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23. Monte Carlo Integration

This chapter describes routines for multidimensional Monte Carlo integration. These include the traditional Monte Carlo method and adaptive algorithms such as VEGAS and MISER which use importance sampling and stratified sampling techniques. Each algorithm computes an estimate of a multidimensional definite integral of the form, over a hypercubic region $((x_l,x_u)$, $(y_l,y_u), ...)$ using a fixed number of function calls. The routines also provide a statistical estimate of the error on the result. This error estimate should be taken as a guide rather than as a strict error bound—random sampling of the region may not uncover all the important features of the function, resulting in an underestimate of the error.

The functions are defined in separate header files for each routine, gsl_monte_plain.h, ‘gsl_monte_miser.h’ and ‘gsl_monte_vegas.h’.

23.1 Interface  
23.2 PLAIN Monte Carlo  
23.3 MISER  
23.4 VEGAS  
23.5 Examples  
23.6 References and Further Reading  
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23.1 Interface

All of the Monte Carlo integration routines use the same general form of interface. There is an allocator to allocate memory for control variables and workspace, a routine to initialize those control variables, the integrator itself, and a function to free the space when done.

Each integration function requires a random number generator to be supplied, and returns an estimate of the integral and its standard deviation. The accuracy of the result is determined by the number of function calls specified by the user. If a known level of accuracy is required this can be achieved by calling the integrator several times and averaging the individual results until the desired accuracy is obtained.

Random sample points used within the Monte Carlo routines are always chosen strictly within the integration region, so that endpoint singularities are automatically avoided.

The function to be integrated has its own datatype, defined in the header file ‘gsl_monte.h’.

Data Type: gsl_monte_function

This data type defines a general function with parameters for Monte Carlo integration.

double (* f) (double * x, size_t dim, void * params)

this function should return the value $f(x,params)$ for the argument x and parameters params, where x is an array of size dim giving the coordinates of the point where the function is to be evaluated.

size_t dim

the number of dimensions for x.

void * params

a pointer to the parameters of the function.

Here is an example for a quadratic function in two dimensions, with $a = 3$, $b = 2$, $c = 1$. The following code defines a gsl_monte_function F which you could pass to an integrator:

 
struct my_f_params { double a; double b; double c; };

double
my_f (double x[], size_t dim, void * p) {
   struct my_f_params * fp = (struct my_f_params *)p;

   if (dim != 2)
      {
        fprintf (stderr, "error: dim != 2");
        abort ();
      }

   return  fp->a * x[0] * x[0] 
             + fp->b * x[0] * x[1] 
               + fp->c * x[1] * x[1];
}

gsl_monte_function F;
struct my_f_params params = { 3.0, 2.0, 1.0 };

F.f = &my_f;
F.dim = 2;
F.params = &params;

The function $f(x)$ can be evaluated using the following macro,

 
#define GSL_MONTE_FN_EVAL(F,x) 
    (*((F)->f))(x,(F)->dim,(F)->params)
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23.2 PLAIN Monte Carlo

The plain Monte Carlo algorithm samples points randomly from the integration region to estimate the integral and its error. Using this algorithm the estimate of the integral $E(f; N)$ for $N$ randomly distributed points $x_i$ is given by, where $V$ is the volume of the integration region. The error on this estimate $\sigma(E;N)$ is calculated from the estimated variance of the mean, For large $N$ this variance decreases asymptotically as $\Var(f)/N$, where $\Var(f)$ is the true variance of the function over the integration region. The error estimate itself should decrease as $\sigma(f)/\sqrt@{N@}$. The familiar law of errors decreasing as $1/\sqrt@{N@}$ applies—to reduce the error by a factor of 10 requires a 100-fold increase in the number of sample points.

The functions described in this section are declared in the header file ‘gsl_monte_plain.h’.

Function: gsl_monte_plain_state * gsl_monte_plain_alloc (size_t dim)

This function allocates and initializes a workspace for Monte Carlo integration in dim dimensions.

Function: int gsl_monte_plain_init (gsl_monte_plain_state* s)

This function initializes a previously allocated integration state. This allows an existing workspace to be reused for different integrations.

