xolotl-psi-uqdev — Uncertainty Quantification List for Xolotl

Re: [Xolotl-psi-uqdev] [EXTERNAL] Ozgur Cekmer is new to the group!

[Habib N. <hn...@sa...>]

Sounds good, and welcome Ozgur.
Sophie, let's stay in touch re visit planning.

Best,
Habib

On 8/4/15 7:24 AM, Sophie Blondel wrote:
> Dear Habib and Omar,
>
> As I told you previously, Ozgur is a new post-doc on the team at Oak
> Ridge, and he will focus on UQ for Xolotl. I just added him to the UQ
> mailing list. He is currently getting familiar with Xolotl and UQ
> concepts, and we will look at the work that has been done with Crystal
> on the formation/binding energies in the next few weeks.
>
> Brian and David believe that a trip to Sandia would be beneficial to him
> later in the fall, so we will have to organize this.
>
> Cheers,
>
> Sophie
>
> PS: I wanted to make sure the mailing list is working.
>
> ------------------------------------------------------------------------------
> _______________________________________________
> Xolotl-psi-uqdev mailing list
> Xol...@li...
> https://lists.sourceforge.net/lists/listinfo/xolotl-psi-uqdev

-- 
===============================================================
Habib N. Najm, PhD            Distinguished Member Tech. Staff
Tel: (925) 294-2054           Sandia National Laboratories
Fax: (925) 294-2595           P.O.Box 969, MS 9051
Email: hn...@sa...      Livermore, CA 94551, USA
http://crf.sandia.gov/index.php/combustion-research-facility/about-us/crf-staff-2/habib-najm
===============================================================

[Xolotl-psi-uqdev] Ozgur Cekmer is new to the group!

[Sophie B. <blo...@or...>]

Dear Habib and Omar,

As I told you previously, Ozgur is a new post-doc on the team at Oak 
Ridge, and he will focus on UQ for Xolotl. I just added him to the UQ 
mailing list. He is currently getting familiar with Xolotl and UQ 
concepts, and we will look at the work that has been done with Crystal 
on the formation/binding energies in the next few weeks.

Brian and David believe that a trip to Sandia would be beneficial to him 
later in the fall, so we will have to organize this.

Cheers,

Sophie

PS: I wanted to make sure the mailing list is working.

[Xolotl-psi-uqdev] Leaving the project

[Jay J. B. <bil...@or...>]

Everyone,

I wanted to let you know that I am leaving the project, for the most 
part, effective immediately. As many of you may know I am returning to 
graduate school in the fall and that required adjustments to my work 
load to make room for coursework.

Sophie Blondel, a Xolotl developer and postdoc on my team, will take 
over. Sophie is the perfect person to lead the development of Xolotl 
and, for those of you who haven't worked with her yet, she has already 
made many great contributions to the project. Please congratulate her!

I will continue to participate in the project in an advisory role and as 
my time allows over the next few months I will start drafting a couple 
of papers on Xolotl. Of course, if any of you need anything I'm happy to 
help to the extent that I can.

Xolotl, for its part, is in pretty good shape. We are currently checking 
it against Karl's work and other sources that Brian has identified for 
He retention in W. We are also doing detailed scaling and performance 
studies on Titan. Finally, we've been investigating alternative 
parallelization schemes to better utilize the hardware.

I have enjoyed this project and working with each of you immensely. See 
you around!

All the Best,
Jay

-- 
Jay Jay Billings
Oak Ridge National Laboratory
Telephone: (865) 272-9420
Email:bil...@or...
Twitter Handle: @jayjaybillings

[Xolotl-psi-uqdev] Comparison between fits including or excluding V = 1, 2

[Sophie B. <blo...@or...>]

Hi Brian,

I attached the plot of the comparison between the fits. I did the ratio 
(top) and the difference (bottom) where the fit excluding V = 1,2 is 
taken as the reference.

Sophie

[Xolotl-psi-uqdev] Sandia Visit

[Crystal J. <jer...@or...>]

Habib,

Jay told Sophie and I that we should plan a trip to visit Sandia soon 
and we were wondering what your schedule was like and/or when the best 
time for us to come would be.

Have a great day!
Crystal

Re: [Xolotl-psi-uqdev] [EXTERNAL] Bayesian Inference of infered parameters

[Habib N. <hn...@sa...>]

Crystal,

There's no reason to cross check the results in that way.
You can check the validity of the fitted uncertain bivariate polynomial 
function by comparison to the data itself using posterior predictive 
testing.

