Re: [Help-gsl] fixed point or adaptive integration for calculating moments using beta PDF?

[Vasu Jaganath]

Yes, I will show you the plots soon, Q is actually 2 variable function but
for my calculations I am treating one of the variables as a parameter,
which is a physically valid assumption. Yes I do encounter alpha and beta
values less than 1.

On Sun, Dec 31, 2017, 9:13 PM Martin Jansche <address@hidden> wrote:

> So you want to find E[f] = \int_0^1 f(x) dbeta(x | a, b) dx. Can you plot
> your typical f? And what are typical values of the parameters a and b? Do
> you encounter a<=1 or b<=1? If so, how does f(x) behave as x approaches 0
> or 1?
>
> On Mon, Jan 1, 2018 at 3:37 AM, Patrick Alken <address@hidden> wrote:
>
> > The question is whether your Q contains any singularities, or is highly
> > oscillatory? Is such cases fixed point quadrature generally doesn't do
> > well. If Q varies fairly smoothly over your interval, you should give
> > fixed point quadrature a try and report back if it works well enough for
> > your problem. The routines you want are documented here:
> >
> > http://www.gnu.org/software/gsl/doc/html/integration.html#
> > fixed-point-quadratures
> >
> > Also, if QAGS isn't working well for you, try also the CQUAD routines.
> > I've found CQUAD is more robust than QAGS in some cases
> >
> > On 12/31/2017 05:28 PM, Vasu Jaganath wrote:
> > > I have attached my entire betaIntegrand function. It is a bit
> complicated
> > > and very boiler-plate, It's OpenFOAM code (where scalar = double etc.)
> I
> > > hope you can get the jist from it.
> > > I can integrate the Q using monte-carlo sampling integration.
> > >
> > > Q is nothing but tabulated values of p,rho, mu etc. I lookup Q using
> the
> > > object "solver" in the snippet.
> > >
> > > I have verified evaluating <Q> when I am not using those <Q> values
> back
> > in
> > > the solution, It works OK.
> > >
> > > Please ask me anything if it seems unclear.
> > >
> > >
> > >
> > >
> > >
> > >
> > > On Sun, Dec 31, 2017 at 3:55 PM, Martin Jansche <address@hidden>
> > wrote:
> > >
> > >> Can you give a concrete example of a typical function Q?
> > >>
> > >> On Sat, Dec 30, 2017 at 3:42 AM, Vasu Jaganath <
> > address@hidden>
> > >> wrote:
> > >>
> > >>> Hi forum,
> > >>>
> > >>> I am trying to integrate moments, basically first order moments <Q>,
> > i.e.
> > >>> averages of some flow fields like temperature, density and mu. I am
> > >>> assuming they distributed according to beta-PDF which is
> parameterized
> > on
> > >>> variable Z, whose mean and variance i am calculating separately and
> > using
> > >>> it to define the shape of the beta-PDF, I have a cut off of not using
> > the
> > >>> beta-PDF when my mean Z value, i.e <Z> is less than a threshold.
> > >>>
> > >>> I am using qags, the adaptive integration routine to calculate the
> > moment
> > >>> integral, however I am restricted to threshold of <Z> = 1e-2.
> > >>>
> > >>> It complains that the integral is too slowly convergent. However
> > >>> physically
> > >>> my threshold should be around 5e-5 atleast.
> > >>>
> > >>> I can integrate these moments with threshold upto 5e-6, using
> > Monte-Carlo
> > >>> integration, by generating random numbers which are beta-distributed.
> > >>>
> > >>> Should I look into fixed point integration routines? What routines
> > would
> > >>> you suggest?
> > >>>
> > >>> Here is the (very simplified) code snippet where, I calculate alpha
> and
> > >>> beta parameter of the PDF
> > >>>
> > >>>                     zvar   = min(zvar,0.9999*zvar_lim);
> > >>>                     alpha = zmean*((zmean*(1 - zmean)/zvar) - 1);
> > >>>                     beta = (1 - zmean)*alpha/zmean;
> > >>>
> > >>>                     // inside the fucntion to be integrated
> > >>>                     // lots of boilerplate for Q(x)
> > >>>                     f = Q(x) * gsl_ran_beta_pdf(x, alpha, beta);
> > >>>
> > >>>                    // my integration call
> > >>>
> > >>>                    helper::gsl_integration_qags (&F, 0, 1, 0, 1e-2,
> > 1000,
> > >>>                                                   w, &result,
> &error);
> > >>>
> > >>> And also, I had to give relative error pretty large, 1e-2. However
> > some of
> > >>> Qs are pretty big order of 1e6.
> > >>>
> > >>> Thanks,
> > >>> Vasu
> > >>>
> > >>
> >
> >
> >
>