/*\ For each element, each entry to be accumulated in the Hessian for
this objective function (\f$ \sum_{e \in E} Q(e)^p \f$ where \f$ E \f$
is the set of all elements in the patch) has the form:
\f$ pQ(e)^{p-1} \nabla^2 Q(e) + p(p-1)Q(e)^{p-2} \nabla Q(e) [\nabla Q(e)]^T \f$.
For \f$ p=2 \f$, this simplifies to
\f$ 2Q(e) \nabla^2 Q(e) + 2 \nabla Q(e) [\nabla Q(e)]^T \f$.
For \f$ p=1 \f$, this simplifies to \f$ \nabla^2 Q(e) \f$.
The \f$ p=1 \f$ simplified version is implemented directly
to speed up computation.
This function does not support vertex-based metrics.
\param pd The PatchData object for which the objective function
hessian is computed.
\param hessian this object must have been previously initialized.
*/
bool LPtoPTemplate::evaluate_with_Hessian( EvalType type,
PatchData& pd,
double& OF_val,
std::vector<Vector3D>& grad,
MsqHessian& hessian,
MsqError& err )
{
QualityMetric* qm = get_quality_metric();
qm->get_evaluations( pd, qmHandles, OF_FREE_EVALS_ONLY, err ); MSQ_ERRFALSE(err);
double negate_flag = qm->get_negate_flag();
// zero gradient and hessian
grad.clear();
grad.resize( pd.num_free_vertices(), 0.0 );
hessian.zero_out();
double QM_val, QM_pow = 1.0;
double fac1, fac2;
Matrix3D elem_outer_product;
bool qm_bool;
size_t i, j, n;
short p;
// Loops over all elements in the patch.
OF_val = 0.0;
std::vector<size_t>::const_iterator k;
for (k = qmHandles.begin(); k != qmHandles.end(); ++k)
{
// Computes \nabla^2 Q(e). Only the free vertices will have non-zero entries.
qm_bool = qm->evaluate_with_Hessian( pd, *k, QM_val, mIndices, mGradient, mHessian, err );
if (MSQ_CHKERR(err) || !qm_bool)
return false;
QM_val = fabs(QM_val);
// **** Computes Hessian ****
const size_t nve = mIndices.size();
if (pVal == 1) {
QM_pow = 1.0;
n=0;
for (i=0; i<nve; ++i) {
for (j=i; j<nve; ++j) {
//negate if necessary
mHessian[n] *= negate_flag;
hessian.add( mIndices[i], mIndices[j], mHessian[n], err ); MSQ_ERRFALSE(err);
++n;
}
}
fac1 = 1;
}
else if (pVal >= 2) {
// Computes the coefficients:
QM_pow = 1.0;
for (p=0; p<pVal-2; ++p)
QM_pow *= QM_val;
// 1 - computes p(p-1)Q(e)^{p-2}
fac2 = pVal* (pVal-1) * QM_pow;
// 2 - computes pQ(e)^{p-1}
QM_pow *= QM_val;
fac1 = pVal * QM_pow;
//fac1 *= qm->get_negate_flag();
//fac2 *= qm->get_negate_flag();
n=0;
for (i=0; i<nve; ++i) {
for (j=i; j<nve; ++j) {
elem_outer_product.outer_product(mGradient[i], mGradient[j]);
elem_outer_product *= fac2;
mHessian[n] *= fac1;
mHessian[n] += elem_outer_product;
mHessian[n] *= negate_flag;
hessian.add( mIndices[i], mIndices[j], mHessian[n], err ); MSQ_ERRFALSE(err);
++n;
}
}
} else {
MSQ_SETERR(err)(" invalid P value.", MsqError::INVALID_STATE);
return false;
}
// **** Computes Gradient ****
// For each vertex in the element ...
for (i=0; i<nve; ++i) {
// ... computes p*q^{p-1}*grad(q) ...
mGradient[i] *= fac1*negate_flag;
// ... and accumulates it in the objective function gradient.
