public SobolVectorGenerator(int d) : base(d)
{
sequences = new SobolSequence[d];
for (int i = 0; i < sequences.Length; i++)
{
SobolSequenceParameters p = SobolSequenceParameters.sobolParameters[i];
sequences[i] = new SobolSequence(p.Dimension, p.Coefficients, p.Seeds);
}
}
/// <summary>
/// Integrates the given function over the given volume.
/// </summary>
/// <param name="integrand">The function to integrate, which maps R<sup>d</sup> to R.</param>
/// <param name="volume">The box defining the volume over with to integrate.</param>
/// <param name="settings">The evaluation settings to use.</param>
/// <returns>The value of the integral.</returns>
public static double Integrate(Func <IList <double>, double> integrand, IList <Interval> volume, EvaluationSettings settings)
{
if (integrand == null)
{
throw new ArgumentNullException("integrand");
}
if (volume == null)
{
throw new ArgumentNullException("volume");
}
// get the dimension of the problem from the volume
int d = volume.Count;
if ((d < 1) || (d > sobolParameters.Length))
{
throw new InvalidOperationException();
}
// if no settings were provided, use defaults
if (settings == null)
{
settings = new EvaluationSettings()
{
RelativePrecision = 1.0 / (1 << (14 - 3 * d / 2)),
AbsolutePrecision = RelativePrecision / 128.0,
EvaluationBudget = 1 << (18 + 3 * d / 2)
};
}
// compute the volume of the box
double V = 1.0;
for (int j = 0; j < volume.Count; j++)
{
V *= volume[j].Width;
}
// generate the appropriate number of Sobol sequences
SobolSequence[] s = new SobolSequence[d];
for (int j = 0; j < d; j++)
{
SobolSequenceParameters p = sobolParameters[j];
s[j] = new SobolSequence(p.Dimension, p.Coefficients, p.Seeds);
}
// create one vector to store the argument
// we don't want to recreate the vector in the
// heap on each iteration
double[] x = new double[d];
// keep track of the average sampled value
double M = 0.0; double M_old = Double.NaN;
// set the first convergence checkpoint to ~4000 points for d=2, increasing
// for higher d
int i_next = 1 << (10 + d);
for (int i = 1; i <= settings.EvaluationBudget; i++)
{
// move to the next value in each dimension's Sobol sequence
// and construct the argument vector
for (int j = 0; j < d; j++)
{
s[j].MoveNext();
x[j] = volume[j].LeftEndpoint + s[j].Current * volume[j].Width;
}
// evaluate the function
double y = integrand(x);
// update the mean
M += (y - M) / i;
// check for convergence at regular intervals
if (i == i_next)
{
// estimate error as change since last check
double dM = 2.0 * Math.Abs(M - M_old);
if (dM < settings.RelativePrecision * Math.Abs(M) || Math.Abs(V) * dM < settings.AbsolutePrecision)
{
return(V * M);
}
// no convergence, so remember current value and set next checkpoint
M_old = M;
i_next = 2 * i_next;
// Consider how to do this better:
// 1. Is there any way we can do better than doubling? Say increasing by 4/3 and then 3/2, for an average increase of
// 40% instead of 100%? If so, how do we estimate error at each step?
// 2. Can we use Romberg extrapolation? I tried using both M and M_old, assuming an error term of 1/N. This gives
// the extrapolated value 2 M - M_old, but experimentally this is usually a worse value than M. I should try out
// 3-value extrapolation.
}
}
throw new NonconvergenceException();
}
