Introduction


NOTE: this page is out-of-date because the option keywords referred to here (indicated in red) are modified or obsolete in 6.8.6+ versions of Perple_X, the page is nonetheless useful if the relation of the current options (indicated in blue) to the former options is understood: 

 

iteration keyword - values i1 and i2, i1 is now (6.8.4+) specified by the resolution_factor keyword and i2 is specified by the refinement_points keyword.

initial_resolution - prior to 6.8.6, initial resolution took one value that specified the static compositional resolution for the exploratory stage of an auto-refine calculation. The static compositional resolution during the auto-refine stage was then computed as initial_resolution/auto_refine_factor_I. In 6.8.6 initial_resolution takes two values, v1 and v2, that correspond to the static compositional resolution during the exploratory and auto-refine stages of an auto-refine calculation. Thus, where the text below refers to a value of auto_refine_factor_I the same effect in the current version of Perple_X is obtained by setting v2 = v1/auto_refine_factor_I. i.e., increasing auto_refine_factor_I had the same affect as decreasing the v2.

 

A brief discussion of adaptive minimization keyword group for the current (6.8.6) version of Perple_X is at Perple_X_why_it_often_does_not_work.


 

The precision and accuracy of phase diagram section and phase fractionation calculations in Perple_X hinges on two groups of runtime keywords specified in the perplex_option.dat file. The adaptive minimization keyword group (iteration, initial_resolution, final_resolution, auto_refine_factor_I, reach_increment, global_reach_increment, solvus_tolerance and solvus_tolerance_II) determines the accuracy and precision with which Perple_X computes phase relations at an individual grid point in a phase diagram section. The grid parameters keyword group (grid_levels, x_nodes, y_nodes, 1d_path, and linear_model) determines the number of grid points used to map the phase relations of a phase diagram section. This page is concerned only with the first group. The relevant grid parameters used in the examples presented here, have been chosen to obviate the mapping of phase relations as a prominent source of error. The first example discussed here primarily illustrates of the role of the final_resolution keyword, while the second example illustrates the role of reach_increment, auto_refine_factor_I, and solvus_tolerance_II keywords in resolving problems related to the spurious and stepped solvi that may appear in Perple_X calculations involving non-ideal solution models. 

 

The challenge in using Perple_X effectively is to find keyword values that produce a high quality result at minimal computational cost. In brief, the keywords have the following roles (refer to perplex_options.html for more detailed descriptions):

 

initial_resolution - is the initial (molar) compositional resolution during the exploratory stage (see auto-refine) of a calculation. Decreasing the value of this keyword always improves the quality of a calculation, but is particularly expensive in terms of memory. The diagnostic **error ver041** during the exploratory stage of a calculation indicates initial_resolution is too fine. Typically a value between 1e-1 and 5e-2 is feasible, small changes in this value have large influence on memory requirements.

 

auto_refine_factor_I - initial_resolution/auto_refine_factor_I is the effective initial resolution during the final (auto-refine) stage of a calculation. This keyword permits finer effective initial resolution during the auto-refine stage than would be possible in the exploratory stage due to memory limitations. Increasing the value of this keyword always improves the quality of a calculation. The diagnostic **error ver041** during the auto-refine stage of a calculation indicates auto_refine_factor_I is too large. Typically a value between 2 and 8 is feasible. Calculations with the same initial_resolution to auto_refine_factor_I ratio should yield essentially identical results. 

 

final_resolution - is the final (molar) compositional resolution. During adaptive minimization, compositions are iteratively refined until their precision is better than the specified value of final_resolution. The final_resolution does not affect memory usage, but because it controls the number of iterations it may have a strong influence on the amount of time required for calculations, particularly for calculations with multicomponent solution phases. Small values of final_resolution may cause numerical instability, the diagnostic **warning ver042** indicates this condition (e.g., Fig 2c). Until this threshold is reached, decreasing final_resolution always improves the quality of computational results. As a rule of thumb, on 32-bit machines, numerical stability becomes problematic if final_resolution <10-4 mol.