//also scale the gradient by the scaling factor
assert (mIndices[i] < pd.num_free_vertices());
grad[mIndices[i]] += mGradient[i];
}
// **** computes Objective Function value \sum_{i=1}^{N_e} |q_i|^P ****
OF_val += QM_pow * QM_val;
}
size_t global_count;
OF_val = negate_flag
* get_value( OF_val, qmHandles.size(), type, global_count, err );
// if (!global_count)
// return false; // invalid mesh
if (dividingByN && global_count) {
const double inv_n = 1.0 / global_count;
std::vector<Vector3D>::iterator g;
for (g = grad.begin(); g != grad.end(); ++g)
*g *= inv_n;
hessian.scale( inv_n );
}
return true;
}
bool PMeanPTemplate::evaluate_with_Hessian( EvalType type,
PatchData& pd,
double& value_out,
std::vector<Vector3D>& grad_out,
MsqHessian& Hessian_out,
MsqError& err )
{
QualityMetric* qm = get_quality_metric();
qm->get_evaluations( pd, qmHandles, OF_FREE_EVALS_ONLY, err ); MSQ_ERRFALSE(err);
// zero gradient and hessian
grad_out.clear();
grad_out.resize( pd.num_free_vertices(), 0.0 );
Hessian_out.zero_out();
// calculate OF value and gradient for just the patch
std::vector<size_t>::const_iterator i;
size_t j, k, n;
double value, working_sum = 0.0;
const double f1 = qm->get_negate_flag() * mPower.value();
const double f2 = f1 * (mPower.value() - 1);
Matrix3D m;
for (i = qmHandles.begin(); i != qmHandles.end(); ++i)
{
bool result = qm->evaluate_with_Hessian( pd, *i, value, mIndices, mGradient, mHessian, err );
if (MSQ_CHKERR(err) || !result)
return false;
if (fabs(value) < DBL_EPSILON)
continue;
const size_t nfree = mIndices.size();
n = 0;
if (mPower.value() == 1.0) {
working_sum += mPower.raise( value );
for (j = 0; j < nfree; ++j) {
mGradient[j] *= f1;
grad_out[mIndices[j]] += mGradient[j];
for (k = j; k < nfree; ++k) {
mHessian[n] *= f1;
Hessian_out.add( mIndices[j], mIndices[k], mHessian[n], err ); MSQ_ERRFALSE(err);
++n;
}
}
}
else {
const double r2 = mPowerMinus2.raise( value );
const double r1 = r2 * value;
working_sum += r1 * value;
const double hf = f2 * r2;
const double gf = f1 * r1;
for (j = 0; j < nfree; ++j) {
for (k = j; k < nfree; ++k) {
m.outer_product( mGradient[j], mGradient[k] );
m *= hf;
mHessian[n] *= gf;
m += mHessian[n];
Hessian_out.add( mIndices[j], mIndices[k], m, err ); MSQ_ERRFALSE(err);
++n;
}
}
for (j = 0; j < nfree; ++j) {
mGradient[j] *= gf;
grad_out[mIndices[j]] += mGradient[j];
}
}
}
// get overall OF value, update member data, etc.
size_t global_count;
value_out = qm->get_negate_flag()
* get_value( working_sum, qmHandles.size(), type, global_count );
const double inv_n = 1.0 / global_count;
std::vector<Vector3D>::iterator g;
for (g = grad_out.begin(); g != grad_out.end(); ++g)
*g *= inv_n;
Hessian_out.scale( inv_n );
return true;
}
/*! \fn LPtoPTemplate::compute_analytical_hessian(PatchData &pd, MsqHessian &hessian, MsqError &err)
For each element, each entry to be accumulated in the Hessian for
this objective function (\f$ \sum_{e \in E} Q(e)^p \f$ where \f$ E \f$
is the set of all elements in the patch) has the form:
\f$ pQ(e)^{p-1} \nabla^2 Q(e) + p(p-1)Q(e)^{p-2} \nabla Q(e) [\nabla Q(e)]^T \f$.
For \f$ p=2 \f$, this simplifies to
\f$ 2Q(e) \nabla^2 Q(e) + 2 \nabla Q(e) [\nabla Q(e)]^T \f$.
For \f$ p=1 \f$, this simplifies to \f$ \nabla^2 Q(e) \f$.
The \f$ p=1 \f$ simplified version is implemented directly
to speed up computation.
This function does not support vertex-based metrics.
\param pd The PatchData object for which the objective function
hessian is computed.
\param hessian: this object must have been previously initialized.
