Provide directly the isoprobabilistic transformation during FORM

Alexis · February 22, 2023, 4:33pm

Hi,

Is it possible to directly provide the isoprobabilistic transformation using FORM instead of defining the distribution and letting Openturns estimates the isoprobabilistic transformation?

Thanks,
Alexis

MichaelBaudin · February 23, 2023, 11:05am

Hi !
As documented by @KieranDelamotte in the example help page the isoprobabilistic transform is a property of the input distribution :

transformation = distribution.getIsoProbabilisticTransformation()
inverseTransformation = distX.getInverseIsoProbabilisticTransformation()

This is why you cannot, with the current interface, provide the iso-probabilistic transformation to the FORM approximation. The way it is implemented is as follows.

The FORM class is based on the run() method of the Analytical class.

github.com

openturns/openturns/blob/8417f12613afef0fca9d0bc5c26cfedc35e7e47e/lib/src/Uncertainty/Algorithm/Analytical/FORM.cxx#L83


      
            OSS oss;
            oss << "class=" << FORM::GetClassName()
                << " " << Analytical::__repr__()
                << " result=" << formResult_;
            return oss;
          }
          
          
/* Function that computes the design point by re-using the Analytical::run() and creates a FORMResult */
          void FORM::run()
          {
            Analytical::run();
            formResult_ = FORMResult(Analytical::getAnalyticalResult().getStandardSpaceDesignPoint(), Analytical::getAnalyticalResult().getLimitStateVariable(), Analytical::getAnalyticalResult().getIsStandardPointOriginInFailureSpace());
            formResult_.setOptimizationResult(Analytical::getAnalyticalResult().getOptimizationResult());
          }
          
          
/* Method save() stores the object through the StorageManager */
          void FORM::save(Advocate & adv) const
          {
          
          
  Analytical::save(adv);
            adv.saveAttribute("formResult_", formResult_);

The Analytical class creates a StandardEvent:

github.com

openturns/openturns/blob/8417f12613afef0fca9d0bc5c26cfedc35e7e47e/lib/src/Uncertainty/Algorithm/Analytical/Analytical.cxx#L111


      
                << " nearestPointAlgorithm=" << nearestPointAlgorithm_.__repr__()
                << " event=" << event_
                << " physicalstartingPoint=" << physicalStartingPoint_;
            return oss;
          }
          
          
/* Performs the actual computation. */
          void Analytical::run()
          {
            /* Build a standard event from the event */
            StandardEvent standardEvent(event_);
          
          
  /* set the level function of the algorithm */
            nearestPointAlgorithm_.setProblem(NearestPointProblem(standardEvent.getImplementation()->getFunction(), standardEvent.getThreshold()));
          
          
  /* set the starting point of the algorithm in the standard space  */
            nearestPointAlgorithm_.setStartingPoint(event_.getImplementation()->getAntecedent().getDistribution().getIsoProbabilisticTransformation().operator()(physicalStartingPoint_));
          
          
  /* solve the nearest point problem */
            nearestPointAlgorithm_.run();

The StandardEvent creates the inverse iso-probabilistic mapping:

github.com

openturns/openturns/blob/8417f12613afef0fca9d0bc5c26cfedc35e7e47e/lib/src/Uncertainty/Model/StandardEvent.cxx#L72


      
              for (UnsignedInteger j = 0; j < i; ++j) if (covariance(i, j) != 0.0) throw InvalidArgumentException(HERE) << "Error: StandardEvent can only be constructed from composite random vectors whose antecedent is standard spherical, here the distribution is " << antecedent.getImplementation()->getAntecedent().getDistribution().getImplementation()->__str__();
            }
          }
          
          
/* Constructor from an Event */
          StandardEvent::StandardEvent(const RandomVector & event)
            : RandomVector()
          {
            // Get the isoprobabilistic inverse transformation from the antecedent distribution
            const Distribution distribution(event.getImplementation()->getAntecedent().getDistribution());
            const InverseIsoProbabilisticTransformation inverse(distribution.getInverseIsoProbabilisticTransformation());
            // Get the function
            const Function function(event.getImplementation()->getFunction());
            // Get the standard distribution from the antecedent distribution
            const Distribution standard(distribution.getStandardDistribution());
            // Build the antecedent
            const RandomVector antecedent(standard);
            const ComposedFunction composed(function, inverse);
            const CompositeRandomVector vect(composed, antecedent);
            // Set the random vector implementation
            (*this) = StandardEvent(vect, event.getOperator(), event.getThreshold());

I cannot see any point where you could be able to provide a inverse mapping. Anyway, what would be the transformation that you need that is not already implemented?

In the case you want to use your own mapping, why not implementing the FORM approximation using your own code? I ask this because the library already provide all the necessary intermediate building blocks, so that this should involved only a minimum amount of code: you can re-use the existing optimization libraries (including AbdoRackwitz), estimate the P_f using the FORM approximation and create your own FORMResult object, using the results that you got from your own code.

Regards,
Michaël

Alexis · February 23, 2023, 3:03pm

Hi,

Thank you for this detailed and quick answer. Yes, I think the easiest is what you suggest: use the OpenTURNS blocks to define my own approximation of the FORM. Thanks

Best regards,
Alexis

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Provide directly the isoprobabilistic transformation during FORM

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