Habib


On 4/30/14 7:08 AM, Crystal Jernigan wrote:
> Habib,
>
> Sophie and I have considered using the bivariate polynomial function 
> as you described in your most recent email, we were just thinking 
> about using the method that Sophie described in order to cross check 
> the results we obtain.
>
> Thanks for your time and have a great day!
> Crystal & Sophie
>
>
>
> On 04/29/2014 06:38 PM, Habib Najm wrote:
>> Dear Sophie
>>
>> The picture is simpler than this. What you are describing is a problem
>> of fitting of a bivariate polynomial function, rather than the simpler
>> initial case of fitting a 1D polynomial.
>>
>> My understanding from our earlier discussions is that you have been 
>> fitting
>>       E = f(x)
>>       with E = formation energy of Helium clusters
>>              x = nHe/nV
>>              nHe = number of atoms in a He cluster
>>              nV    = vacancy number
>>
>> Also, you've said,
>>       E = f(x) = a + bx + cx^2 + dx^3
>>       where
>>           a = a(nV)
>>           b = b(nV)
>>           c = c(nV)
>>           d = d(nV)
>>
>> So far, you've evaluated a posterior density on (a,b,c,d) for a given 
>> nV.
>>
>> Now, you'd like to include the nV dependence. In this case, you can
>> simply define (e.g.) the proposed quadratic models
>>           a = a0 + a1*nV + a2*nV^2
>>           b = b0 + b1*nV + b2*nV^2
>>           c  = ...
>>           d = ...
>>
>> So that, now, your model has the parameters
>> (a0,a1,a2,b0,b1,b2,c0,c1,c2,d0,d1,d2) and you simply do inference on
>> these parameters, instead of the original four (a,b,c,d). In this
>> context, your data are the triplets (nV_i, nHe_i/nV_i, E_i), and your
>> forward model is
>>           E =     a0 + a1*nV + a2*nV^2  +   (b0 + b1*nV + b2*nV^2)*x +
>> (c0 + c1*nV + c2*nV^2)*x^2
>>                 + (d0 + d1*nV + d2*nV^2)*x^3
>>
>> Hope this is clear?
>> Habib
>>
>>
>> On 4/29/14 2:31 PM, Sophie Blondel wrote:
>>> I am going to try to explain what we are trying to do in more details
>>> and I hope it will be clearer after (for me also):
>>>
>>> - we are studying formation energies of Helium and Vacancy clusters, 
>>> for
>>> which we have data in the form of files for specific Vacancy numbers
>>> (not all of them) containing the energy value for different Helium 
>>> number.
>>>
>>> - with Bayesian Inference we model the energy as a function of the
>>> Helium number only. So we obtain a set of parameters (lets say a, b 
>>> , c,
>>> d if our model is a 3rd order polynomials) for each Vacancy number that
>>> we have.
>>>
>>> - we are now interested in the behavior of these parameters as a
>>> function of the Vacancy number. We could only take the mean value of
>>> them and perform a new inference giving us a(V), b(V), c(V), and d(V)
>>> but we would loose some information on the way because we obtained the
>>> joint density of a, b, c, and d from the first Bayesian Inference.
>>>
>>> - with that in mind, as opposed to using a single y value per x, we
>>> would want to use a range of y values for each x.
>>>
>>> In a nut shell, we want to perform Bayesian Inference on the Bayesian
>>> Inference we have already done (we are trying to fit the fit 
>>> parameters,
>>> since there will be a different set of parameters for each Vacancy 
>>> number).
>>>
>>> Any help you may be able to offer us would be greatly appreciated.
>>>
>>> Sophie
>>>
>>> ------------------------------------------------------------------------------ 
>>>
>>> "Accelerate Dev Cycles with Automated Cross-Browser Testing - For FREE
>>> Instantly run your Selenium tests across 300+ browser/OS combos.  Get
>>> unparalleled scalability from the best Selenium testing platform 
>>> available.
>>> Simple to use. Nothing to install. Get started now for free."
>>> http://p.sf.net/sfu/SauceLabs
>>> _______________________________________________
>>> Xolotl-psi-uqdev mailing list
>>> Xol...@li...
>>> https://lists.sourceforge.net/lists/listinfo/xolotl-psi-uqdev
>

-- 
Habib N. Najm, PhD         Distinguished Member Tech. Staff
Tel: (925) 294-2054            Sandia National Laboratories
Fax: (925) 294-2595                    P.O.Box 969, MS 9051
Email: hn...@sa...           Livermore, CA 94551, USA

Re: [Xolotl-psi-uqdev] [EXTERNAL] Bayesian Inference of infered parameters

[Crystal J. <jer...@or...>]

Habib,

Sophie and I have considered using the bivariate polynomial function as 
you described in your most recent email, we were just thinking about 
using the method that Sophie described in order to cross check the 
results we obtain.

