static double pmean_of_triangle_corner_hessians( double inner_power,
double outer_power,
const double* v,
const Vector3D* cg,
const Matrix3D* ch,
Vector3D* tg,
Matrix3D* th,
bool scale )
{
Matrix3D op;
double gf[3], hf[3];
double nm, m = 0;
double den = scale ? 3.0: 1.0;
for (unsigned i = 0; i < 3; ++i) {
nm = pow(v[i], inner_power);
m += nm;
gf[i] = inner_power*nm / v[i] / den;
hf[i] = (inner_power-1)*gf[i] / v[i];
}
nm = m / den;
tg[0] = gf[0] * cg[0] + gf[1] * cg[5] + gf[2] * cg[7];
tg[1] = gf[0] * cg[1] + gf[1] * cg[3] + gf[2] * cg[8];
tg[2] = gf[0] * cg[2] + gf[1] * cg[4] + gf[2] * cg[6];
th[0] = hf[0]*op.outer_product( cg[0], cg[0] ) + gf[0]*ch[ 0]
+ hf[1]*op.outer_product( cg[5], cg[5] ) + gf[1]*ch[11]
+ hf[2]*op.outer_product( cg[7], cg[7] ) + gf[2]*ch[15];
th[3] = hf[0]*op.outer_product( cg[1], cg[1] ) + gf[0]*ch[ 3]
+ hf[1]*op.outer_product( cg[3], cg[3] ) + gf[1]*ch[ 6]
+ hf[2]*op.outer_product( cg[8], cg[8] ) + gf[2]*ch[17];
th[5] = hf[0]*op.outer_product( cg[2], cg[2] ) + gf[0]*ch[ 5]
+ hf[1]*op.outer_product( cg[4], cg[4] ) + gf[1]*ch[ 9]
+ hf[2]*op.outer_product( cg[6], cg[6] ) + gf[2]*ch[12];
th[1] = hf[0]*op.outer_product( cg[0], cg[1] ) + gf[0]*ch[ 1]
+ hf[1]*op.outer_product( cg[5], cg[3] ) + gf[1]*transpose(ch[ 8])
+ hf[2]*op.outer_product( cg[7], cg[8] ) + gf[2]*ch[16];
th[2] = hf[0]*op.outer_product( cg[0], cg[2] ) + gf[0]*ch[ 2]
+ hf[1]*op.outer_product( cg[5], cg[4] ) + gf[1]*transpose(ch[10])
+ hf[2]*op.outer_product( cg[7], cg[6] ) + gf[2]*transpose(ch[13]);
th[4] = hf[0]*op.outer_product( cg[1], cg[2] ) + gf[0]*ch[ 4]
+ hf[1]*op.outer_product( cg[3], cg[4] ) + gf[1]*ch[ 7]
+ hf[2]*op.outer_product( cg[8], cg[6] ) + gf[2]*transpose(ch[14]);
m = pow(nm, outer_power);
double g = m * outer_power / nm;
double h = (outer_power - 1.0) * g / nm;
for (unsigned r = 0; r < 3; ++r) {
for (unsigned c = r; c < 3; ++c) {
*th = g * *th + h * op.outer_product( tg[r], tg[c] );
++th;
}
tg[r] *= g;
}
return m;
}
bool PowerQualityMetric::evaluate_with_Hessian( PatchData& pd,
size_t handle,
double& value,
std::vector<size_t>& indices,
std::vector<Vector3D>& gradient,
std::vector<Matrix3D>& Hessian,
MsqError& err )
{
indices.clear();
bool rval = mMetric.evaluate_with_Hessian( pd, handle, value, indices, gradient, Hessian, err );
const double v = mPower.raise(value);
const double g = fabs(value) > DBL_EPSILON ? mPower.value() * v / value : 0.0;
const double h = fabs(value) > DBL_EPSILON ? g * (mPower.value() - 1) / value : 0.0;
value = v;
Matrix3D op;
unsigned idx = 0;
unsigned n = indices.size();
for (unsigned r = 0; r < n; ++r) {
for (unsigned c = r; c < n; ++c) {
Hessian[idx] *= g;
op.outer_product( gradient[r], gradient[c] );
op *= h;
Hessian[idx] += op;
++idx;
}
gradient[r] *= g;
}
return !MSQ_CHKERR(err) && rval;
}
void test_outer_product()
{
Matrix3D mat;
Vector3D vec1(2, 7, 3);
Vector3D vec2(5, 8, 9);
