed recommendations.

Example: {"HybridCostDecrementTolerance":1.e-7}

HybridMaximumNumberOfIterations

Integer value. This key indicates the maximum number of internal iterations allowed for hybrid optimization, for the variational part. The default is 15000, which is very similar to no limit on iterations. It is then recommended to adapt this parameter to the needs on real problems. For some optimizers, the effective stopping step can be slightly different of the limit due to algorithm internal control requirements. One can refer to the section describing ways for Convergence control for calculation cases and iterative algorithms for more detailed recommendations.

Example: {"HybridMaximumNumberOfIterations":100}

HybridNumberOfLocalHunters

Integer value. This key indicates the number of insects on which the local search will be conducted. Insects are chosen as the best from the current iteration of the global search. With a default value of 1, the local search is performed on the best insect only. It is then recommended to adapt this parameter to the needs on real problems.

Example : {"HybridNumberOfLocalHunters":1}

HybridNumberOfWarmupIterations

Integer value. This key indicates the number of initial iterations of the global search performed before establishing a local search on the best insects. By default, with a value of 0, the local search is performed from the first iterative step of the global search.

Example : {"HybridNumberOfWarmupIterations":0}

InertiaWeight

Real value. This key indicates the part of the insect velocity which is imposed by the swarm, named “inertia weight”. It is a positive floating point value. It is a floating point value between 0 and 1. The default value is about $1/(2*ln(2))$ and it is recommended to adapt it to the physical case that is being treated.

Example : {"InertiaWeight":0.72135}

InitializationPoint

Vector. The variable specifies one vector to be used as the initial state around which an iterative algorithm starts. By default, this initial state is not required and is equal to the background $\mathbf{x}^b$ . Its value must allow to build a vector of the same size as the background. If provided, it replaces the background only for initialization.

Example : {"InitializationPoint":[1, 2, 3, 4, 5]}

MaximumNumberOfFunctionEvaluations

Integer value. This key indicates the maximum number of evaluation of the cost function to be optimized. The default is 15000, which is an arbitrary limit. It is then recommended to adapt this parameter to the needs on real problems. For some optimizers, the effective number of function evaluations can be slightly different of the limit due to algorithm internal control requirements.

Example: {"MaximumNumberOfFunctionEvaluations":50}

MaximumNumberOfIterations

Integer value. This key indicates the maximum number of internal iterations allowed for iterative optimization. The default is 50, which is an arbitrary limit. It is then recommended to adapt this parameter to the needs on real problems.

Example: {"MaximumNumberOfIterations":50}

NumberOfInsects

Integer value. This key indicates the number of insects or particles in the swarm. The default is 100, which is a usual default for this algorithm.

Example : {"NumberOfInsects":100}

QualityCriterion

Predefined name. This key indicates the quality criterion, minimized to find the optimal state estimate. The default is the usual data assimilation criterion named “DA”, the augmented weighted least squares. The possible criterion has to be in the following list, where the equivalent names are indicated by the sign “<=>”: [“AugmentedWeightedLeastSquares” <=> “AWLS” <=> “DA”, “WeightedLeastSquares” <=> “WLS”, “LeastSquares” <=> “LS” <=> “L2”, “AbsoluteValue” <=> “L1”, “MaximumError” <=> “ME” <=> “Linf”]. See the section for Going further in the state estimation by optimization methods to have a detailed definition of these quality criteria.

Example: {"QualityCriterion":"DA"}

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}

SocialAcceleration

Real value. This key indicates the recall rate towards the best insect in the current insect’s neighborhood, which by default is the complete swarm. It is a floating point positive value. The default value is about $1/2+ln(2)=1.19315$ and it is recommended to adapt it, rather by reducing it, to the physical case that is being treated.

In the standard (non-adaptive) case, this rate is constant and is equal to the indicated value. In the ASAPSO adaptive case [Wang09], the value of this key indicates the initial recall rate, which then increases linearly with the number of generations and the associated “SocialAccelerationControl” factor.

Example : {"SocialAcceleration":1.19315}

SocialAccelerationControl

Real value. This key indicates the factor of change in the recall rate towards the best insect in the current insect’s neighborhood, which by default is the complete swarm. It is a positive real value whose default is 0, that is, by default, there is no change in the recall rate.