Function: int gsl_monte_plain_integrate (gsl_monte_function * f, double * xl, double * xu, size_t dim, size_t calls, gsl_rng * r, gsl_monte_plain_state * s, double * result, double * abserr)

This routines uses the plain Monte Carlo algorithm to integrate the function f over the dim-dimensional hypercubic region defined by the lower and upper limits in the arrays xl and xu, each of size dim. The integration uses a fixed number of function calls calls, and obtains random sampling points using the random number generator r. A previously allocated workspace s must be supplied. The result of the integration is returned in result, with an estimated absolute error abserr.

Function: void gsl_monte_plain_free (gsl_monte_plain_state * s)

This function frees the memory associated with the integrator state s.

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23.3 MISER

The MISER algorithm of Press and Farrar is based on recursive stratified sampling. This technique aims to reduce the overall integration error by concentrating integration points in the regions of highest variance.

The idea of stratified sampling begins with the observation that for two disjoint regions $a$ and $b$ with Monte Carlo estimates of the integral $E_a(f)$ and $E_b(f)$ and variances $\sigma_a^2(f)$ and $\sigma_b^2(f)$, the variance $\Var(f)$ of the combined estimate $E(f) = (1/2) (E_a(f) + E_b(f))$ is given by, It can be shown that this variance is minimized by distributing the points such that, Hence the smallest error estimate is obtained by allocating sample points in proportion to the standard deviation of the function in each sub-region.

The MISER algorithm proceeds by bisecting the integration region along one coordinate axis to give two sub-regions at each step. The direction is chosen by examining all $d$ possible bisections and selecting the one which will minimize the combined variance of the two sub-regions. The variance in the sub-regions is estimated by sampling with a fraction of the total number of points available to the current step. The same procedure is then repeated recursively for each of the two half-spaces from the best bisection. The remaining sample points are allocated to the sub-regions using the formula for $N_a$ and $N_b$. This recursive allocation of integration points continues down to a user-specified depth where each sub-region is integrated using a plain Monte Carlo estimate. These individual values and their error estimates are then combined upwards to give an overall result and an estimate of its error.

The functions described in this section are declared in the header file ‘gsl_monte_miser.h’.

Function: gsl_monte_miser_state * gsl_monte_miser_alloc (size_t dim)

This function allocates and initializes a workspace for Monte Carlo integration in dim dimensions. The workspace is used to maintain the state of the integration.

Function: int gsl_monte_miser_init (gsl_monte_miser_state* s)

This function initializes a previously allocated integration state. This allows an existing workspace to be reused for different integrations.

Function: int gsl_monte_miser_integrate (gsl_monte_function * f, double * xl, double * xu, size_t dim, size_t calls, gsl_rng * r, gsl_monte_miser_state * s, double * result, double * abserr)

This routines uses the MISER Monte Carlo algorithm to integrate the function f over the dim-dimensional hypercubic region defined by the lower and upper limits in the arrays xl and xu, each of size dim. The integration uses a fixed number of function calls calls, and obtains random sampling points using the random number generator r. A previously allocated workspace s must be supplied. The result of the integration is returned in result, with an estimated absolute error abserr.

Function: void gsl_monte_miser_free (gsl_monte_miser_state * s)

This function frees the memory associated with the integrator state s.

The MISER algorithm has several configurable parameters. The following variables can be accessed through the gsl_monte_miser_state struct,

Variable: double estimate_frac

This parameter specifies the fraction of the currently available number of function calls which are allocated to estimating the variance at each recursive step. The default value is 0.1.

Variable: size_t min_calls

This parameter specifies the minimum number of function calls required for each estimate of the variance. If the number of function calls allocated to the estimate using estimate_frac falls below min_calls then min_calls are used instead. This ensures that each estimate maintains a reasonable level of accuracy. The default value of min_calls is 16 * dim.

Variable: size_t min_calls_per_bisection

This parameter specifies the minimum number of function calls required to proceed with a bisection step. When a recursive step has fewer calls available than min_calls_per_bisection it performs a plain Monte Carlo estimate of the current sub-region and terminates its branch of the recursion. The default value of this parameter is 32 * min_calls.

