// * * * * * * * * * * * * * * * Member Functions * * * * * * * * * * * * * //
void Foam::extendedMomentInversion::invert(const univariateMomentSet& moments)
{
univariateMomentSet m(moments);
reset();
// Terminate execution if negative number density is encountered
if (m[0] < 0.0)
{
FatalErrorIn
(
"Foam::extendedMomentInversion::invert\n"
"(\n"
" const univariateMomentSet& moments\n"
")"
) << "The zero-order moment is negative."
<< abort(FatalError);
}
// Exclude cases where the zero-order moment is very small to avoid
// problems in the inversion due to round-off error
if (m[0] < SMALL)
{
sigma_ = 0.0;
nullSigma_ = true;
return;
}
label nRealizableMoments = m.nRealizableMoments();
if (nRealizableMoments % 2 == 0)
{
// If the number of realizable moments is even, we apply the standard
// QMOM directly to maximize the number of preserved moments.
// Info << "Even number of realizable moments: using QMOM" << endl;
// Info << "Moments: " << m << endl;
// Info << "Invertible: " << m.nInvertibleMoments() << endl;
// Info << "Realizable: " << m.nRealizableMoments() << endl;
m.invert();
sigma_ = 0.0;
nullSigma_ = true;
secondaryQuadrature(m);
}
else
{
// Resizing the moment set to avoid copying again
m.resize(nRealizableMoments);
// Local set of starred moments
univariateMomentSet mStar(nRealizableMoments, 0);
// Compute target function for sigma = 0
scalar sigmaLow = 0.0;
scalar fLow = targetFunction(sigmaLow, m, mStar);
sigma_ = sigmaLow;
// Check if sigma = 0 is root
if (mag(fLow) <= targetFunctionTol_)
{
m.invert();
sigma_ = 0.0;
nullSigma_ = true;
secondaryQuadrature(m);
return;
}
// Compute target function for sigma = sigmaMax
scalar sigMax = sigmaMax(m);
scalar sigmaHigh = sigMax;
scalar fHigh = targetFunction(sigmaHigh, m, mStar);
if (fLow*fHigh > 0)
{
// Root not found. Minimize target function in [0, sigma_]
sigma_ = minimizeTargetFunction(0, sigmaHigh, m, mStar);
// Check if sigma is small and use QMOM
if (mag(sigma_) < sigmaTol_)
{
m.invert();
sigma_ = 0.0;
nullSigma_ = true;
secondaryQuadrature(m);
return;
}
targetFunction(sigma_, m, mStar);
secondaryQuadrature(mStar);
return;
}
// Apply Ridder's algorithm to find sigma
for (label iter = 0; iter < maxSigmaIter_; iter++)
{
scalar fMid, sigmaMid;
sigmaMid = (sigmaLow + sigmaHigh)/2.0;
fMid = targetFunction(sigmaMid, m, mStar);
scalar s = sqrt(sqr(fMid) - fLow*fHigh);
if (s == 0.0)
{
FatalErrorIn
(
"Foam::extendedMomentInversion::invert\n"
"(\n"
" const univariateMomentSet& moments\n"
")"
)<< "Singular value encountered while attempting to find root."
<< "Moment set = " << m << endl
<< "sigma = " << sigma_ << endl
<< "fLow = " << fLow << endl
<< "fMid = " << fMid << endl
<< "fHigh = " << fHigh
<< abort(FatalError);
}
sigma_ = sigmaMid + (sigmaMid - sigmaLow)*sign(fLow - fHigh)*fMid/s;
momentsToMomentsStar(sigma_, m, mStar);
scalar fNew = targetFunction(sigma_, m, mStar);
scalar dSigma = (sigmaHigh - sigmaLow)/2.0;
// Check for convergence
if (mag(fNew) <= targetFunctionTol_ || mag(dSigma) <= sigmaTol_)
{
if (mag(sigma_) < sigmaTol_)
{
m.invert();
sigma_ = 0.0;
nullSigma_ = true;
secondaryQuadrature(m);
return;
}
scalar momentError = normalizedMomentError(sigma_, m, mStar);
if
(
momentError < momentsTol_
)
{
// Found a value of sigma that preserves all the moments
secondaryQuadrature(mStar);
return;
}
else
{
// Root not found. Minimize target function in [0, sigma_]
sigma_ = minimizeTargetFunction(0, sigma_, m, mStar);
if (mag(sigma_) < sigmaTol_)
{
m.invert();
sigma_ = 0.0;
nullSigma_ = true;
secondaryQuadrature(m);
return;
}
targetFunction(sigma_, m, mStar);
secondaryQuadrature(mStar);
return;
}
}
else
{
if (fNew*fMid < 0 && sigma_ < sigmaMid)
{
sigmaLow = sigma_;
fLow = fNew;
sigmaHigh = sigmaMid;
fHigh = fMid;
}
else if (fNew*fMid < 0 && sigma_ > sigmaMid)
{
sigmaLow = sigmaMid;
fLow = fMid;
sigmaHigh = sigma_;
fHigh = fNew;
}
else if (fNew*fLow < 0)
{
sigmaHigh = sigma_;
fHigh = fNew;
}
else if (fNew*fHigh < 0)
{
sigmaLow = sigma_;
fLow = fNew;
}
}
}
FatalErrorIn
(
"Foam::extendedMomentInversion::invert\n"
"(\n"
" const univariateMomentSet& moments\n"
")"
) << "Number of iterations exceeded."
<< abort(FatalError);
}
}
// * * * * * * * * * * * * * * * Member Functions * * * * * * * * * * * * * //
void Foam::extendedMomentInversion::invert(const univariateMomentSet& moments)
{
univariateMomentSet m(moments);
reset();
// Terminate execution if negative number density is encountered
if (m[0] < 0.0)
{
FatalErrorInFunction
<< "The zero-order moment is negative."
