ODE IV Control and Variable Scale

ODE IV Control and Variable Scale

[Kevin H. Hobbs]

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I'm trying make sure that I am using the gsl_odeiv_control functions
correctly.

I'm modelling neurons and my variables have different scales. Voltage is
usually between -100 and 50 mV. The membrane current gating variables
stay between 0 and 1. Internal calcium concentration is usually between
0 and a few hundred micro molar.

Each time I read the descriptions of the gsl_odeiv_control functions I
become more convinced that the parameters eps_rel, a_y, and a_dydt are
there to excuse some error in variables and derivatives that are far
from zero. Is that correct?

Since I have no reason to care more about values close to zero (unless I
do some complicated remapping of the variable values) should I set
eps_rel, a_y, and a_dydt to 0?

Since my variables have different ranges should I use
gsl_odeiv_control_scaled_new with an appropriate eps_abs and scale_abs
set with something like 1 for internal calcium concentration, 0.3 for
membrane voltage, and 0.001 for all of the gating variables?


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Re: ODE IV Control and Variable Scale

[Tuomo Keskitalo]

Hi,

about ODE solver error tolerances: you _never_ want to set eps_rel to 
zero. It would mean that you allow no error for the ODE solver, which is 
something no numerical method can give you.

As an example: gsl_odeiv_control_y_new (1e-12, 1e-8) means that if the 
absolute value of a variable drops below 1e-12, you don't really care of 
it's value any more (it is essentially zero for you). eps_rel = 1e-8 
means that you want at least 7 decimals of each variable to be accurate 
on each ODE solver step. If you need to control the level of error with 
more detail, then you can use a_y, a_dydt and scale_abs.

A practical way to choose tolerances is to make a test and compare 
results calculated with "strict" and "loose" eps_abs and eps_rel (order 
of magnitude difference between strict and loose). If you get nearly 
same values as result, you are probably safe to use your loose 
tolerances. Of course, this depends on your problem, so be sure to test 
it with your final computation case again.

If you are not interested in optimizing the performance, I'd suggest you 
simply try to use control_y_new and see how it works for you.

Regards,
Tuomo

On 03/18/2010 08:27 PM, Kevin H. Hobbs wrote:

> I'm trying make sure that I am using the gsl_odeiv_control functions
> correctly.
> 
> I'm modelling neurons and my variables have different scales. Voltage is
> usually between -100 and 50 mV. The membrane current gating variables
> stay between 0 and 1. Internal calcium concentration is usually between
> 0 and a few hundred micro molar.
> 
> Each time I read the descriptions of the gsl_odeiv_control functions I
> become more convinced that the parameters eps_rel, a_y, and a_dydt are
> there to excuse some error in variables and derivatives that are far
> from zero. Is that correct?
> 
> Since I have no reason to care more about values close to zero (unless I
> do some complicated remapping of the variable values) should I set
> eps_rel, a_y, and a_dydt to 0?
> 
> Since my variables have different ranges should I use
> gsl_odeiv_control_scaled_new with an appropriate eps_abs and scale_abs
> set with something like 1 for internal calcium concentration, 0.3 for
> membrane voltage, and 0.001 for all of the gating variables?
> 


-- 
Tuomo.Keskitalo@iki.fi
http://iki.fi/tuomo.keskitalo

Re: ODE IV Control and Variable Scale

[Kevin H. Hobbs]

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On 03/20/2010 03:18 PM, Tuomo Keskitalo wrote:
> Hi,
> 
> about ODE solver error tolerances: you _never_ want to set eps_rel to
> zero. It would mean that you allow no error for the ODE solver, which is
> something no numerical method can give you.

The ODE example program sets eps_rel to 0.0.

> 
> As an example: gsl_odeiv_control_y_new (1e-12, 1e-8) means that if the
> absolute value of a variable drops below 1e-12, you don't really care of
> it's value any more (it is essentially zero for you). 

Right now I'm setting eps_abs to something like 1.0e-3.

As far as I can tell from the documentation there's nothing special
about a variable value of 0.0 when eps_rel is 0.0. 1.0e-3 is the same as
0.0 and 100 + 1.0e-3 is the same as 100.

I want a constant error across the range of the variables.

eps_rel = 1e-8
> means that you want at least 7 decimals of each variable to be accurate
> on each ODE solver step. If you need to control the level of error with
> more detail, then you can use a_y, a_dydt and scale_abs.