Variable: double alpha

This parameter controls how the estimated variances for the two sub-regions of a bisection are combined when allocating points. With recursive sampling the overall variance should scale better than $1/N$, since the values from the sub-regions will be obtained using a procedure which explicitly minimizes their variance. To accommodate this behavior the MISER algorithm allows the total variance to depend on a scaling parameter $\alpha$, The authors of the original paper describing MISER recommend the value $\alpha = 2$ as a good choice, obtained from numerical experiments, and this is used as the default value in this implementation.

Variable: double dither

This parameter introduces a random fractional variation of size dither into each bisection, which can be used to break the symmetry of integrands which are concentrated near the exact center of the hypercubic integration region. The default value of dither is zero, so no variation is introduced. If needed, a typical value of dither is 0.1.

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23.4 VEGAS

The VEGAS algorithm of Lepage is based on importance sampling. It samples points from the probability distribution described by the function $\vert f\vert$, so that the points are concentrated in the regions that make the largest contribution to the integral.

In general, if the Monte Carlo integral of $f$ is sampled with points distributed according to a probability distribution described by the function $g$, we obtain an estimate $E_g(f; N)$, with a corresponding variance, If the probability distribution is chosen as $g = \vert f\vert/I(\vert f\vert)$ then it can be shown that the variance $V_g(f; N)$ vanishes, and the error in the estimate will be zero. In practice it is not possible to sample from the exact distribution $g$ for an arbitrary function, so importance sampling algorithms aim to produce efficient approximations to the desired distribution.

The VEGAS algorithm approximates the exact distribution by making a number of passes over the integration region while histogramming the function $f$. Each histogram is used to define a sampling distribution for the next pass. Asymptotically this procedure converges to the desired distribution. In order to avoid the number of histogram bins growing like $K^d$ the probability distribution is approximated by a separable function: $g(x_1, x_2, ...) = g_1(x_1) g_2(x_2) ...$ so that the number of bins required is only $Kd$. This is equivalent to locating the peaks of the function from the projections of the integrand onto the coordinate axes. The efficiency of VEGAS depends on the validity of this assumption. It is most efficient when the peaks of the integrand are well-localized. If an integrand can be rewritten in a form which is approximately separable this will increase the efficiency of integration with VEGAS.

VEGAS incorporates a number of additional features, and combines both stratified sampling and importance sampling. The integration region is divided into a number of “boxes”, with each box getting a fixed number of points (the goal is 2). Each box can then have a fractional number of bins, but if the ratio of bins-per-box is less than two, Vegas switches to a kind variance reduction (rather than importance sampling).

Function: gsl_monte_vegas_state * gsl_monte_vegas_alloc (size_t dim)

This function allocates and initializes a workspace for Monte Carlo integration in dim dimensions. The workspace is used to maintain the state of the integration.

Function: int gsl_monte_vegas_init (gsl_monte_vegas_state* s)

This function initializes a previously allocated integration state. This allows an existing workspace to be reused for different integrations.

Function: int gsl_monte_vegas_integrate (gsl_monte_function * f, double * xl, double * xu, size_t dim, size_t calls, gsl_rng * r, gsl_monte_vegas_state * s, double * result, double * abserr)

This routines uses the VEGAS Monte Carlo algorithm to integrate the function f over the dim-dimensional hypercubic region defined by the lower and upper limits in the arrays xl and xu, each of size dim. The integration uses a fixed number of function calls calls, and obtains random sampling points using the random number generator r. A previously allocated workspace s must be supplied. The result of the integration is returned in result, with an estimated absolute error abserr. The result and its error estimate are based on a weighted average of independent samples. The chi-squared per degree of freedom for the weighted average is returned via the state struct component, s->chisq, and must be consistent with 1 for the weighted average to be reliable.

Function: void gsl_monte_vegas_free (gsl_monte_vegas_state * s)

This function frees the memory associated with the integrator state s.

The VEGAS algorithm computes a number of independent estimates of the integral internally, according to the iterations parameter described below, and returns their weighted average. Random sampling of the integrand can occasionally produce an estimate where the error is zero, particularly if the function is constant in some regions. An estimate with zero error causes the weighted average to break down and must be handled separately. In the original Fortran implementations of VEGAS the error estimate is made non-zero by substituting a small value (typically 1e-30). The implementation in GSL differs from this and avoids the use of an arbitrary constant—it either assigns the value a weight which is the average weight of the preceding estimates or discards it according to the following procedure,

current estimate has zero error, weighted average has finite error

The current estimate is assigned a weight which is the average weight of the preceding estimates.

current estimate has finite error, previous estimates had zero error

The previous estimates are discarded and the weighted averaging procedure begins with the current estimate.

current estimate has zero error, previous estimates had zero error

The estimates are averaged using the arithmetic mean, but no error is computed.

The VEGAS algorithm is highly configurable. The following variables can be accessed through the gsl_monte_vegas_state struct,

Variable: double result
Variable: double sigma

These parameters contain the raw value of the integral result and its error sigma from the last iteration of the algorithm.

Variable: double chisq

This parameter gives the chi-squared per degree of freedom for the weighted estimate of the integral. The value of chisq should be close to 1. A value of chisq which differs significantly from 1 indicates that the values from different iterations are inconsistent. In this case the weighted error will be under-estimated, and further iterations of the algorithm are needed to obtain reliable results.

Variable: double alpha

The parameter alpha controls the stiffness of the rebinning algorithm. It is typically set between one and two. A value of zero prevents rebinning of the grid. The default value is 1.5.

Variable: size_t iterations

The number of iterations to perform for each call to the routine. The default value is 5 iterations.

Variable: int stage

Setting this determines the stage of the calculation. Normally, stage = 0 which begins with a new uniform grid and empty weighted average. Calling vegas with stage = 1 retains the grid from the previous run but discards the weighted average, so that one can “tune” the grid using a relatively small number of points and then do a large run with stage = 1 on the optimized grid. Setting stage = 2 keeps the grid and the weighted average from the previous run, but may increase (or decrease) the number of histogram bins in the grid depending on the number of calls available. Choosing stage = 3 enters at the main loop, so that nothing is changed, and is equivalent to performing additional iterations in a previous call.

Variable: int mode

The possible choices are GSL_VEGAS_MODE_IMPORTANCE, GSL_VEGAS_MODE_STRATIFIED, GSL_VEGAS_MODE_IMPORTANCE_ONLY. This determines whether VEGAS will use importance sampling or stratified sampling, or whether it can pick on its own. In low dimensions VEGAS uses strict stratified sampling (more precisely, stratified sampling is chosen if there are fewer than 2 bins per box).

Variable: int verbose
Variable: FILE * ostream

These parameters set the level of information printed by VEGAS. All information is written to the stream ostream. The default setting of verbose is -1, which turns off all output. A verbose value of 0 prints summary information about the weighted average and final result, while a value of 1 also displays the grid coordinates. A value of 2 prints information from the rebinning procedure for each iteration.

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23.5 Examples

The example program below uses the Monte Carlo routines to estimate the value of the following 3-dimensional integral from the theory of random walks, The analytic value of this integral can be shown to be $I = \Gamma(1/4)^4/(4 \pi^3) = 1.393203929685676859...$. The integral gives the mean time spent at the origin by a random walk on a body-centered cubic lattice in three dimensions.

For simplicity we will compute the integral over the region $(0,0,0)$ to $(\pi,\pi,\pi)$ and multiply by 8 to obtain the full result. The integral is slowly varying in the middle of the region but has integrable singularities at the corners $(0,0,0)$, $(0,\pi,\pi)$, $(\pi,0,\pi)$ and $(\pi,\pi,0)$. The Monte Carlo routines only select points which are strictly within the integration region and so no special measures are needed to avoid these singularities.

 
#include <stdlib.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_monte.h>
#include <gsl/gsl_monte_plain.h>
#include <gsl/gsl_monte_miser.h>
#include <gsl/gsl_monte_vegas.h>

/* Computation of the integral,

      I = int (dx dy dz)/(2pi)^3  1/(1-cos(x)cos(y)cos(z))

   over (-pi,-pi,-pi) to (+pi, +pi, +pi).  The exact answer
   is Gamma(1/4)^4/(4 pi^3).  This example is taken from
   C.Itzykson, J.M.Drouffe, "Statistical Field Theory -
   Volume 1", Section 1.1, p21, which cites the original
   paper M.L.Glasser, I.J.Zucker, Proc.Natl.Acad.Sci.USA 74
   1800 (1977) */

/* For simplicity we compute the integral over the region 
   (0,0,0) -> (pi,pi,pi) and multiply by 8 */

double exact = 1.3932039296856768591842462603255;

double
g (double *k, size_t dim, void *params)
{
  double A = 1.0 / (M_PI * M_PI * M_PI);
  return A / (1.0 - cos (k[0]) * cos (k[1]) * cos (k[2]));
}

void
display_results (char *title, double result, double error)
{
  printf ("%s ==================\n", title);
  printf ("result = % .6f\n", result);
  printf ("sigma  = % .6f\n", error);
  printf ("exact  = % .6f\n", exact);
  printf ("error  = % .6f = %.1g sigma\n", result - exact,
          fabs (result - exact) / error);
}

int
main (void)
{
  double res, err;

  double xl[3] = { 0, 0, 0 };
  double xu[3] = { M_PI, M_PI, M_PI };

  const gsl_rng_type *T;
  gsl_rng *r;

  gsl_monte_function G = { &g, 3, 0 };

  size_t calls = 500000;

  gsl_rng_env_setup ();

  T = gsl_rng_default;
  r = gsl_rng_alloc (T);

  {
    gsl_monte_plain_state *s = gsl_monte_plain_alloc (3);
    gsl_monte_plain_integrate (&G, xl, xu, 3, calls, r, s, 
                               &res, &err);
    gsl_monte_plain_free (s);

    display_results ("plain", res, err);
  }

  {
    gsl_monte_miser_state *s = gsl_monte_miser_alloc (3);
    gsl_monte_miser_integrate (&G, xl, xu, 3, calls, r, s,
                               &res, &err);
    gsl_monte_miser_free (s);

    display_results ("miser", res, err);
  }

  {
    gsl_monte_vegas_state *s = gsl_monte_vegas_alloc (3);

    gsl_monte_vegas_integrate (&G, xl, xu, 3, 10000, r, s,
                               &res, &err);
    display_results ("vegas warm-up", res, err);

    printf ("converging...\n");

    do
      {
        gsl_monte_vegas_integrate (&G, xl, xu, 3, calls/5, r, s,
                                   &res, &err);
        printf ("result = % .6f sigma = % .6f "
                "chisq/dof = %.1f\n", res, err, s->chisq);
      }
    while (fabs (s->chisq - 1.0) > 0.5);

    display_results ("vegas final", res, err);

    gsl_monte_vegas_free (s);
  }

  gsl_rng_free (r);

  return 0;
}

With 500,000 function calls the plain Monte Carlo algorithm achieves a fractional error of 0.6%. The estimated error sigma is consistent with the actual error, and the computed result differs from the true result by about one standard deviation,

 
plain ==================
result =  1.385867
sigma  =  0.007938
exact  =  1.393204
error  = -0.007337 = 0.9 sigma

The MISER algorithm reduces the error by a factor of two, and also correctly estimates the error,

 
miser ==================
result =  1.390656
sigma  =  0.003743
exact  =  1.393204
error  = -0.002548 = 0.7 sigma

In the case of the VEGAS algorithm the program uses an initial warm-up run of 10,000 function calls to prepare, or “warm up”, the grid. This is followed by a main run with five iterations of 100,000 function calls. The chi-squared per degree of freedom for the five iterations are checked for consistency with 1, and the run is repeated if the results have not converged. In this case the estimates are consistent on the first pass.

 
vegas warm-up ==================
result =  1.386925
sigma  =  0.002651
exact  =  1.393204
error  = -0.006278 = 2 sigma
converging...
result =  1.392957 sigma =  0.000452 chisq/dof = 1.1
vegas final ==================
result =  1.392957
sigma  =  0.000452
exact  =  1.393204
error  = -0.000247 = 0.5 sigma

If the value of chisq had differed significantly from 1 it would indicate inconsistent results, with a correspondingly underestimated error. The final estimate from VEGAS (using a similar number of function calls) is significantly more accurate than the other two algorithms.

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23.6 References and Further Reading

The MISER algorithm is described in the following article by Press and Farrar,

The VEGAS algorithm is described in the following papers,

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