In the ASAPSO adaptive case [Wang09], the value of this key indicates the linear growth factor of the recall rate with the number of generations ( with respect to the requested total number of generations), given that the initial value of the rate is indicated by the associated “SocialAcceleration” factor. There is no recommended value, but you can use the initial value $1.19315$ of the associated factor, for example, if you want to double the recall towards the best known value of the neighborhood at the end of the iterations.

Example : {"SocialAccelerationControl":0.}

StoreInitialState

Boolean value. This variable defines whether the algorithm’s initial state is stored (with True) or not (with False, by default) as the first state in the iterative sequence of found states. This makes algorithmic iterative storage identical to temporal iterative storage (similar, for example, to Kalman filters).

Example : {"StoreInitialState":False}

SwarmInitialization

Predefined name. The name defines the particle swarm initialization mode for the “ParticleSwarmOptimization” algorithm. The particle series is initialized by specifying the distribution per component, which can be identical for all components (this is the case for all values except “DistributionByComponents”), or component-specific with “DistributionByComponents”. In the latter case, it is also necessary to specify the “DistributionByComponents” keyword content. The default value is “UniformByComponents”.

The possible name is therefore in the following list: [“UniformByComponents”, “LogUniformByComponents”, “LogarithmicByComponents”, “DistributionByComponents”].

Example : {"SwarmInitialization":"UniformByComponents"}

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”): [ “Analysis”, “APosterioriCorrelations”, “APosterioriCovariance”, “APosterioriStandardDeviations”, “APosterioriVariances”, “BMA”, “CostFunctionJ”, “CostFunctionJb”, “CostFunctionJo”, “CurrentIterationNumber”, “CurrentState”, “EnsembleOfStates”, “EnsembleOfSimulations”, “Innovation”, “InternalCostFunctionJ”, “InternalCostFunctionJb”, “InternalCostFunctionJo”, “OMA”, “OMB”, “SimulatedObservationAtBackground”, “SimulatedObservationAtCurrentState”, “SimulatedObservationAtOptimum”, ].

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

SwarmTopology

Predefined name. This key indicates how the particles (or insects) communicate information to each other during the evolution of the particle swarm. The most classical method consists in exchanging information between all particles (called “gbest” or “FullyConnectedNeighborhood”). But it is often more efficient to exchange information on a reduced neighborhood, as in the classical method “lbest” (or “RingNeighborhoodWithRadius1”) exchanging information with the two neighboring particles in numbering order (the previous one and the next one), or the method “RingNeighborhoodWithRadius2” exchanging with the 4 neighbors (the two previous ones and the two following ones). A variant of reduced neighborhood consists in exchanging with 3 neighbors (method “AdaptativeRandomWith3Neighbors”) or 5 neighbors (method “AdaptativeRandomWith5Neighbors”) chosen randomly (the particle can be drawn several times). The default value is “FullyConnectedNeighborhood”, and it is advisable to change it carefully depending on the properties of the simulated physical system. The possible communication topology is to be chosen from the following list, in which the equivalent names are indicated by a “<=>” sign: [“FullyConnectedNeighborhood” <=> “FullyConnectedNeighbourhood” <=> “gbest”, “RingNeighborhoodWithRadius1” <=> “RingNeighbourhoodWithRadius1” <=> “lbest”, “RingNeighborhoodWithRadius2” <=> “RingNeighbourhoodWithRadius2”, “AdaptativeRandomWith3Neighbors” <=> “AdaptativeRandomWith3Neighbours” <=> “abest”, “AdaptativeRandomWith5Neighbors” <=> “AdaptativeRandomWith5Neighbours”].