mat.outer_product(vec1, vec2);
Matrix3D correct(" 10 16 18 "
" 35 56 63 "
" 15 24 27 ");
CPPUNIT_ASSERT( mat == correct );
}
void CompositeOFTest::test_multiply_hess_diagonal()
{
CompositeOFMultiply OF( &LP1, &LP2 );
std::vector<SymMatrix3D> hess1, hess2, hess;
MsqPrintError err(cout);
PatchData pd;
create_twelve_hex_patch( pd, err );
ASSERT_NO_ERROR( err );
std::vector<Vector3D> grad1, grad2, grad;
bool rval;
double value1, value2, value;
rval = LP1.evaluate_with_Hessian_diagonal( ObjectiveFunction::CALCULATE, pd, value1, grad1, hess1, err );
ASSERT_NO_ERROR(err);
CPPUNIT_ASSERT(rval);
rval = LP2.evaluate_with_Hessian_diagonal( ObjectiveFunction::CALCULATE, pd, value2, grad2, hess2, err );
ASSERT_NO_ERROR(err);
CPPUNIT_ASSERT(rval);
rval = OF .evaluate_with_Hessian_diagonal( ObjectiveFunction::CALCULATE, pd, value, grad, hess , err );
ASSERT_NO_ERROR(err);
CPPUNIT_ASSERT(rval);
CPPUNIT_ASSERT_EQUAL( pd.num_free_vertices(), grad1.size() );
CPPUNIT_ASSERT_EQUAL( pd.num_free_vertices(), grad2.size() );
CPPUNIT_ASSERT_EQUAL( pd.num_free_vertices(), grad .size() );
CPPUNIT_ASSERT_EQUAL( pd.num_free_vertices(), hess1.size() );
CPPUNIT_ASSERT_EQUAL( pd.num_free_vertices(), hess2.size() );
CPPUNIT_ASSERT_EQUAL( pd.num_free_vertices(), hess .size() );
CPPUNIT_ASSERT_DOUBLES_EQUAL( value1 * value2, value, 1e-6 );
for (size_t i = 0; i < pd.num_free_vertices(); ++i) {
const Vector3D expected_grad = value2 * grad1[i] + value1 * grad2[i];
CPPUNIT_ASSERT_VECTORS_EQUAL( expected_grad, grad[i], 1e-6 );
Matrix3D o;
o.outer_product( grad1[i], grad2[i] );
Matrix3D expect = o + transpose(o);
expect += value2 * hess1[i];
expect += value1 * hess2[i];
CPPUNIT_ASSERT_MATRICES_EQUAL( expect, Matrix3D(hess[i]), 1e-6 );
}
}
void PMeanPMetricTest::test_hessian()
{
MsqError err;
FauxMetric m;
ElementPMeanP m1( 1.0, &m );
ElementPMeanP m2( 2.0, &m );
// get vertices for later
std::vector<size_t> vertices;
pd.element_by_index(0).get_vertex_indices( vertices );
// evaluate gradient
double v1, v2, v3, v4;
std::vector<size_t> indices1, indices2, indices3, indices4, tmpi;
std::vector<Vector3D> grad1, grad2, grad3, grad4;
std::vector<Matrix3D> hess3, hess4;
m1.evaluate_with_gradient( pd, 0, v1, indices1, grad1, err );
CPPUNIT_ASSERT(!err);
m2.evaluate_with_gradient( pd, 0, v2, indices2, grad2, err );
CPPUNIT_ASSERT(!err);
// evaluate with Hessian
m1.evaluate_with_Hessian( pd, 0, v3, indices3, grad3, hess3, err );
CPPUNIT_ASSERT(!err);
m2.evaluate_with_Hessian( pd, 0, v4, indices4, grad4, hess4, err );
CPPUNIT_ASSERT(!err);
// compare value and indices to eval w/out gradient
CPPUNIT_ASSERT_DOUBLES_EQUAL( v1, v3, 1e-6 );
CPPUNIT_ASSERT_DOUBLES_EQUAL( v2, v4, 1e-6 );
// It isn't a requirement that the index order remain the same
// for both eval_with_grad and eval_with_Hess, but assuming it
// does simplifies a lot of stuff in this test. Check that the
// assumption remains valid.