The trouble I have with these parameters is they depend on 0.0 being a
special value, but in my model there is nothing special about 0.0 for
most of the variables.

Voltage is the major example. Membrane voltage ranges from the reversal
potential for potassium E_K = -80 mV to the reversal potential of sodium
E_Na = 50 mV. The values of membrane voltage where the various membrane
currents turn on and off are the critical values, but there are many of
them and they are spread across the voltage range.

> 
> A practical way to choose tolerances is to make a test and compare
> results calculated with "strict" and "loose" eps_abs and eps_rel (order
> of magnitude difference between strict and loose). If you get nearly
> same values as result, you are probably safe to use your loose
> tolerances. Of course, this depends on your problem, so be sure to test
> it with your final computation case again.

We have a whole graduate student devoted to this testing right now.

> 
> If you are not interested in optimizing the performance, I'd suggest you
> simply try to use control_y_new and see how it works for you.
> 

We are very much interested in performance since we're trying to
evaluate millions of models per day.

It appears that by using gsl_odeiv_control_scaled_new with scales
proportional to the range of each variable, eps_rel = 0.0, a_y = 0.0,
and a_dydt = 0.0 we can relax eps_abs and evaluate more models with
greater accuracy.


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Re: ODE IV Control and Variable Scale

[Tuomo Keskitalo]

Hi,

sorry, I did not understand your case correctly the first time. I often 
want to make sure that (fast) changes in variable values are followed 
precisely (my problems tend to be stiff) and therefore use eps_rel 
dominantly. You are right that eps_rel may well be 0. Depends on your 
problem.

On 03/22/2010 04:48 PM, Kevin H. Hobbs wrote:

> On 03/20/2010 03:18 PM, Tuomo Keskitalo wrote:
>> Hi,
>>
>> about ODE solver error tolerances: you _never_ want to set eps_rel to
>> zero. It would mean that you allow no error for the ODE solver, which is
>> something no numerical method can give you.
> 
> The ODE example program sets eps_rel to 0.0.
> 
>> As an example: gsl_odeiv_control_y_new (1e-12, 1e-8) means that if the
>> absolute value of a variable drops below 1e-12, you don't really care of
>> it's value any more (it is essentially zero for you). 
> 
> Right now I'm setting eps_abs to something like 1.0e-3.
> 
> As far as I can tell from the documentation there's nothing special
> about a variable value of 0.0 when eps_rel is 0.0. 1.0e-3 is the same as
> 0.0 and 100 + 1.0e-3 is the same as 100.
> 
> I want a constant error across the range of the variables.
> 
> eps_rel = 1e-8
>> means that you want at least 7 decimals of each variable to be accurate
>> on each ODE solver step. If you need to control the level of error with
>> more detail, then you can use a_y, a_dydt and scale_abs.
> 
> The trouble I have with these parameters is they depend on 0.0 being a
> special value, but in my model there is nothing special about 0.0 for
> most of the variables.
> 
> Voltage is the major example. Membrane voltage ranges from the reversal
> potential for potassium E_K = -80 mV to the reversal potential of sodium
> E_Na = 50 mV. The values of membrane voltage where the various membrane
> currents turn on and off are the critical values, but there are many of
> them and they are spread across the voltage range.
> 
>> A practical way to choose tolerances is to make a test and compare
>> results calculated with "strict" and "loose" eps_abs and eps_rel (order
>> of magnitude difference between strict and loose). If you get nearly
>> same values as result, you are probably safe to use your loose
>> tolerances. Of course, this depends on your problem, so be sure to test
>> it with your final computation case again.
> 
> We have a whole graduate student devoted to this testing right now.
> 
>> If you are not interested in optimizing the performance, I'd suggest you
>> simply try to use control_y_new and see how it works for you.
>>
> 
> We are very much interested in performance since we're trying to
> evaluate millions of models per day.
> 
> It appears that by using gsl_odeiv_control_scaled_new with scales
> proportional to the range of each variable, eps_rel = 0.0, a_y = 0.0,
> and a_dydt = 0.0 we can relax eps_abs and evaluate more models with
> greater accuracy.
> 


-- 
Tuomo.Keskitalo@iki.fi
http://iki.fi/tuomo.keskitalo