CoolDwarf.EOS.invert package

Submodules

CoolDwarf.EOS.invert.EOSInverter module

EOSInverter.py – Inverter class for EOS tables

This module contains the Inverter class for EOS tables. The class is designed to be used with the CoolDwarf Stellar Structure code, and provides the necessary functions to invert the EOS tables. Because the inversion problem is non-linear, the Inverter class uses the scipy.optimize.minimize function to find the solution.

Further, because EOSs may not be truley invertible, the Inverter class uses a loss function to find the closest solution to the target energy. over a limited range of temperatures and densities. This is intended to be a range centered around the initial guess for the inversion and limited in size by some expected maximum deviation from the initial guess.

Dependencies

• CoolDwarf.utils.misc.backend

• CoolDwarf.err.EOSInverterError

Example usage

>>> from CoolDwarf.EOS.invert.EOSInverter import Inverter
>>> from CoolDwarf.EOS.ChabrierDebras2021.EOS import CH21EOS
>>> eos = CH21EOS("path/to/eos/table")
>>> inverter = Inverter(eos)
>>> logTInit, logRhoInit = 7.0, -2.0
>>> energy = 1e15
>>> logT, logRho = inverter.temperature_density(energy, logTInit, logRhoInit)

class CoolDwarf.EOS.invert.EOSInverter.Inverter(EOS, tol=1e-07, maxDepth=1000)

Bases: object

Inverter – Inverter class for EOS tables

This class is designed to be used with the CoolDwarf Stellar Structure code, and provides the necessary functions to invert the EOS tables. The Inverter class uses PyTorch optimizers to find the solution to the non-linear inversion problem. Because EOSs may not be truly invertible, the Inverter class uses a loss function to find the closest solution to the target energy over a limited range of temperatures and densities. This is intended to be a range centered around the initial guess for the inversion and limited in size by some expected maximum deviation from the initial guess.

Parameters:

EOSEOS

EOS object to invert

tolfloat, default=1e-6

The minimum allowed median fractional energy error

maxDepthint, default=1000

The maximum number of recursions allowed.

Attributes:

EOSEOS

EOS object to invert

tolfloat

The minimum allowed median fractional energy error

maxDepthint

The maximum number of recursions allowed

Methods

temperature_density(energy, logTInit, logRhoInit,f=0.01)

Inverts the EOS to find the temperature and density that gives the target energy

temperature_density(energy: ndarray, temperature: ndarray, density: ndarray, f: float = 0.01, _rDepth: int = 0) → ndarray

Given the target energy, temperature and density, find the temperature and density that gives the target energy. This is dones by makining the assumption that the EOS is linear in the range of temperatures and densities given by the bounds. If this is true then a function rho(E) at some constant temperature is well defined.

We define two functions: rho(E)_{T0} and rho(E)_{T1} as the density as a function of energy at two constant temperatures. These temperatures are taken as some fraction (f) less than the initial temperature guess and that same fraction greater than the initial temperature guess.

Once these two linear functions have been found we evaluate them at the target energy. This gives us the density which results in the target energy at two different constant temperaratures. We can then fit a third linear function rho(T) using these two points to pull out a linear approximation for the isoenergy curve over the search domain.

The question then becomes: where along this isoenergy curve will the grid point move to. Any temperature and density on that curve will result in the same final energy. We can think here about some arbitrary path from the initial conditions to a point on the isoenergy curve. Every path has some path integral in energy. The most likeley destination is the path which minimizes the path integral.

Because there is an infinite search space and we do not have an analytic function we need to make some simplifying assumptions to actually solve this. We observe that over a limited search domain the equation of state is continous and smooth. Further, it monotonically increases with temperature and density. This means that the path which minimizes the path integral of energy should be the shortest distance (in temperature, density space) between the initial condition and the isoenergy curve.

Finding this path is then as simple as finding the line perpendicular to the isoenergy curve which pases through the initial condition and then solving for where this line is equal to the isoenercy curve.

Once we have found the point on the isoenergy curve which is closest to the initial condition we can then evaluate the EOS at that point to find the final energy. We then check the error against the target energy and if the error is greater than the tolerance we recurse with a smaller search domain and a new initial guess for the temperature and density based on the previous optimization. This continues until the error is less than the tolerance or the maximum recursion depth is reached.

The procedure described above is preformed simultaneously for all grid points and has been formulated as a pure matrix problem. Because of this is is very efficient.

Parameters:

energyxp.ndarray

Target energy to invert the EOS to

temperaturexp.ndarray

Initial guess for the temperature. This should be in linear space NOT log space.

densityxp.ndarray

Initial guess for the density. This should be in linear space NOT in log space.

ffloat, default=0.01

Fraction of the initial guess to use for the bounds

_rDepthint, default=0

Current recursion depth

Returns:

xp.ndarray

New temperature

xp.ndarray

New density

Raises:

EOSInverterError

If the maximum recursion depth is reached before the error tolerance is met

Notes

If you are using an equation of state which is not as well behaved as the Chabrier Debras 2021 EOS the assumptions I made here may not work. Notebaly, you will need to check if, within the search domain, the energy varies linearly with density at a constant temperature. And if, again within the search domain, if the isoenercy curve is linear in density and temperature space. If these are true then the algorithm to find the isoenergy curve should still be valid. Secondly, you will need to validate that the shortest path between the initial condition and the isoenergy curve is the one which minimizes the energy path integral. If that is also true then this method should reliably find the target energy.

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