 

iteration - this keyword takes two values, i1 and i2, the default values for this keyword are probably optimal. i1 is the factor by which compositional precision increases in each iteration, the lowest value permitted for i1 is two, increasing i1 from this value decreases the amount of time required for adaptive minimization (Connolly 2009), but increases memory consumption (diagnostic **error ver058**). i2 controls Perple_X's ability to distinguish stable phases from nearly stable phases and is the number of (perceived) metastable phases considered during adaptive minimization. As a consequence i2 has a strong influence on the accuracy of phase relations in close proximity to phase boundaries. Experience suggests that increasing i2 beyond its default (4) yields little if any improvement to computational results, while decreasing this value does not have major benefits in terms of computational resources. 

 

solvus_tolerance - this keyword specifies the minimum distance (in normalized composition space) between two compositions of a solution phase at which the compositions are considered to represent distinct (immiscible) phases of the solution. Compositions that are separated by a distance less than solvus_tolerance are considered to represent a single phase whose composition is computed as the weighted average of the different compositions of the solution. For reasons discussed below, the default value for this tolerance is much larger than the theoretical resolution of the adaptive minimization algorithm. The effect of this parameter is cosmetic.

 

solvus_tolerance_IIthis keyword specifies a parameter that is used as a criterion for selecting the metastable phase compositions that are to be refined during adaptive minimization. This parameter can be adjusted to eliminate stepped solvus boundaries along which the compositions of the immiscible phases of a solution change by increments more closely related to the effective initial_resolution than the requested final resolution for the calculation. This keyword is discussed in greater detail at perplex_options.html, see also Example 2 below. 

 

reach_increment (and/or global_reach_increment) - this keyword can be used to force Perple_X to widen the interval over which the estimated composition of a phase may change between successive iterations during adaptive minimization. This parameter can be adjusted to eliminate spurious solvi that may occur in calculations than involve solution models that predict immiscibility. This keyword is discussed in greater detail below. 


Example 1: Pelite Model, Effect of final_resolution, auto_refine_factor_II and reach_increment

 

The phase diagram sections below illustrate the effect of changing the final_resolution, auto_refine_factor_II and reach_increment keywords on the calculation of phase relations involving immiscible feldspar phases. Unless otherwise indicated the adaptive minimization keyword group have their default values, the grid parameters keyword group values (grid_levels = 4, x_nodes = y_nodes = 100) are slightly finer than the default. The input files for this calculation are here.

 

 

Figure 1. The phase diagram section obtained with the default adaptive minimization keyword values (initial_resolution = 0.067, auto_refine_factor_I = 2, iteration = [3, 4], solvus_tolerance = auto, solvus_tolerance_II = 0.25). The phase names have been edited for clarity. Phase relations in this section are complicated by the existence of a solvus for the feldspar (Fsp) solution model. This solvus is indicated in the output by the phase fields in which two (or more) coexisting phases are represented by the same solution model (e.g., the phase field Chl+Pheng+St+Fsp+Fsp+Bio). The boundaries of the solvus are recognized by the conditions at which the only one feldspar is stable (e.g., the boundary between the Chl+Pheng+St+Fsp+Fsp+Bio and  Chl+Pheng+St+Fsp+Bio phase fields). With the default adaptive minimization keyword settings some of these phase boundaries are poorly resolved, additionally some spurious solvi are also present (these flaws are circled by red curves in Fig 2b). In Perple_X, characteristically, spurious solvi have more phases than are stable in the correct result and are often patchy or have geometries that look unnatural (e.g., they often form striped patterns). Spurious solvi can also be recognized by the fact that their location is usually sensitive to computational parameters such as initial_resolution and final_resolution. Spurious solvi can always be eliminated by increasing the value of the solvus_tolerance keyword, however this approach to eliminating the solvi hides rather than solves the underlying problem. Moreover using large values of the solvus_tolerance keyword increase the probability that truly immiscible phase compositions will be homogenized.