CPPUNIT_ASSERT( indices3 == indices1 );
CPPUNIT_ASSERT( indices4 == indices2 );
// It isn't a requirement that the index order remain the same
// for any value of P, but assuming it does simplifies a lot
// of stuff in this test, so check that the assumption is valid.
CPPUNIT_ASSERT( indices1 == indices2 );
// check that gradient values match
for (size_t i = 0; i < indices1.size(); ++i) {
CPPUNIT_ASSERT_VECTORS_EQUAL( grad1[i], grad3[i], 1e-5 );
CPPUNIT_ASSERT_VECTORS_EQUAL( grad2[i], grad4[i], 1e-5 );
}
// setup evaluation of underying metric
std::vector<size_t> handles;
m.get_element_evaluations( pd, 0, handles, err );
CPPUNIT_ASSERT(!err);
// calculate expected Hessians
std::vector<Vector3D> g;
std::vector<Matrix3D> expected1, expected2, h;
std::vector<Matrix3D>::iterator h_iter;
const unsigned N = vertices.size();
expected1.resize( N*(N+1)/2, Matrix3D(0,0,0,0,0,0,0,0,0) );
expected2 = expected1;
Matrix3D outer;
for (unsigned i = 0; i < handles.size(); ++i) {
double v;
m.evaluate_with_Hessian( pd, handles[i], v, tmpi, g, h, err );
CPPUNIT_ASSERT(!err);
h_iter = h.begin();
for (unsigned r = 0; r < tmpi.size(); ++r) {
unsigned R = index_of( vertices, tmpi[r] );
CPPUNIT_ASSERT( R < N );
for (unsigned c = r; c < tmpi.size(); ++c ,++h_iter) {
unsigned C = index_of( vertices, tmpi[c] );
CPPUNIT_ASSERT( C < N );
if (R <= C) {
unsigned idx = N*R - R*(R+1)/2 + C;
expected1[idx] += 1.0 / handles.size() * *h_iter;
expected2[idx] += 2.0 * v / handles.size() * *h_iter;
outer.outer_product( g[r], g[c] );
expected2[idx] += 2.0 / handles.size() * outer;
}
else {
unsigned idx = N*C - C*(C+1)/2 + R;
expected1[idx] += 1.0 / handles.size() * transpose(*h_iter);
expected2[idx] += 2.0 * v / handles.size() * transpose(*h_iter);
outer.outer_product( g[c], g[r] );
expected2[idx] += 2.0 / handles.size() * outer;
}
}
}
}
// compare Hessians
unsigned H_idx = 0;
for (unsigned R = 0; R < vertices.size(); ++R) {
if (vertices[R] >= pd.num_free_vertices())
continue;
unsigned r = index_of( indices3, vertices[R] );
CPPUNIT_ASSERT(r < indices3.size() );
for (unsigned C = R; C < vertices.size(); ++C, ++H_idx) {
if (vertices[C] >= pd.num_free_vertices())
continue;
unsigned c = index_of( indices3, vertices[C] );
CPPUNIT_ASSERT( c < indices3.size() );
if (r <= c) {
unsigned idx = indices3.size()*r - r*(r+1)/2 + c;
CPPUNIT_ASSERT_MATRICES_EQUAL( expected1[H_idx], hess3[idx], 1e-5 );
CPPUNIT_ASSERT_MATRICES_EQUAL( expected2[H_idx], hess4[idx], 1e-5 );
}
else {
unsigned idx = indices3.size()*c - c*(c+1)/2 + r;
CPPUNIT_ASSERT_MATRICES_EQUAL( transpose(expected1[H_idx]), hess3[idx], 1e-5 );
CPPUNIT_ASSERT_MATRICES_EQUAL( transpose(expected2[H_idx]), hess4[idx], 1e-5 );
}
}
}
}