Example : {"SwarmTopology":"FullyConnectedNeighborhood"}

Variant

Predefined name. This key allows to choose one of the possible variants for the main algorithm. The default variant is the original “CanonicalPSO”, and the possible choices are “CanonicalPSO” (Canonical Particle Swarm Optimization), “OGCR” (Simple Particle Swarm Optimization), “SPSO-2011” (Standard Standard Particle Swarm Optimization 2011) identical to “SPSO-2011-AIS” (Standard Standard Particle Swarm Optimisation 2011 with Asynchronous Iteration Strategy), “SPSO-2011-SIS” (Standard Particle Swarm Optimisation 2011 with Synchronous Iteration Strategy), “SPSO-2011-PSIS” (Standard Particle Swarm Optimisation 2011 with Parallel Synchronous Iteration Strategy). The local accelerated “VLS” (Variational Local Search) versions are obtained by adding the suffix “-VLS” to each method: “CanonicalPSO-VLS”, “OGCR-VLS”, “SPSO-2011-AIS-VLS”, “SPSO-2011-SIS-VLS”, “SPSO-2011-PSIS-VLS”.

It is recommended to try the “CanonicalPSO” variant with about 100 particles for robust performance, and to reduce the number of particles to about 40 for all variants other than the original “CanonicalPSO” formulation.

Example : {"Variant":"CanonicalPSO"}

VelocityClampingFactor

Real value. This key indicates the rate of group velocity attenuation in the update for each insect, useful to avoid swarm explosion, i.e. uncontrolled growth of insect velocity. It is a floating point value between 0+ and 1. The default value is 0.3.

Example : {"VelocityClampingFactor":0.3}

13.13.4. 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:

Analysis

List of vectors. Each element of this variable is an optimal state $\mathbf{x}^*$ in optimization, an interpolate or an analysis $\mathbf{x}^a$ in data assimilation.

Example: xa = ADD.get("Analysis")[-1]

CostFunctionJ

List of values. Each element is a value of the chosen error function $J$ .

Example: J = ADD.get("CostFunctionJ")[:]

CostFunctionJb

List of values. Each element is a value of the error function $J^b$ , that is of the background difference part. If this part does not exist in the error function, its value is zero.

Example: Jb = ADD.get("CostFunctionJb")[:]

CostFunctionJo

List of values. Each element is a value of the error function $J^o$ , that is of the observation difference part.

Example: Jo = ADD.get("CostFunctionJo")[:]

Set of on-demand outputs (conditional or not)

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

Analysis

List of vectors. Each element of this variable is an optimal state $\mathbf{x}^*$ in optimization, an interpolate or an analysis $\mathbf{x}^a$ in data assimilation.

Example: xa = ADD.get("Analysis")[-1]

APosterioriCorrelations

List of matrices. Each element is an a posteriori error correlations matrix of the optimal state, coming from the $\mathbf{A}$ covariance matrix. In order to get them, this a posteriori error covariances calculation has to be requested at the same time.

Example: apc = ADD.get("APosterioriCorrelations")[-1]

APosterioriCovariance

List of matrices. Each element is an a posteriori error covariance matrix $\mathbf{A}$ of the optimal state.

Example: apc = ADD.get("APosterioriCovariance")[-1]

APosterioriStandardDeviations

List of matrices. Each element is an a posteriori error standard errors diagonal matrix of the optimal state, coming from the $\mathbf{A}$ covariance matrix. In order to get them, this a posteriori error covariances calculation has to be requested at the same time.

Example: aps = ADD.get("APosterioriStandardDeviations")[-1]

APosterioriVariances

List of matrices. Each element is an a posteriori error variance errors diagonal matrix of the optimal state, coming from the $\mathbf{A}$ covariance matrix. In order to get them, this a posteriori error covariances calculation has to be requested at the same time.

Example: apv = ADD.get("APosterioriVariances")[-1]

BMA

List of vectors. Each element is a vector of difference between the background and the optimal state.

Example: bma = ADD.get("BMA")[-1]

CostFunctionJ

List of values. Each element is a value of the chosen error function $J$ .

Example: J = ADD.get("CostFunctionJ")[:]

CostFunctionJb

List of values. Each element is a value of the error function $J^b$ , that is of the background difference part. If this part does not exist in the error function, its value is zero.

Example: Jb = ADD.get("CostFunctionJb")[:]

CostFunctionJo

List of values. Each element is a value of the error function $J^o$ , that is of the observation difference part.

Example: Jo = ADD.get("CostFunctionJo")[:]

CurrentIterationNumber

List of integers. Each element is the iteration index at the current step during the iterative algorithm procedure. There is one iteration index value per assimilation step corresponding to an observed state.

