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From: Alan W. Irwin <irwin@be...>  20061221 22:11:47

On 20061221 14:430500 Aaron Dotter wrote: > Hello Alan, > > I found your report from earlier this week intriguing. Making stellar models > down to the Hburning limit is a particular goal of mine and I've had limited > success doing it with FreeEOS but only by cheating in a couple of different > ways. > Hi Aaron: I would like to encourage more list discussion of FreeEOS problems by everybody that is using that software since that is an excellent way to motivate improvements to it. I also strongly encourage those who are using FreeEOS to give a short summary of their various research projects where FreeEOS is currently used or where there may be future plans to use it. Thus, if there is anything list members wish to to discuss about FreeEOS, I strongly encourage them to post their email to freeeosgeneral@... rather than me. In that spirit, Aaron, I hope that you don't mind that I am moving my reply to your comment above to the list. Thanks, Aaron, for your continuing interest in the solution of the highdensity problem for FreeEOS. Your results (you have to cheat to get much below 0.1 solar masses with the current FreeEOS) confirms results Don VandenBerg calculated with the original version of FreeEOS and also results by Pietrinferni with a later version of FreeEOS that he used for his thesis work. The problem is FreeEOS (like other equations of state) yields at least a double solution for high densities, i.e., there is simultaneously a valid mostly unionized solution and a valid mostly ionized solution for a range of high densities for a given isotherm. If you move along an isotherm toward higher densities you eventually run into density where the low ionization solution is no longer valid, and the whole solution wildly diverges because the initial solution (inferred from a Taylor series based on a slightly higher density) is so far from the valid highionization solution. This discontinuity is also found in other equations of state, but typically ~1 dex higher in density than for FreeEOS. So sub0.1 solarmass models are affected for FreeEOS while higherdensity models (i.e. hot browndwarf models beyond the hydrogenburning limit) are affected for the other equations of state. Ideally I would like move the FreeEOS discontinuity ~1 dex higher in density while simultaneously preserving the good fit I now have to OPAL work at lower (especially solar) densities. To do that I need to calculate EOS grids that extend to high density to see how the location of the discontinuity in the rhoT plane is affected by the various Coulomb and pressureionization coefficients I use in FreeEOS to fit the OPAL work. Of course, calculating those grids is extremely laborious if you have to hand intervene to get the FreeEOS solution to converge each time it hits a discontinuity for _every_ isotherm. So that is my motivation for trying yet again to get the BFGS technique to work in the robust manner that all optimization books claim it fundamentally must have compared to NewtonRaphson techniques. I believe I now understand all the negative results I have had before so I hope this time I have the BFGS answer with regard to the FreeEOS formulation of the EOS. Thus, with luck, I hope to to report a positive result to this list in the next few weeks, but there is lots of derivative programming I have to do before I know for sure. Alan __________________________ Alan W. Irwin Astronomical research affiliation with Department of Physics and Astronomy, University of Victoria (astrowww.phys.uvic.ca). Programming affiliations with the FreeEOS equationofstate implementation for stellar interiors (freeeos.sf.net); PLplot scientific plotting software package (plplot.org); the Yorick frontend to PLplot (yplot.sf.net); the Loads of Linux Links project (loll.sf.net); and the Linux Brochure Project (lbproject.sf.net). __________________________ Linuxpowered Science __________________________ 
From: Alan W. Irwin <irwin@be...>  20061220 03:51:43

This is a status report for FreeEOS for extreme conditions (i.e., the conditions in the envelopes of extreme LMS stars) where it has convergence difficulties. When I released FreeEOS2.0.0 last January I thought I had nearly finished a BFGS implementation that promised to give robust initial convergence wherever my normally trusty NewtonRaphson iteration was diverging. I am now sadder but wiser. Here are some issues I encountered: * The BFGS algorithm for finding the minimum of a function should do that task robustly. All it should need are the function and gradient as functions of the independent variables. However, the BFGS algorithm implementation I first tried turned out to be nonrobust. Fortunately, "Practical Methods of Optimization" by Fletcher (where Fletcher is the "F" in the author initials making up the BFGS acronym) gives a clear description of the BFGS algorithm which allowed me to implement my own independent version which I completely understand. Fletcher already discusses certain robustness issues and how his algorithm avoids them, but if any more turn up during my FreeEOS experiments, I should be in a good position to deal with them. * If incorrect independent variables are