private static void pce_ode_hermite_test01()
//****************************************************************************80
//
// Purpose:
//
// PCE_ODE_HERMITE_TEST01 runs a test problem with PCE_ODE_HERMITE.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 17 March 2012
//
// Author:
//
// John Burkardt
//
{
int i;
const int np = 4;
const int nt = 200;
double[] t = new double[nt + 1];
double[] u = new double[(nt + 1) * (np + 1)];
double[] uex = new double[nt + 1];
Console.WriteLine("");
Console.WriteLine("PCE_ODE_HERMITE_TEST01:");
Console.WriteLine(" Call PCE_ODE_HERMITE to compute a polynomial chaos expansion");
Console.WriteLine(" for the ODE:");
Console.WriteLine("");
Console.WriteLine(" u' = - alpha * u,");
Console.WriteLine(" u(0) = 1.");
const double ti = 0.0;
const double tf = 2.0;
const double ui = 1.0;
const double alpha_mu = 0.0;
const double alpha_sigma = 1.0;
Console.WriteLine("");
Console.WriteLine(" Initial time TI = " + ti + "");
Console.WriteLine(" Final time TF = " + tf + "");
Console.WriteLine(" Number of time steps NT = " + nt + "");
Console.WriteLine(" Initial condition UI = " + ui + "");
Console.WriteLine(" Expansion degree NP = " + np + "");
Console.WriteLine(" E(ALPHA) ALPHA_MU = " + alpha_mu + "");
Console.WriteLine(" STD(ALPHA) ALPHA_SIGMA = " + alpha_sigma + "");
HermitePolyChaosExpansion.pce_ode_hermite(ti, tf, nt, ui, np, alpha_mu, alpha_sigma, ref t, ref u);
//
// Evaluate the exact expected value function.
//
for (i = 0; i <= nt; i++)
{
uex[i] = ui * Math.Exp(t[i] * t[i] / 2.0);
}
//
// Compare the first computed component against the exact expected value.
//
Console.WriteLine("");
Console.WriteLine(" i T(i) E(U(T(i))) U(T(i),0)");
Console.WriteLine("");
for (i = 0; i <= nt; i += 10)
{
Console.WriteLine(" " + i.ToString(CultureInfo.InvariantCulture).PadLeft(4)
+ " " + t[i].ToString(CultureInfo.InvariantCulture).PadLeft(6)
+ " " + uex[i].ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + u[i + 0 * (nt + 1)].ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + Math.Abs(uex[i] - u[i + 0 * (nt + 1)]).ToString(CultureInfo.InvariantCulture).PadLeft(14) + "");
}
}
private static void pce_ode_hermite_test02()
//****************************************************************************80
//
// Purpose:
//
// PCE_ODE_HERMITE_TEST02 looks at convergence behavior for a fixed time.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 18 March 2012
//
// Author:
//
// John Burkardt
//
{
double[] ep = new double[6];
int np;
const int nt = 2000;
double[] t = new double[nt + 1];
Console.WriteLine("");
Console.WriteLine("PCE_ODE_HERMITE_TEST02:");
Console.WriteLine(" Examine convergence behavior as the PCE degree increases:");
Console.WriteLine("");
Console.WriteLine(" u' = - alpha * u,");
Console.WriteLine(" u(0) = 1.");
const double ti = 0.0;
const double tf = 2.0;
const double ui = 1.0;
const double alpha_mu = 0.0;
const double alpha_sigma = 1.0;
Console.WriteLine("");
Console.WriteLine(" Initial time TI = " + ti + "");
Console.WriteLine(" Final time TF = " + tf + "");
Console.WriteLine(" Number of time steps NT = " + nt + "");
Console.WriteLine(" Initial condition UI = " + ui + "");
Console.WriteLine(" E(ALPHA) ALPHA_MU = " + alpha_mu + "");
Console.WriteLine(" STD(ALPHA) ALPHA_SIGMA = " + alpha_sigma + "");
double uexf = ui * Math.Exp(tf * tf / 2.0);
for (np = 0; np <= 5; np++)
{
double[] u = new double[(nt + 1) * (np + 1)];
HermitePolyChaosExpansion.pce_ode_hermite(ti, tf, nt, ui, np, alpha_mu, alpha_sigma, ref t, ref u);
ep[np] = Math.Abs(uexf - u[nt + 0 * (nt + 1)]);
}
//
// Plot error in expected value as a function of the PCE degree.
//
Console.WriteLine("");
Console.WriteLine(" NP Error(NP) Log(Error(NP))");
Console.WriteLine("");
for (np = 0; np <= 5; np++)
{
Console.WriteLine(" " + np.ToString(CultureInfo.InvariantCulture).PadLeft(4)
+ " " + ep[np].ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + Math.Log(ep[np]).ToString(CultureInfo.InvariantCulture).PadLeft(14) + "");
}
}