Thanks for your time and have a great day!
Crystal & Sophie



On 04/29/2014 06:38 PM, Habib Najm wrote:
> Dear Sophie
>
> The picture is simpler than this. What you are describing is a problem
> of fitting of a bivariate polynomial function, rather than the simpler
> initial case of fitting a 1D polynomial.
>
> My understanding from our earlier discussions is that you have been fitting
>       E = f(x)
>       with E = formation energy of Helium clusters
>              x = nHe/nV
>              nHe = number of atoms in a He cluster
>              nV    = vacancy number
>
> Also, you've said,
>       E = f(x) = a + bx + cx^2 + dx^3
>       where
>           a = a(nV)
>           b = b(nV)
>           c = c(nV)
>           d = d(nV)
>
> So far, you've evaluated a posterior density on (a,b,c,d) for a given nV.
>
> Now, you'd like to include the nV dependence. In this case, you can
> simply define (e.g.) the proposed quadratic models
>           a = a0 + a1*nV + a2*nV^2
>           b = b0 + b1*nV + b2*nV^2
>           c  = ...
>           d = ...
>
> So that, now, your model has the parameters
> (a0,a1,a2,b0,b1,b2,c0,c1,c2,d0,d1,d2) and you simply do inference on
> these parameters, instead of the original four (a,b,c,d). In this
> context, your data are the triplets (nV_i, nHe_i/nV_i, E_i), and your
> forward model is
>           E =     a0 + a1*nV + a2*nV^2  +   (b0 + b1*nV + b2*nV^2)*x +
> (c0 + c1*nV + c2*nV^2)*x^2
>                 + (d0 + d1*nV + d2*nV^2)*x^3
>
> Hope this is clear?
> Habib
>
>
> On 4/29/14 2:31 PM, Sophie Blondel wrote:
>> I am going to try to explain what we are trying to do in more details
>> and I hope it will be clearer after (for me also):
>>
>> - we are studying formation energies of Helium and Vacancy clusters, for
>> which we have data in the form of files for specific Vacancy numbers
>> (not all of them) containing the energy value for different Helium number.
>>
>> - with Bayesian Inference we model the energy as a function of the
>> Helium number only. So we obtain a set of parameters (lets say a, b , c,
>> d if our model is a 3rd order polynomials) for each Vacancy number that
>> we have.
>>
>> - we are now interested in the behavior of these parameters as a
>> function of the Vacancy number. We could only take the mean value of
>> them and perform a new inference giving us a(V), b(V), c(V), and d(V)
>> but we would loose some information on the way because we obtained the
>> joint density of a, b, c, and d from the first Bayesian Inference.
>>
>> - with that in mind, as opposed to using a single y value per x, we
>> would want to use a range of y values for each x.
>>
>> In a nut shell, we want to perform Bayesian Inference on the Bayesian
>> Inference we have already done (we are trying to fit the fit parameters,
>> since there will be a different set of parameters for each Vacancy number).
>>
>> Any help you may be able to offer us would be greatly appreciated.
>>
>> Sophie
>>
>> ------------------------------------------------------------------------------
>> "Accelerate Dev Cycles with Automated Cross-Browser Testing - For FREE
>> Instantly run your Selenium tests across 300+ browser/OS combos.  Get
>> unparalleled scalability from the best Selenium testing platform available.
>> Simple to use. Nothing to install. Get started now for free."
>> http://p.sf.net/sfu/SauceLabs
>> _______________________________________________
>> Xolotl-psi-uqdev mailing list
>> Xol...@li...
>> https://lists.sourceforge.net/lists/listinfo/xolotl-psi-uqdev

Re: [Xolotl-psi-uqdev] [EXTERNAL] Bayesian Inference of infered parameters

[Habib N. <hn...@sa...>]

Dear Sophie

The picture is simpler than this. What you are describing is a problem 
of fitting of a bivariate polynomial function, rather than the simpler 
initial case of fitting a 1D polynomial.

