protected override double EvaluateAt(LinearPSI <Square> Psi, NodeSet Node, int cell)
{
Debug.Assert(Node.NoOfNodes == 1);
NodeSet nodeOnPsi = Psi.ProjectOnto(Node);
double value = lsData.GetLevSetValues(nodeOnPsi, cell, 1)[0, 0];
return(value);
}
/// <summary>
/// First order approximation of delta >= sup_x|psi(x) - psi(x_center)|
/// </summary>
/// <param name="Arg"></param>
/// <param name="psi"></param>
/// <param name="x_center"></param>
/// <param name="cell"></param>
/// <returns></returns>
protected override double EvaluateBounds(LinearSayeSpace <Cube> Arg, LinearPSI <Cube> psi, NodeSet x_center, int cell)
{
double[] arr = new double[RefElement.SpatialDimension];
Arg.Diameters.CopyTo(arr, 0);
psi.SetInactiveDimsToZero(arr);
MultidimensionalArray diameters = MultidimensionalArray.CreateWrapper(arr, 3);
NodeSet nodeOnPsi = psi.ProjectOnto(x_center);
MultidimensionalArray grad = ReferenceGradient(nodeOnPsi, cell);
grad.ApplyAll(x => Math.Abs(x));
double delta = grad.InnerProduct(diameters);
return(delta);
}
protected override bool HeightDirectionIsSuitable(SayeSquare arg, LinearPSI <Square> psi, NodeSet x_center,
int heightDirection, MultidimensionalArray gradient, int cell)
{
//Determine bounds
//-----------------------------------------------------------------------------------------------------------------
double[] arr = arg.Diameters;
psi.SetInactiveDimsToZero(arr);
MultidimensionalArray diameters = MultidimensionalArray.CreateWrapper(arr, new int[] { 2, 1 });
NodeSet nodeOnPsi = psi.ProjectOnto(x_center);
MultidimensionalArray hessian = lsData.GetLevelSetReferenceHessian(nodeOnPsi, cell, 1);
hessian = hessian.ExtractSubArrayShallow(new int[] { 0, 0, -1, -1 });
MultidimensionalArray jacobian = grid.InverseJacobian.GetValue_Cell(nodeOnPsi, cell, 1);
jacobian = jacobian.ExtractSubArrayShallow(new int[] { 0, 0, -1, -1 });
hessian = jacobian * hessian;
hessian.ApplyAll(x => Math.Abs(x));
//abs(Hessian) * diameters = delta
MultidimensionalArray delta = hessian * diameters;
delta = delta.ExtractSubArrayShallow(new int[] { -1, 0 });
//Check if suitable
//-----------------------------------------------------------------------------------------------------------------
//|gk| > δk
if (Math.Abs(gradient[heightDirection]) > delta[heightDirection])
{
bool suitable = true;
// Sum_j( g_j + delta_j)^2 / (g_k - delta_k)^2 < 20
double sum = 0;
for (int j = 0; j < delta.Length; ++j)
{
sum += Math.Pow(gradient[j] + delta[j], 2);
}
sum /= Math.Pow(gradient[heightDirection] - delta[heightDirection], 2);
suitable &= sum < 20;
return(suitable);
}
return(false);
}
/// <summary>
/// First order approximation of delta >= sup_x|psi(x) - psi(x_center)|
/// </summary>
/// <param name="Arg"></param>
/// <param name="psi"></param>
/// <param name="x_center"></param>
/// <param name="cell"></param>
/// <returns></returns>
protected override double EvaluateBounds(SayeSquare Arg, LinearPSI <Square> psi, NodeSet x_center, int cell)
{
double[] arr = new double[Arg.Dimension];
Arg.Diameters.CopyTo(arr, 0);
psi.SetInactiveDimsToZero(arr);
MultidimensionalArray diameters = MultidimensionalArray.CreateWrapper(arr, new int[] { 2 });
NodeSet nodeOnPsi = psi.ProjectOnto(x_center);
MultidimensionalArray grad = ScaledReferenceGradient(nodeOnPsi, cell);
grad.ApplyAll(x => Math.Abs(x));