Figure 2. [Below]. Results obtained with inadequate (a), default (b), and excessive (c) values of the final_resolution keyword (in b numbers indicate the number of feldspar phases stable in each field), text labels have been suppressed for clarity. Inadequate final_resolution leads to irregular phase boundaries and spurious solvi. With the default final_resolution (b) the extent of spurious solvi is reduced and all non-solvus phase boundaries are well resolved. In (c) excessive final_resolution has caused numeric instability manifest by irregular white patches where the adapative minimization algorithm has failed to converge (these patches can be distinguished from spurious solvi from the indicated phase assemblage, e.g., if Perple_X does not converge, WERAMI will indicate that there is no data at the relevant conditions). 

 

 

The calculations illustrated in Fig 2 indicate that the default value of the final_resolution keyword is appropriate for the calculation. The fact that the prominent flaws in Fig 2b are associated with a solvus boundary indicates that these flaws are related to difficulties in converging to the correct compositions of the immiscible solution. These difficulties arise in Perple_X because the adaptive minimization algorithm assumes that if phase compositions have been discretized with resolution dn, then the accuracy for any compositional variable xn is dn. Therefore during iterative refinement the compositional estimate is not allowed to move outside the interval xndn, this has the consequence that if two compositions are in reality miscible, but separated by a distance > dn, the compositional estimates cannot converge during iteration. Thus, although Perple_X usually resolves phase compositions with accuracy ~final_resolution, in the case of solvi the final resolution may be limited by the effective initial resolution (i.e., initial_resolution/auto_refine_factor_I), hence if solvus_tolerance is set to automatic, then the value of the tolerance is generously estimated as 3/2*initial_resolution/auto_refine_factor_I. These considerations suggest two ways of resolving solvus related problems, the first is to use finer effective initial resolution, and the second is to widen the interval over which phase compositions are allowed to vary. Unfortunately, there is no universal solution for these problems, so trial and error may be necessary to achieve high quality results. From the perspective of minimizing computational cost, it is less expensive (and only slightly less accurate) to obtain finer effective initial resolution by increasing auto_refine_factor_I than it is by reducing the value of initial_resolution. For the present example, increasing the effective resolution by doubling auto_refine_factor (Fig 3a) does little to improve the computational result. The range over which compositions are allowed to vary during iteration can be changed for all phases by specifying a non-zero value for the global_reach_increment keyword, in which case the range is xnglobal_reach_increment*dn. However, for many problems the cost of increasing the range globally is prohibitive, an inexpensive alternative is to only widen the range of the solution phase(s) that exhibits immiscibility by specifying a reach_increment for that solution. A complication in this regard is that the reach_increment must be specified in the solution model file (e.g., solution_model.dat, Fig 4) rather than the runtime option file (e.g., perplex_option.dat). For the present example, applying a reach_increment of 6 to the feldspar solution model eliminates the spurious solvi, but does not completely eliminate rough solvus boundaries. However, increasing auto_refine_factor_I together with reach_increment does lead to a good quality result (Fig 3c).

 

Figure 3. Neither increasing auto_refine_factor_I (a) nor the reach_increment for the feldspar model (b) in isolation is completely successful in eliminating the flaws apparent in the calculation depicted in Fig 2a, but applying both changes in tandem produces a high quality result (c).

 

 

 

begin_model                
                           | ternary feldsar (fuhrman & lindsley, am min, 1988)
feldspar                   
2                          | model type: Margules or Ideal
3                          | 3 endmembers
abh      
an       
san      
0 0 0                      | endmember flags = 0 if the endmember is part of the solution.
0. 1. 0.1   0              | range and resolution for albite, imod = 0 -> cartesian subdivision
0. 1. 0.1   0              | range and resolution for anorthite, imod = 0 -> cartesian subdivision

begin_excess_function 
w(abh abh san) 27320.   -10.3  .394 
w(abh san san) 18810.   -10.3  .394 
w(an  an  san) 52468.      .0   .0   
w(an  san san) 47396.      .0  -.12  
w(an  an  abh) 28226.      .0   .0   
w(an  abh abh) 8471.       .0   .0   
w(an  abh san) 100045.5 -10.3 -0.76 
end_excess_function 