My understanding from our earlier discussions is that you have been fitting
     E = f(x)
     with E = formation energy of Helium clusters
            x = nHe/nV
            nHe = number of atoms in a He cluster
            nV    = vacancy number

Also, you've said,
     E = f(x) = a + bx + cx^2 + dx^3
     where
         a = a(nV)
         b = b(nV)
         c = c(nV)
         d = d(nV)

So far, you've evaluated a posterior density on (a,b,c,d) for a given nV.

Now, you'd like to include the nV dependence. In this case, you can 
simply define (e.g.) the proposed quadratic models
         a = a0 + a1*nV + a2*nV^2
         b = b0 + b1*nV + b2*nV^2
         c  = ...
         d = ...

So that, now, your model has the parameters 
(a0,a1,a2,b0,b1,b2,c0,c1,c2,d0,d1,d2) and you simply do inference on 
these parameters, instead of the original four (a,b,c,d). In this 
context, your data are the triplets (nV_i, nHe_i/nV_i, E_i), and your 
forward model is
         E =     a0 + a1*nV + a2*nV^2  +   (b0 + b1*nV + b2*nV^2)*x +   
(c0 + c1*nV + c2*nV^2)*x^2
               + (d0 + d1*nV + d2*nV^2)*x^3

Hope this is clear?
Habib


On 4/29/14 2:31 PM, Sophie Blondel wrote:
> I am going to try to explain what we are trying to do in more details
> and I hope it will be clearer after (for me also):
>
> - we are studying formation energies of Helium and Vacancy clusters, for
> which we have data in the form of files for specific Vacancy numbers
> (not all of them) containing the energy value for different Helium number.
>
> - with Bayesian Inference we model the energy as a function of the
> Helium number only. So we obtain a set of parameters (lets say a, b , c,
> d if our model is a 3rd order polynomials) for each Vacancy number that
> we have.
>
> - we are now interested in the behavior of these parameters as a
> function of the Vacancy number. We could only take the mean value of
> them and perform a new inference giving us a(V), b(V), c(V), and d(V)
> but we would loose some information on the way because we obtained the
> joint density of a, b, c, and d from the first Bayesian Inference.
>
> - with that in mind, as opposed to using a single y value per x, we
> would want to use a range of y values for each x.
>
> In a nut shell, we want to perform Bayesian Inference on the Bayesian
> Inference we have already done (we are trying to fit the fit parameters,
> since there will be a different set of parameters for each Vacancy number).
>
> Any help you may be able to offer us would be greatly appreciated.
>
> Sophie
>
> ------------------------------------------------------------------------------
> "Accelerate Dev Cycles with Automated Cross-Browser Testing - For FREE
> Instantly run your Selenium tests across 300+ browser/OS combos.  Get
> unparalleled scalability from the best Selenium testing platform available.
> Simple to use. Nothing to install. Get started now for free."
> http://p.sf.net/sfu/SauceLabs
> _______________________________________________
> Xolotl-psi-uqdev mailing list
> Xol...@li...
> https://lists.sourceforge.net/lists/listinfo/xolotl-psi-uqdev

-- 
Habib N. Najm, PhD         Distinguished Member Tech. Staff
Tel: (925) 294-2054            Sandia National Laboratories
Fax: (925) 294-2595                    P.O.Box 969, MS 9051
Email: hn...@sa...           Livermore, CA 94551, USA

Re: [Xolotl-psi-uqdev] [EXTERNAL] Bayesian Inference of infered parameters

[Sophie B. <blo...@or...>]

I am going to try to explain what we are trying to do in more details 
and I hope it will be clearer after (for me also):

- we are studying formation energies of Helium and Vacancy clusters, for 
which we have data in the form of files for specific Vacancy numbers 
(not all of them) containing the energy value for different Helium number.

- with Bayesian Inference we model the energy as a function of the 
Helium number only. So we obtain a set of parameters (lets say a, b , c, 
d if our model is a 3rd order polynomials) for each Vacancy number that 
we have.