double delta = grad.InnerProduct(diameters) * sqrt_2;
return(delta);
}
protected override int FindPromisingHeightDirection(LinearPSI <Cube> Psi, NodeSet Node, int Cell)
{
NodeSet nodeOnPsi = Psi.ProjectOnto(Node);
MultidimensionalArray gradient = ReferenceGradient(nodeOnPsi, Cell);
int heightDirection = 0;
double max = double.MinValue;
for (int i = 0; i < RefElement.SpatialDimension; ++i)
{
if ((Math.Abs(gradient[i]) > max) && (!Psi.DirectionIsFixed(i)))
{
max = Math.Abs(gradient[i]);
heightDirection = i;
}
}
return(heightDirection);
}
protected override double[] FindRoots(LinearPSI <Cube> psi, MultidimensionalArray X, int heightDirection, double[] bounds, int cell)
{
MultidimensionalArray XonPsi = psi.ProjectOnto(X);
XonPsi = XonPsi.ExtractSubArrayShallow(0, -1);
double[] start = XonPsi.To1DArray();
double[] end = XonPsi.To1DArray();
start[heightDirection] = -1;
end[heightDirection] = 1;
HMF.LineSegment line = new HMF.LineSegment(3, RefElement, start, end);
LevelSet levelSet = lsData.LevelSet as LevelSet;
line.ProjectBasisPolynomials(levelSet.Basis);
double[] roots = rootFinder.GetRoots(line, levelSet, cell, this.iKref);
return(roots);
}
protected override bool HeightDirectionIsSuitable(
LinearSayeSpace <Cube> arg,
LinearPSI <Cube> psi, NodeSet
x_center,
int heightDirection,
MultidimensionalArray agradient, int cell)
{
//throw new NotImplementedException();
//Determine bounds
//-----------------------------------------------------------------------------------------------------------------
NodeSet nodeOnPsi = psi.ProjectOnto(x_center);
MultidimensionalArray jacobian = grid.Jacobian.GetValue_Cell(nodeOnPsi, cell, 1);
jacobian = jacobian.ExtractSubArrayShallow(new int[] { 0, 0, -1, -1 });
LevelSet levelSet = lsData.LevelSet as LevelSet;
MultidimensionalArray hessian = MultidimensionalArray.Create(1, 1, 3, 3);
levelSet.EvaluateHessian(cell, 1, nodeOnPsi, hessian);
hessian = hessian.ExtractSubArrayShallow(new int[] { 0, 0, -1, -1 }).CloneAs();
//hessian = jacobian * hessian;
hessian.ApplyAll(x => Math.Abs(x));
MultidimensionalArray gradient = lsData.GetLevelSetGradients(nodeOnPsi, cell, 1);
gradient = gradient.ExtractSubArrayShallow(0, 0, -1).CloneAs();
//abs(Hessian) * 0,5 * diameters.^2 = delta ,( square each entry of diameters) ,
//this bounds the second error term from taylor series
//+ + + +
double[] arr = arg.Diameters.CloneAs();
psi.SetInactiveDimsToZero(arr);
MultidimensionalArray diameters = MultidimensionalArray.CreateWrapper(arr, 3, 1);
diameters = jacobian * diameters;
diameters.ApplyAll(x => 0.5 * x * x);
MultidimensionalArray delta = hessian * diameters;
delta = delta.ExtractSubArrayShallow(-1, 0);
//Check if suitable
//-----------------------------------------------------------------------------------------------------------------
//|gk| > δk
//Gradient should be able to turn arround
psi.SetInactiveDimsToZero(gradient.Storage);
if (Math.Abs(gradient[heightDirection]) > delta[heightDirection])
{
bool suitable = true;
// ||Grad + maxChange|| should be smaller than 20 *
// Sum_j( g_j + delta_j)^2 / (g_k - delta_k)^2 < 20
double sum = 0;
for (int j = 0; j < delta.Length; ++j)
{
sum += Math.Pow(Math.Abs(gradient[j]) + delta[j], 2);
}
sum /= Math.Pow(Math.Abs(gradient[heightDirection]) - delta[heightDirection], 2);
suitable &= sum < 20;
return(suitable);
}
return(false);
}