2                          | 2 site (O-site and T-site) entropy model

3 1.                       | 3 species on O-site, 1 site per formula unit.
z(Na) = 1 abh
z(Ca) = 1 an

2 2.                       | 2 species on T-site, 2. sites per formula (al-avoidance model)
z(Al) = 1/2 + 1/2 an 

reach_increment 6         

end_of_model       

Figure 4. To widen the range over which the composition of a specific solution is allowed to vary during adaptive minimization, the reach_increment keyword must be added to the solution model (e.g., in solution_model.dat) immediately before the end_model keyword, as illustrated here (and highlighted in red) for the feldspar model (NOTE: there is no equals sign between a keyword and its value).



Example 2: Spurious and Stepped Solvi: the Effect of reach_increment and solvus_tolerance_II

 

The phase diagram sections below illustrate the effect of changing the reach_increment, auto_refine_factor_II, and solvus_tolerance_II keywords on the calculation of phase relations involving immiscible clinopyroxene phases represented by the Omph(GHP) solution model. Unless otherwise indicated the adaptive minimization keyword group have their default values, the grid parameters keyword group values (grid_levels = 4, x_nodes = y_nodes = 40) lead to somewhat coarser mapping of the phase relations than illustrated in the previous example. The input files for this calculation are here

 

 

Figure 5. The phase diagram section obtained with the default adaptive minimization keyword values (initial_resolution = 0.067, auto_refine_factor_I = 2, iteration = [3, 4], solvus_tolerance = auto, solvus_tolerance_II = 0.25). The phase names have been edited for clarity. Phase relations in this section are complicated by immiscibility predicted by the clinopyroxene (Omph) solution model. The immiscibility is manifest in the Perple_X calculation by both real and spurious solvi (solvi are phase fields in which two, or more, coexisting phases are represented by the same solution model, e.g., Omph+Omph+F+Mica+Do+zo+sph+q+arag). The red numerals indicate the number of stable phases in each phase field that correspond to the clinopyroxene model. In Perple_X, characteristically, spurious solvi have more phases than are stable in the correct result and are often patchy or have geometries that look unnatural (e.g., they often form striped patterns, e.g., within the area circled in red). Spurious solvi can also be recognized by the fact that their location is usually sensitive to computational parameters such as initial_resolution and final_resolution. In addition to the spurious solvi, this calculation also illustrates a second problem referred to here as a "stepped" solvus (circled in blue). In Perple_X, characteristically, stepped solvi are recognized by highly irregular (though often periodic) boundaries, these irregularities correspond to conditions at which one of the immiscible phases abruptly changes composition. Fig 6 illustrates how the reach_increment parameter can be used to eliminate the spurious solvi, while Fig 7 illustrates how solvus_tolerance_II can be manipulated to eliminate the stepped solvus. Spurious solvi can always be eliminated by increasing the value of the solvus_tolerance keyword, however this approach to eliminating the solvi hides rather than solves the underlying problem. Moreover using large values of the solvus_tolerance keyword increase the probability that truly immiscible phase compositions will be homogenized.