- we are now interested in the behavior of these parameters as a 
function of the Vacancy number. We could only take the mean value of 
them and perform a new inference giving us a(V), b(V), c(V), and d(V) 
but we would loose some information on the way because we obtained the 
joint density of a, b, c, and d from the first Bayesian Inference.

- with that in mind, as opposed to using a single y value per x, we 
would want to use a range of y values for each x.

In a nut shell, we want to perform Bayesian Inference on the Bayesian 
Inference we have already done (we are trying to fit the fit parameters, 
since there will be a different set of parameters for each Vacancy number).

Any help you may be able to offer us would be greatly appreciated.

Sophie

Re: [Xolotl-psi-uqdev] [EXTERNAL] Bayesian Inference of infered parameters

[Habib N. <hn...@sa...>]

Hi Sophie,

Some comments below.

On 4/29/14 11:51 AM, Sophie Blondel wrote:
> Hi everybody,
>
> I have a general question about Bayesian Inference.
>
> Right now, Crystal and I are using UQTk to do Bayesian Inference in a
> "simple" way, i.e. we have data with no uncertainty on it (no error bars
> on each point) and we use a simple model (polynomials with gaussian
> noise) to reproduce it.
The data that you have been using now is noisy. Noisy data is the reason 
you have error bars in the first place. When you consider measurements 
from an instrument, the most fundamental measurement is simply noisy 
data, it does not come with error bars. It's only processed data 
products that are assigned error bars, ideally based on this type of 
Bayesian inference that you've been doing. So, for example, if I am 
measuring temperature with a thermocouple, I can read voltages from the 
voltmeter (neglecting errors in the high-precision digital voltmeter :)) 
and when I fit them with a thermocouple model (with Bayes) I get a 
temperature "measurement" with error bars.
> But for our next step, we are thinking of using the parameters that were
> infered in that "simple" way (which means that we have PDF and
> correlation for them) as our input data for Bayesian Inference. How can
> we use this additional information (and not just use the mean of the
> parameters) to compute the correct posteriors?
Now, as to this next step, I am not sure first what you're describing. 
Are you talking about a forward or inverse problem? If you mean to take 
parameters q with a given joint density p(q) and, given y=f(q), you want 
to estimate p(y), then this is a forward UQ problem, and you don't need 
Bayes. Rather you need sparse-quadrature non-intrusive PC methods in the 
present context. On the other hand, if you mean that you *do* have a 
Bayesian inference problem, with some new measured data set, and where 
the uncertain parameters q appear in the underlying forward model, this 
is fine also. Yes, this is an inverse problem, and the uncertain 
parameters q can be implemented as nuisance parameters in the Bayesian 
fitting.

Hope this helps? Pls let me know re questions.

Habib

> Cheers,
>
> Sophie
>
> ------------------------------------------------------------------------------
> "Accelerate Dev Cycles with Automated Cross-Browser Testing - For FREE
> Instantly run your Selenium tests across 300+ browser/OS combos.  Get
> unparalleled scalability from the best Selenium testing platform available.
> Simple to use. Nothing to install. Get started now for free."
> http://p.sf.net/sfu/SauceLabs
> _______________________________________________
> Xolotl-psi-uqdev mailing list
> Xol...@li...
> https://lists.sourceforge.net/lists/listinfo/xolotl-psi-uqdev

-- 
Habib N. Najm, PhD         Distinguished Member Tech. Staff
Tel: (925) 294-2054            Sandia National Laboratories
Fax: (925) 294-2595                    P.O.Box 969, MS 9051
Email: hn...@sa...           Livermore, CA 94551, USA

[Xolotl-psi-uqdev] Bayesian Inference of infered parameters

[Sophie B. <blo...@or...>]

Hi everybody,

I have a general question about Bayesian Inference.

Right now, Crystal and I are using UQTk to do Bayesian Inference in a 
"simple" way, i.e. we have data with no uncertainty on it (no error bars 
on each point) and we use a simple model (polynomials with gaussian 
noise) to reproduce it.

But for our next step, we are thinking of using the parameters that were 
infered in that "simple" way (which means that we have PDF and 
correlation for them) as our input data for Bayesian Inference. How can 
we use this additional information (and not just use the mean of the 
parameters) to compute the correct posteriors?

