The AmIteratorGrid algorithm available to the pscf_pc or pscf_pg programs uses an Anderson mixing algorithm in which residual equations and field updates are formulated using values defined on the nodes of a regular spatial grid. The algorithm can be used to either solve the SCFT equations for a rigid unit cell or to solve the SCFT equations and also optimize the unit cell parameters of a flexible unit cell so as to minimize the free energy density. Closely analogous classes to implement this algorithm are defined in the Pscf::Rpc and Pscf::Rpg namespaces.
Class API documentation:
- See also
- Anderson mixing algorithms
Parameter file
The full format of an AmIteratorGrid parameter block is described below:
Here, as elsewhere, labels followed by an asterisk (*) represent optional parameters. The only required parmeter is the error tolerance, epsilon. The meaning of most of the other parameters is explained in a general discussion of Anderson mixing algorithms, and summarized below:
| Label | Description |
| epsilon | error tolerance - iteration stops if the magnitude of the scalar error becomes less than epsilon. |
| maxItr* | Maximum number of iterations that will be attempted (200 by default) |
| maxHist* | Maximum number of previous trial solultions that will be retained in memory for used by the AM algorithm (50 by default) |
| verbose* | Integer level 0-2 for verbosity of log output during iteration. (0 by default) |
| errorType* | String identifer for the type of variable used to define scalar error . The only allowed values are "norm", "rms", "max", and "relNorm". ("relNorm" by default) |
| isFlexible* | Set isFlexible true to enable or false to disable iterative optimization of the unit cell parameters so as to minimize the free energy. (True by default) |
| scaleStress* | Constant factor by which stress components are multipled in the definition of the residual vector if isFlexible is true (10.0 by default). |
| lambda* | Mixing parameter \( \lambda \) in a standard mixing correction step for an Anderson mixing algorithm. (1.0 by default) |
| useLambdaRamp* | If true, use slow ramp-up of mixing parameter, as discussed here. (true by default) |
| r* | Ratio \( r \) used in formula for ramp factor \( \alpha \) that multiplies the mixing parameter, as discussed here. (only allowed if useLambdaRamp is true, 0.9 by default) |
The iterative loop exits if the number of iterations has reaches maxItr or if the magnitude of the scalar error drops below epsilon.
Residual definition
The AmIteratorGrid iterator uses a definition of the residual vector that is analogous to that used by AmIteratorBasis, except that residal elements are associated with grid points, rather than with coefficients of basis functions in a symmetry-adapted Fourier series. The residual vector is thus a real-space version of that described by Eqs. (10-13) of the article by Arora et al. (J. Chem. Phys. 2017). In what follows, we use zero based indexing for monomer types, and use \( N_{\rm m} \) to denote the number of monomer types (i.e., the parameter nMonomer).
Let \( w_{i}({\bf r}) \) and \( \phi_{i}({\bf r)} \) denote values of the chemical potential field and volume fraction for monomer type i at a grid point located at position \( {\bf r} \). The residual component \( R_{i}({\bf r})\) associated with grid point \( {\bf r} \) and monomer type i is given by
\[ R_{i}({\bf r}) = \sum_{j=0}^{N_{\rm m}-1} \left ( \chi_{ij} \phi_{j}({\bf r}) - P_{ij}w_{j}({\bf r}) \right ) \quad. \]
Here, \( P_{ij} \) is an element of an idempotent \( N_{\rm m} \times N_{\rm m} \) matrix \( P \) that can be expressed in matrix notation as
\[ P = I - \frac{\epsilon \epsilon^{T} \chi^{-1} } { \epsilon^{T} \chi^{-1} \epsilon } \quad. \]
where \( \chi^{-1} \) is the matrix inverse of the symmetric \( \chi \) matrix, and \(\epsilon\) is a \( N_{m} \)-component column vector with elements given by \( \epsilon_{i} = 1 \) for all \( i = 0, \ldots, N_{\rm m}-1 \).
The vector of residuals for a calculation with a flexible unit cell (when parameter isFleixble set true) has one additional stress residual associated with each flexible unit cell parameter. The residual \( R_{m} \) associated with flexible unit cell parameter \( \theta_{m} \) is given by
\[ R_{m} = -S\frac{\partial f}{\partial \theta_{m}} \]
where \( f \) is the free energy per monomer (i.e., the free energy density times the monomer reference volume) in units in which \( kT = 1 \), while \(S\) is an arbitrary prefactor whose value is given by the parameter scaleStress, which is set to 10.0 by default.
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