Spurious solvi arise in Perple_X because the adaptive minimization algorithm assumes that if phase compositions have been discretized with resolution dn, then the accuracy for any compositional variable xn is dn. (where the subscript n indicates the values in the nth iteration of an adaptive minimization). During iterative refinement the compositional estimate is not allowed to move outside the interval xndn-1, this has the consequence that if two compositions are in reality miscible, but separated by a distance > dn, the compositional estimates cannot converge during iteration. Thus, although Perple_X usually resolves phase compositions with accuracy ~final_resolution, in the case of solvi the final resolution may be limited by the effective initial resolution (i.e., initial_resolution/auto_refine_factor_I). For this reason, if solvus_tolerance is set to automatic (as recommended), then the value of the tolerance is generously estimated as 3/2*initial_resolution/auto_refine_factor_I. These considerations suggest two ways of eliminating spurious solvi (aside from hiding them by increasing solvus_tolerance), the first is to use finer effective initial resolution, and the second is to widen the interval over which phase compositions are allowed to vary. In general the default settings in the runtime option file are such that the effective initial resolution cannot be increased significantly (see Fig 3a), accordingly here the range over which compositions are allowed to vary during iteration is changed. This range can be changed for all phases by specifying a non-zero value for the global_reach_increment keyword, in which case the range is xnglobal_reach_increment*dn. However, for many problems the cost of increasing the range globally is prohibitive, an inexpensive alternative is to widen the range of only the solution phase(s) that exhibits immiscibility by specifying a reach_increment for that solution. A minor complication in this regard is that reach_increment must be specified in the solution model file (e.g., solution_model.dat, Fig 4) rather than the runtime option file (e.g., perplex_option.dat). For the present example, applying a reach_increment of six to the clinopyroxene solution model eliminates the spurious solvi (Fig 6) that were initially present (Fig 5). 

 

Even with a reach_increment of six the calculated section retains a number of flaws, these include some wavy or irregular phase boundaries (circled in red in Fig 6), the stepped solvus (circled in blue), and the seemingly univariant boundary between sphene-bearing assemblages at low pressure and rutile-bearing assemblages at high pressure (circled in green). The univariant rutile-sphene boundary is plausible topologically, but improbable because it implies that the corresponding reaction does not effect the composition of omphacite as it evidently does at higher temperature where the transition corresponds to a narrow divariant field that extends to the (stepped) omphacite solvus which then degenerates back to a univariant boundary. There is no universal rule that governs the course of action to be taken to eliminate/resolve these flaws, but my personal guiding principal is to maximize both the initial and final resolution (in this regard I remark that there is no reason to use the high mapping resolution specified by the grid parameters keyword group for this problem, thus if the time required by high compositional resolution is a concern, this concern can be alleviated by using much lower mapping resolution). In the present case, finer final_resolution is not an option because values below 2e-4 cause numerical instabilities (warning ver042), the effective initial resolution can be improved by either reducing initial_resolution or increasing auto_refine_factor_I. Here I opt for the latter as it is generally more efficient, and after some experimentation find that I can increase auto_refine_factor_I to at least 17.5 (corresponding to an effective initial resolution of 0.0038 mol) before VERTEX runs out of memory (error ver041). The result of this high resolution calculation (Fig 7a) is satisfying in that the sphene-rutile boundary (circled in green) is now adequately resolved, but the stepped solvus (circled in blue) persists and small spurious solvi (circled in red) have reappeared. The spurious solvi are eliminated by increasing the reach_increment for the Omph(GHP) model to eight (Fig 7b), but because the stepped solvus is extremely narrow in composition (~0.03 mol) eliminating this flaw by further increases in initial resolution is prohibitively expensive. A less expensive alternative course of action is to reduce solvus_tolerance_II. The disadvantage of adjusting solvus_tolerance_II is that while it improves the resolution of solvus boundaries, it may degrade the quality of normal phase boundaries. Thus, reducing solvus_tolerance_II to 0.1 mol (Fig 7c) and still further to 0.05 mol (Fig 7d) improve the resolution of the solvus, but introduce minor flaws (circled in red) that were not present in the initial version of the high resolution calculation (Fig 7b). 

 

 

Figure 6. Increasing the reach_increment specified for the clinopyroxene model, Omph(GHP), ultimately eliminates the spurious solvi evident in Fig 5. Increasing reach_increment always improves the accuracy of computational results, but the cosmetic quality of the section (i.e., how the section appears rather than the information embodied within it) does not necessarily improve until the reach_increment is sufficiently large to completely eliminate all spurious solvi. Increasing reach_increment also influences the stepped solvus (circled in blue in Fig 5) because the increased range permitted for the clinopyroxene composition allows Perple_X greater freedom to adjust the phase compositions during iteration; however the shape of the solvus is clearly unnatural.

 

Figure 7. Stepped solvi and solvus_tolerance_II.