<< abort(FatalError);
}
// Exclude cases where the zero-order moment is very small to avoid
// problems in the inversion due to round-off error
if (m[0] < SMALL)
{
sigma_ = 0.0;
nullSigma_ = true;
return;
}
label nRealizableMoments = m.nRealizableMoments();
// If the moment set is on the boundary of the moment space, the
// distribution will be reconstructed by a summation of Dirac delta,
// and no attempt to use the extended quadrature method of moments is made.
if (m.isOnMomentSpaceBoundary())
{
sigma_ = 0.0;
nullSigma_ = true;
m.invert();
secondaryQuadrature(m);
return;
}
if (nRealizableMoments % 2 == 0)
{
// If the number of realizable moments is even, we apply the standard
// QMOM directly to maximize the number of preserved moments.
m.invert();
sigma_ = 0.0;
nullSigma_ = true;
secondaryQuadrature(m);
}
else
{
// Do not attempt the EQMOM reconstruction if mean or variance of the
// moment set are small to avoid numerical problems. These problems are
// particularly acute in the calculation of the recurrence relationship
// of the Jacobi orthogonal polynomials used for the beta kernel density
// function.
if (m[1]/m[0] < minMean_ || (m[2]/m[0] - sqr(m[1]/m[0])) < minVariance_)
{
sigma_ = 0.0;
nullSigma_ = true;
m.invert();
secondaryQuadrature(m);
return;
}
// Resizing the moment set to avoid copying again
m.resize(nRealizableMoments);
// Local set of starred moments
univariateMomentSet mStar(nRealizableMoments, 0, m.support());
// Compute target function for sigma = 0
scalar sigmaLow = 0.0;
scalar fLow = targetFunction(sigmaLow, m, mStar);
sigma_ = sigmaLow;
// Check if sigma = 0 is root
if (mag(fLow) <= targetFunctionTol_)
{
m.invert();
sigma_ = 0.0;
nullSigma_ = true;
secondaryQuadrature(m);
return;
}
// Compute target function for sigma = sigmaMax
scalar sigMax = sigmaMax(m);
scalar sigmaHigh = sigMax;
scalar fHigh = targetFunction(sigmaHigh, m, mStar);
// This should not happen with the new algorithm
if (fLow*fHigh > 0)
{
// Root not found. Minimize target function in [0, sigma_]
sigma_ = minimizeTargetFunction(0, sigmaHigh, m, mStar);
// If sigma_ is small, use QMOM
if (mag(sigma_) < sigmaMin_)
{
m.invert();
sigma_ = 0.0;
nullSigma_ = true;
secondaryQuadrature(m);
return;
}
targetFunction(sigma_, m, mStar);
secondaryQuadrature(mStar);
return;
}
// Apply Ridder's algorithm to find sigma
for (label iter = 0; iter < maxSigmaIter_; iter++)
{
scalar fMid, sigmaMid;
sigmaMid = (sigmaLow + sigmaHigh)/2.0;
fMid = targetFunction(sigmaMid, m, mStar);
scalar s = sqrt(sqr(fMid) - fLow*fHigh);
if (s == 0.0)
{
FatalErrorInFunction
<< "Singular value encountered searching for root.\n"
<< "Moment set = " << m << endl
<< "sigma = " << sigma_ << endl
<< "fLow = " << fLow << endl
<< "fMid = " << fMid << endl
<< "fHigh = " << fHigh
<< abort(FatalError);
}
sigma_ = sigmaMid + (sigmaMid - sigmaLow)*sign(fLow - fHigh)*fMid/s;
momentsToMomentsStar(sigma_, m, mStar);
scalar fNew = targetFunction(sigma_, m, mStar);
scalar dSigma = (sigmaHigh - sigmaLow)/2.0;
// Check for convergence
if (mag(fNew) <= targetFunctionTol_ || mag(dSigma) <= sigmaTol_)
{
// Root finding converged
// If sigma_ is small, use QMOM
if (mag(sigma_) < sigmaMin_)
{
m.invert();
sigma_ = 0.0;
nullSigma_ = true;
secondaryQuadrature(m);
return;
}
scalar momentError = normalizedMomentError(sigma_, m, mStar);
if
(
momentError < momentsTol_
)
{
// Found a value of sigma that preserves all the moments
secondaryQuadrature(mStar);
return;
}
else
{
// Root not found. Minimize target function in [0, sigma_]
sigma_ = minimizeTargetFunction(0, sigma_, m, mStar);
// If sigma_ is small, use QMOM
if (mag(sigma_) < sigmaMin_)
{
m.invert();
sigma_ = 0.0;
nullSigma_ = true;
secondaryQuadrature(m);
return;
}
targetFunction(sigma_, m, mStar);
secondaryQuadrature(mStar);
return;
}
}
else
{
// Root finding did not converge. Refine search.
if (fNew*fMid < 0 && sigma_ < sigmaMid)
{
sigmaLow = sigma_;
fLow = fNew;
sigmaHigh = sigmaMid;
fHigh = fMid;
}
else if (fNew*fMid < 0 && sigma_ > sigmaMid)
{
sigmaLow = sigmaMid;
fLow = fMid;
sigmaHigh = sigma_;
fHigh = fNew;
}
else if (fNew*fLow < 0)
{
sigmaHigh = sigma_;
fHigh = fNew;
}
else if (fNew*fHigh < 0)
{
sigmaLow = sigma_;
fLow = fNew;
}
}
}
FatalErrorInFunction
<< "Number of iterations exceeded.\n"
<< "Max allowed iterations = " << maxSigmaIter_
<< abort(FatalError);
}
}