chosen the minimization of the free energy obviously does not correspond to the solution of the EOS. I was bitten twice by this problem with lots of useless gradient programming as a result. * Even when correct independent variables are chosen, you can run into problems with the robustness of maintaining the constraints (such as the abundance conservation and overall charge neutrality equations) that must be satisfied by the EOS solution. This bit me several times with much useless programming as a result. So I am now in the middle of doing a "final" BFGS implementation that has the same auxiliary variables I iterate to convergence in my NewtonRaphson scheme as the independent variables. These variables are welldesigned to be free of limit issues (aside from the usual numerical underflow and overflow floatingpoint limits). An iteration on the degeneracy parameter insures minimization of the free energy using these variables is consistent with the solution of the EOS. Here are some additional details of that last absolutely essential point. Normally for fixed T and abundance, the FreeEOS solution determines number densities and mass density, rho, as functions of the degeneracy parameter, fl, and a set of auxiliary variables. (See Paper II at http://freeeos.sourceforge.net/solution.pdf for details.) For FreeEOS2.0.0 both fl and the set of auxiliary variables are simultaneously iterated (using the NewtonRaphson technique) to convergence defined by consistency between the calculated rho and input rho, and consistency between auxiliary variables calculated from the number densities and the input auxiliary variables. When that NewtonRaphson technique fails to converge, then the formulation will be changed as follows: for fixed input auxiliary variables, iterate just on fl for consistency between the calculated and input rho. In this way, fl is implicitly eliminated in terms of the input rho so the solution of the EOS and in particular the number densities become implicit functions of just the input auxiliary variables for fixed (input) rho, T, and abundance. Under these special conditions (and only these conditions) with fl implicitly eliminated, it is straightforward to show that minimizing the free energy as a function of auxiliary variables is equivalent to minimizing the free energy as a function of number densities which (see Paper II) is exactly equivalent to the solution of the EOS using the NewtonRaphson technique. The only conditions on this general theoretical result are (1) the iteration on fl with fixed input auxiliary variables must converge, and (2) the derivatives of number densities with respect to those auxiliary variables must not be zero. These two conditions are also necessary for the NewtonRaphson technique to converge so a BFGS minimization based on this formulation should not be less robust than the NewtonRaphson technique in this respect, and should be considerably more robust when the NewtonRaphson technique is diverging for other reasons (such as having a poor starting approximation). This fundamental theoretical analysis has encouraged me to move ahead with the implementation I have outlined above, and recently I have been making good progress on it (the fl iteration is done), but there is considerable programming and a lot of testing still to go to express the free energy and its gradient in terms of the auxiliary variables with fl implicitly eliminated. If this final BFGS implementation performs as expected (and I am obviously somewhat concerned about that issue since I have already spent quite an effort on the BFGS idea with few positive results, yet), then I should be able to calculate FreeEOS for much more dense conditions automatically without any hand intervention. That in turn should make it much more convenient to adjust the semiempirical Coulomb and pressureionization coefficients in FreeEOS to get a better match with Forrest Rogers' recent OPAL EOS results for his highest tabulated densities (roughly corresponding to conditions in a ~0.1 solarmass model) while constraining the fit in the following two ways: * Maintain the existing excellent fit to OPAL results for conditions in a solarmass model (and by implication all models with relatively low densities). * Give a physically reasonable and smooth extrapolation at higher densities than covered by OPAL so at least the hydrogen burning limit of the main sequence near 0.07 solar masses can be reached with FreeEOS. Anyhow, that constrained OPAL fit is the dream that has motivated this BFGS effort since FreeEOS2.0.0 was released. I feel I am finally on the right track, but some substantial gradient programming still needs to be completed before I will know for sure. Alan __________________________ Alan W. Irwin Astronomical research affiliation with Department of Physics and Astronomy, University of Victoria (astrowww.phys.uvic.ca). Programming affiliations with the FreeEOS equationofstate implementation for stellar interiors (freeeos.sf.net); PLplot scientific plotting software package (plplot.org); the Yorick frontend to PLplot (yplot.sf.net); the Loads of Linux Links project (loll.sf.net); and the Linux Brochure Project (lbproject.sf.net). __________________________ Linuxpowered Science __________________________ 