bool MultiplyQualityMetric::evaluate_with_Hessian( PatchData& pd,
size_t handle,
double& value,
std::vector<size_t>& indices,
std::vector<Vector3D>& gradient,
std::vector<Matrix3D>& Hessian,
MsqError& err )
{
std::vector<size_t>::iterator i;
size_t j, r, c, n, h;
double val1, val2;
bool rval1, rval2;
rval1 = metric1.evaluate_with_Hessian( pd, handle, val1, indices1, grad1, Hess1, err ); MSQ_ERRZERO(err);
rval2 = metric2.evaluate_with_Hessian( pd, handle, val2, indices2, grad2, Hess2, err ); MSQ_ERRZERO(err);
// merge index lists
indices.resize( indices1.size() + indices2.size() );
i = std::copy( indices1.begin(), indices1.end(), indices.begin() );
std::copy( indices2.begin(), indices2.end(), i );
std::sort( indices.begin(), indices.end() );
indices.erase( std::unique( indices.begin(), indices.end() ), indices.end() );
// calculate grads and convert index lists to indices into output list
gradient.clear();
gradient.resize( indices.size(), Vector3D(0.0) );
for (j = 0; j < indices1.size(); ++j)
{
i = std::lower_bound( indices.begin(), indices.end(), indices1[j] );
indices1[j] = i - indices.begin();
gradient[indices1[j]] += val2 * grad1[j];
}
for (j = 0; j < indices2.size(); ++j)
{
i = std::lower_bound( indices.begin(), indices.end(), indices2[j] );
indices2[j] = i - indices.begin();
gradient[indices2[j]] += val1 * grad2[j];
}
// allocate space for hessians, and zero it
const size_t N = indices.size();
Hessian.clear();
Hessian.resize( N * (N+1) / 2, Matrix3D(0.0) );
// add hessian terms from first metric
n = indices1.size();
h = 0;
for (r = 0; r < n; ++r)
{
const size_t nr = indices1[r];
for (c = r; c < n; ++c)
{
const size_t nc = indices1[c];
Hess1[h] *= val2;
if (nr <= nc)
Hessian[N*nr - nr*(nr+1)/2 + nc] += Hess1[h];
else
Hessian[N*nc - nc*(nc+1)/2 + nr].plus_transpose_equal( Hess1[h] );
++h;
}
}
// add hessian terms from second metric
n = indices2.size();
h = 0;
for (r = 0; r < n; ++r)
{
const size_t nr = indices2[r];
for (c = r; c < n; ++c)
{
const size_t nc = indices2[c];
Hess2[h] *= val1;
if (nr <= nc)
Hessian[N*nr - nr*(nr+1)/2 + nc] += Hess2[h];
else
Hessian[N*nc - nc*(nc+1)/2 + nr].plus_transpose_equal( Hess2[h] );
++h;
}
}
// add gradient outer products
n = indices1.size();
size_t m = indices2.size();
Matrix3D outer;
for (r = 0; r < n; ++r)
{
const size_t nr = indices1[r];
for (c = 0; c < m; ++c)
{
const size_t nc = indices2[c];
outer.outer_product( grad1[r], grad2[c] );
if (nr == nc)
Hessian[N*nr - nr*(nr+1)/2 + nc] += outer.plus_transpose_equal(outer);
else if (nr < nc)
Hessian[N*nr - nr*(nr+1)/2 + nc] += outer;
else
Hessian[N*nc - nc*(nc+1)/2 + nr].plus_transpose_equal(outer);
}
}
value = val1 * val2;
return rval1 && rval2;
}
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;
}