Example: cin = ADD.get("CurrentIterationNumber")[-1]

CurrentState

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

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

EnsembleOfStates

List of vectors or matrix. Each element is an ordered collection of physical or parameter state vectors $\mathbf{x}$ . These are $H$ operator entries, i.e. current states before observation. 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 : {"EnsembleOfStates":[x1, x2, x3...]}

EnsembleOfSimulations

List of vectors or matrix. This key contains an ordered collection of physical state vectors or simulated state vectors $\mathbf{y}$ that may be observed. These are $H$ operator outputs, i.e. simulated observation states (called “snapshots” in reduced-base 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 : {"EnsembleOfSimulations":[y1, y2, y3...]}

Innovation

List of vectors. Each element is an innovation vector, which is in static the difference between the optimal and the background, and in dynamic the evolution increment.

Example: d = ADD.get("Innovation")[-1]

OMA

List of vectors. Each element is a vector of difference between the observation and the optimal state in the observation space.

Example: oma = ADD.get("OMA")[-1]

OMB

List of vectors. Each element is a vector of difference between the observation and the background state in the observation space.

Example: omb = ADD.get("OMB")[-1]

SimulatedObservationAtBackground

List of vectors. Each element is a vector of observation simulated by the observation operator from the background $\mathbf{x}^b$ . It is the forecast from the background, and it is sometimes called “Dry”.

Example: hxb = ADD.get("SimulatedObservationAtBackground")[-1]

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]

SimulatedObservationAtOptimum

List of vectors. Each element is a vector of observation obtained by the observation operator from simulation on the analysis or optimal state $\mathbf{x}^a$ . It is the observed forecast from the analysis or the optimal state, and it is sometimes called “Forecast”.

Example: hxa = ADD.get("SimulatedObservationAtOptimum")[-1]

13.13.5. Python (TUI) use examples¶

Here is one or more very simple examples of the proposed algorithm and its parameters, written in [DocR] Textual User Interface for ADAO (TUI/API). Moreover, when it is possible, the information given as input also allows to define an equivalent case in [DocR] Graphical User Interface for ADAO (GUI/EFICAS).

This example describes the calibration of parameters $\mathbf{x}$ of a quadratic observation model $H$ . This model is here represented as a function named QuadFunction. This function get as input the coefficients vector $\mathbf{x}$ , and return as output the evaluation vector $\mathbf{y}$ of the quadratic model at the predefined internal control points. The calibration is done using an initial coefficient set (background state specified by Xb in the code), and with the information $\mathbf{y}^o$ (specified by Yobs in the code) of 5 measures obtained in these same internal control points. We set twin experiments (see To test a data assimilation chain: the twin experiments) and the measurements are supposed to be perfect. We choose to emphasize the observations versus the background by setting a great variance for the background error, here of $10^{6}$ .

The adjustment is carried out by displaying intermediate results during iterative optimization.