*/
bool LPtoPTemplate::compute_analytical_hessian(PatchData &pd,
MsqHessian &hessian,
Vector3D *const &grad,
double &OF_val,
MsqError &err)
{
double scaling_value=1.0;
MSQ_FUNCTION_TIMER( "LPtoPTemplate::compute_analytical_hessian" );
MsqMeshEntity* elements = pd.get_element_array(err); MSQ_ERRZERO(err);
MsqVertex* vertices = pd.get_vertex_array(err); MSQ_ERRZERO(err);
size_t num_elems = pd.num_elements();
//if scaling divide by the number of elements.
if(dividingByN){
if(num_elems<=0) {
MSQ_SETERR(err)("LPtoP is attempting to divide by zero in analytical Hessian.",
MsqError::INVALID_MESH);
return false;
}
scaling_value/=num_elems;
}
size_t num_vertices = pd.num_vertices();
Matrix3D elem_hessian[MSQ_MAX_NUM_VERT_PER_ENT*(MSQ_MAX_NUM_VERT_PER_ENT+1)/2];
Matrix3D elem_outer_product;
Vector3D grad_vec[MSQ_MAX_NUM_VERT_PER_ENT];
double QM_val;
double fac1, fac2;
Matrix3D grad_outprod;
bool qm_bool;
QualityMetric* currentQM = get_quality_metric();
MsqVertex* ele_free_vtces[MSQ_MAX_NUM_VERT_PER_ENT];
short i;
for (i=0; i<MSQ_MAX_NUM_VERT_PER_ENT; ++i) ele_free_vtces[i]=NULL;
const size_t* vtx_indices;
size_t e, v;
size_t nfve; // number of free vertices in element
short j,n;
hessian.zero_out();
for (v=0; v<num_vertices; ++v) grad[v] = 0.;
OF_val = 0.;
// Loops over all elements in the patch.
for (e=0; e<num_elems; ++e) {
short nve = elements[e].vertex_count();
// Gets a list of free vertices in the element.
vtx_indices = elements[e].get_vertex_index_array();
nfve=0;
for (i=0; i<nve; ++i) {
if ( vertices[vtx_indices[i]].is_free_vertex() ) {
ele_free_vtces[nfve] = vertices + vtx_indices[i];
++nfve;
}
}
// Computes \nabla^2 Q(e). Only the free vertices will have non-zero entries.
qm_bool = currentQM->compute_element_hessian(pd,
elements+e, ele_free_vtces,
grad_vec, elem_hessian,
nfve, QM_val, err);
if (MSQ_CHKERR(err) || !qm_bool) return false;
// **** Computes Hessian ****
double QM_pow=1.;
if (pVal == 1) {
n=0;
for (i=0; i<nve; ++i) {
for (j=i; j<nve; ++j) {
//negate if necessary
elem_hessian[n] *= (scaling_value * get_negate_flag());
++n;
}
}
hessian.accumulate_entries(pd, e, elem_hessian, err);
fac1 = 1;
}
else if (pVal >= 2) {
// Computes the coefficients:
QM_val = fabs(QM_val);
QM_pow = 1;
for (i=0; i<pVal-2; ++i)
QM_pow *= QM_val;
// 1 - computes p(p-1)Q(e)^{p-2}
fac2 = pVal* (pVal-1) * QM_pow;
// 2 - computes pQ(e)^{p-1}
QM_pow *= QM_val;
fac1 = pVal * QM_pow;
//fac1 *= get_negate_flag();
//fac2 *= get_negate_flag();
n=0;
for (i=0; i<nve; ++i) {
for (j=i; j<nve; ++j) {
if ( vertices[vtx_indices[i]].is_free_vertex() &&
vertices[vtx_indices[j]].is_free_vertex() ) {
// Computes \nabla Q(e) [\nabla Q(e)]^T
elem_outer_product.outer_product(grad_vec[i], grad_vec[j]);
elem_outer_product *= fac2;
elem_hessian[n] *= fac1;
elem_hessian[n] += elem_outer_product;
} else {
// elem_outer_product is nul
elem_hessian[n] *= fac1;
}
//scale the hessian by the scaling factor
elem_hessian[n] *= (scaling_value * get_negate_flag());
++n;
}
}
hessian.accumulate_entries(pd, e, elem_hessian, err); MSQ_ERRZERO(err);
} else {
MSQ_SETERR(err)(" invalid P value.", MsqError::INVALID_STATE);
return false;
}
// **** Computes Gradient ****
// For each vertex in the element ...
for (i=0; i<nve; ++i) {
if ( vertices[vtx_indices[i]].is_free_vertex() ) {
// ... computes p*q^{p-1}*grad(q) ...
grad_vec[i] *= fac1*get_negate_flag();
// ... and accumulates it in the objective function gradient.
//also scale the gradient by the scaling factor
grad[vtx_indices[i]] += (scaling_value * grad_vec[i]);
}
}
// **** computes Objective Function value \sum_{i=1}^{N_e} |q_i|^P ****
//and scale by 1/num if necessary
OF_val += (scaling_value * QM_pow * QM_val);
}
OF_val *= get_negate_flag();
return true;
}