Cheers,

Sophie

[Xolotl-psi-uqdev] Formation energy fits

[Sophie B. <blo...@or...>]

Dear Brian,

Here are the fit parameters you were interested in (x = He/V, y = V) 
when all the data are used up to V = 50:
- For the 2D fit (the separation is actually at 1.1 here, and 
polynomials of order 3 below and above the separation)
*He/V <= 1.1
                                                    y               
y^2                  y^3
           [  1.86453745e+00   2.43282904e+00  -3.45705570e-02 
3.03260401e-04]
   x      [  6.71149092e+00  -1.37833394e+00   7.14656373e-02 
-9.80111040e-04]
  x^2  [  1.42931236e+00   7.29520166e-01  -6.87522351e-02 1.10114140e-03]
  x^3  [ -6.30490963e+00   5.49444370e-01   3.54878543e-03 -2.20925602e-04]

*He/V > 1.1
                                                   y         
y^2                      y^3
          [ -1.79715297e-01   1.47850485e+00   5.50033059e-02 
-1.00391190e-03]
   x     [  5.28044060e+00   5.25227305e-01  -1.11264208e-01 1.67360562e-03]
  x^2  [ -7.49710725e-01   1.54470949e-01   4.32566976e-02 -7.07270501e-04]
  x^3  [  1.09668671e-02   3.10141664e-02  -6.25568851e-03 1.02553956e-04]

  - For the two-step fit (separation at 1, 2nd order polynomial fit for 
He/V <= 1 and then 3rd order polynomial fit on the parameters, 3rd order 
polynomial fit for He/V >= 1 and then 3rd order polynomial fit on the 
parameters)
*He/V <= 1.0
                                                      y                
y^2                   y^3
           [  2.15810899e+00,   2.34162124e+00,  -2.89642755e-02, 
2.21530499e-04]
    x     [  1.97296491e+00,  -2.20092580e-01,   2.56190417e-03, 
1.28566949e-05]
   x^2  [ -3.91437205e-02,   1.48025700e-01,  -1.53822938e-04, 
-4.72980005e-05]

*He/V > 1.0
y                          y^2                      y^3
           [  6.55775789e-01,   1.92961236e+00,   9.79478220e-03, 
-2.02674593e-04]
    x     [  3.84737696e+00,  -7.43702330e-02,  -4.99468581e-02, 
5.81612123e-04]
   x^2  [ -4.89069326e-01,   4.75506021e-01,   1.42564728e-02, 
-2.05582261e-04]
   x^3  [  1.84446642e-02,  -1.86291619e-02,  -2.11845922e-03, 
3.24952110e-05]

  - Now the same for Crystal's two-step fit for only V = 1, 2 (with the 
same characteristics as the previous one):
*He/V <= 1.0
                                        y y^2              y^3
            [ 2.23767515  1.04443605  0.39504942  0.14927938]
    x      [-0.05562908  0.28497843  0.26454075  0.1637199 ]
   x^2  [ 0.85929827  0.06573789 -0.14418381 -0.12324235]

*He/V > 1.0
                                        y y^2           y^3
           [ 1.53160457  0.63126522  0.19662118  0.05715427]
    x     [ 1.42108884  0.72537583  0.30566365  0.12823212]
   x^2  [ 0.1431226   0.03525138 -0.00257175 -0.00744112]
   x^3  [-0.00775551 -0.00112535  0.00083187  0.00082583]

We hope the order is the right one, it can be confusing sometimes... 
Don't hesitate if you have any questions.

After what was said this morning, we tried to use this obtained fit (the 
two-step one with all the data up to V = 50) to extrapolate the higher 
vacancy energies.
Between V = 50 and 100 we only have V = 52 and 87, and there are two 
files for V = 52 ("form_52vac_s5p75n30_skipped_2_Y" and 
"form_52vac_s6n30_skipped_1_Y"). Not knowing which one should be used we 
decided to skip it and only try for V = 87. The plots obtained are 
attached, where dots are the data and the line is from the fit. Both 
formation and binding energies are plotted. The same for V = 44 is 
attached to compare with a vacancy number where the fit is supposed to 
be good because the data were used to make the fit.

This extrapolation doesn't seem to be really good, and it gets worse if 
we go to higher vacancy numbers (another interesting thing appeared in 
our data file: for some with V > 100, He numbers are periodically 
missing in the file).
Crystal suggested to try to fit only the data with V = [6..44] and then 
extrapolate to V = 87. The result is attached ("improved") too and looks 
better.

We will most certainly try other things by next Monday.

Cheers,

Crystal and Sophie