# -*- coding: utf-8 -*-
#
from numpy import array, ravel
def QuadFunction( coefficients ):
    """
    Quadratic simulation in x points: y = a x^2 + b x + c
    """
    a, b, c = list(ravel(coefficients))
    x_points = (-5, 0, 1, 3, 10)
    y_points = []
    for x in x_points:
        y_points.append( a*x*x + b*x + c )
    return array(y_points)
#
Xb   = array([1., 1., 1.])
Yobs = array([57, 2, 3, 17, 192])
#
NumberOfInsects = 40
#
print("Resolution of the calibration problem")
print("-------------------------------------")
print("")
from adao import adaoBuilder
case = adaoBuilder.New()
case.setBackground( Vector = Xb, Stored=True )
case.setBackgroundError( ScalarSparseMatrix = 1.e6 )
case.setObservation( Vector = Yobs, Stored=True )
case.setObservationError( ScalarSparseMatrix = 1. )
case.setObservationOperator( OneFunction = QuadFunction )
case.setAlgorithmParameters(
    Algorithm="ParticleSwarmOptimization",
    Parameters={
        "NumberOfInsects":NumberOfInsects,
        "MaximumNumberOfIterations": 20,
        "StoreSupplementaryCalculations": [
            "CurrentState",
            ],
        "Bounds":[[0,5],[-2,2],[0,5]],
        "SetSeed":123456789,
        },
    )
case.setObserver(
    Info="  Intermediate state at the current iteration:",
    Template="ValuePrinter",
    Variable="CurrentState",
    )
case.execute()
print("")
#
#-------------------------------------------------------------------------------
#
print("Calibration of %i coefficients in a 1D quadratic function on %i measures"%(
    len(case.get("Background")),
    len(case.get("Observation")),
    ))
print("----------------------------------------------------------------------")
print("")
print("Observation vector.................:", ravel(case.get("Observation")))
print("A priori background state..........:", ravel(case.get("Background")))
print("")
print("Expected theoretical coefficients..:", ravel((2,-1,2)))
print("")
print("Number of iterations...............:", len(case.get("CurrentState")))
print("Number of simulations..............:", NumberOfInsects*len(case.get("CurrentState")))
print("Calibration resulting coefficients.:", ravel(case.get("Analysis")[-1]))
#
Xa = case.get("Analysis")[-1]
import matplotlib.pyplot as plt
plt.rcParams["figure.figsize"] = (10, 4)
#
plt.figure()
plt.plot((-5,0,1,3,10),QuadFunction(Xb),"b--",label="Simulation at background")
plt.plot((-5,0,1,3,10),Yobs,            "kX", label="Observation",markersize=10)
plt.plot((-5,0,1,3,10),QuadFunction(Xa),"r-", label="Simulation at optimum")
plt.legend()
plt.title("Coefficients calibration", fontweight="bold")
plt.xlabel("Arbitrary coordinate")
plt.ylabel("Observations")
plt.savefig("simple_ParticleSwarmOptimization1.png")

The execution result is the following:

Resolution of the calibration problem
-------------------------------------

  Intermediate state at the current iteration: [ 1.76770856 -1.2054263   1.22625259]
  Intermediate state at the current iteration: [ 2.04776518 -1.17449716  3.14493347]
  Intermediate state at the current iteration: [ 2.04776518 -1.17449716  3.14493347]
  Intermediate state at the current iteration: [ 2.01933291 -1.          3.15162067]
  Intermediate state at the current iteration: [ 1.96384202 -0.74855119  3.40642058]
  Intermediate state at the current iteration: [ 1.96384202 -0.74855119  3.40642058]
  Intermediate state at the current iteration: [ 1.96384202 -0.74855119  3.40642058]
  Intermediate state at the current iteration: [ 1.96384202 -0.74855119  3.40642058]
  Intermediate state at the current iteration: [ 1.95417745 -0.73191939  3.14451887]
  Intermediate state at the current iteration: [ 1.96217646 -0.7883895   2.91127919]
  Intermediate state at the current iteration: [ 1.97610485 -1.00254825  2.89582746]
  Intermediate state at the current iteration: [ 2.0007262  -1.06443275  2.69825603]
  Intermediate state at the current iteration: [ 1.9934285  -1.02432071  2.39823319]
  Intermediate state at the current iteration: [ 1.9934285  -1.02432071  2.39823319]
  Intermediate state at the current iteration: [ 1.9942533  -0.99256953  2.30174702]
  Intermediate state at the current iteration: [ 1.9942533  -0.99256953  2.30174702]
  Intermediate state at the current iteration: [ 1.99742923 -0.99796085  2.1278678 ]
  Intermediate state at the current iteration: [ 1.99742923 -0.99796085  2.1278678 ]
  Intermediate state at the current iteration: [ 1.99742923 -0.99796085  2.1278678 ]
  Intermediate state at the current iteration: [ 1.99742923 -0.99796085  2.1278678 ]
  Intermediate state at the current iteration: [ 2.00166149 -1.0012696   2.02137857]

Calibration of 3 coefficients in a 1D quadratic function on 5 measures
----------------------------------------------------------------------

Observation vector.................: [ 57.   2.   3.  17. 192.]
A priori background state..........: [1. 1. 1.]

Expected theoretical coefficients..: [ 2 -1  2]

Number of iterations...............: 21
Number of simulations..............: 840
Calibration resulting coefficients.: [ 2.00166149 -1.0012696   2.02137857]

The figures illustrating the result of its execution are as follows:

_images/simple_ParticleSwarmOptimization1.png

13.13.6. See also¶

References to other sections:

Bibliographical references: