![\"\"](\"http:\/\/aristotle2digital.blogwyrm.com\/wp-content\/uploads\/2021\/11\/f_p_look.png\")
<\/figure>\n\n\n\nThis function has the antiderivative<\/p>\n\n\n\n
\\[ F_p(x) = \\frac{3.4}{4} x^4 – \\frac{5.9}{3} x^3 + \\frac{0.3}{2} x^2 – 15.2 x \\; . \\]<\/p>\n\n\n\n
The integral of $f_p$ from $x_{min} = -2$ to $x_{max} = 3.5$ has the exact value<\/p>\n\n\n\n
\\[ \\int_{x_{min}}^{x_{max}} dx f_p = F_p(x_{max}) \u2013 F_p(x_{min}) = -68.46354166666669 \\; . \\]<\/p>\n\n\n\n
A simple script in python using numpy, scipy, and a Jupyter notebook was implemented to explore numerical techniques to perform the integral. The scipy.integrate.quad<\/a> function, which uses traditional quadrature based on the Fortran QUADPACK routines, returned the value of -68.46354166666667, which is exactly (to within machine precision) the analytic result. The Monte Carlo integration, which is implemented with the following lines<\/p>\n\n\n\n returned a value of -68.57203483190794 using 100,000 samples (generally called a trial \u2013 a terminology that will be used hereafter) with a relative error of 0.00158. This is not bad agreement but clearly orders of magnitude worse than the deterministic algorithm. We\u2019ll return to this point later.<\/p>\n\n\n\n One more example is worth discussing. This time the function is a nasty, nested function given by<\/p>\n\n\n\n \\[ f_n = 4 \\sin \\left( 2 \\pi e^{\\cos x^2} \\right) \\; , \\]<\/p>\n\n\n\n which looks like<\/p>\n\n\n\n over the range $[-2,3.5]$. Even though this function doesn\u2019t have an anti-derivative, its integral clearly exists since the function is continuous with no divergences. The deterministic algorithm found in scipy.integrate.quad generates the value of -2.824748541066965, which we will take as truth. The same Monte Carlo scheme as above yields a value -2.8020549889975164 or relative error 0.008034.<\/p>\n\n\n\n So, we can see that the method is versatile and reasonably accurate. However, there are two questions that one might immediately pose: 1) why use it if there are more accurate methods available and 2) can the estimates be made more accurate? The remainder of this post will deal with answering question 1 while the subsequent post will answer the second question.<\/p>\n\n\n\n To answer question 1, we will cite, without proof, the answer to question 2 that states the error $\\epsilon$ in the estimate of the integral goes to zero as the number of samples according to<\/p>\n\n\n\n \\[ \\epsilon \\sim \\frac{1}{\\sqrt{N}} \\; \\]<\/p>\n\n\n\n independent of the number of dimensions. <\/p>\n\n\n\n To determine the error for a conventional quadrature method (e.g. trapezoidal<\/a> or Simpson\u2019s<\/a>), we follow the argument from Steve Koonin\u2019s Computational Physics p 201. Assume that we will evaluate the integrand $N$ times (corresponding to the $N$ times it will be evaluated with the Monte Carlo scheme). The $d$-dimensional region over for the integral consists of a volume ${V} ~ {h}^d$. Distributing the $N$ points over the volume means that the grid spacing $h ~ N^{-1\/d}$. <\/p>\n\n\n\n Koonin states that an analysis similar to those used for the usual one-dimensional formulae gives the error as going $O\\left(h^{d+2}\\right)$. The total error of the quadrature thus goes as<\/p>\n\n\n\n \\[ N O\\left(h^{d+2}\\right) = O\\left(N^{-2\/d}\\right) \\; .\\]<\/p>\n\n\n\n When $d \\geq 4$, the error in estimating the integral via Monte Carlo becomes smaller than that for standard quadrature techniques for a given computing time invested. This reason alone justifies why the Monte Carlo technique is so highly favored. In particular, Monte Carlo proves invaluable in propagating variations through dynamical systems whose dimensionality precludes using a gridded method like traditional quadrature.<\/p>\n\n\n\n The next post will derive why the Monte Carlo error goes as $1\/\\sqrt(N)$.<\/p>\n","protected":false},"excerpt":{"rendered":"N = 100_000\nrand_x = x_min + (x_max - x_min)*np.random.random(size=N)\nrand_y = f(rand_x)\nI_poly_est = (x_max - x_min)*np.average(rand_y)<\/pre>\n<\/div>\n\n\n\n
![\"\"](\"http:\/\/aristotle2digital.blogwyrm.com\/wp-content\/uploads\/2021\/11\/f_n_look.png\")
<\/figure>\n\n\n\n![\"\"](\"http:\/\/aristotle2digital.blogwyrm.com\/wp-content\/uploads\/2021\/11\/Grid_size.png\")
<\/figure>\n\n\n\n