git-svn-id: svn://svn.icms.temple.edu/lammps-ro/trunk@12769 f3b2605a-c512-4ea7-a41b-209d697bcdaa

tools/i-pi/ipi/utils/nmtransform.py

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+"""Contains functions for doing the inverse and forward normal mode transforms.
+
+Copyright (C) 2013, Joshua More and Michele Ceriotti
+
+This program is free software: you can redistribute it and/or modify
+it under the terms of the GNU General Public License as published by
+the Free Software Foundation, either version 3 of the License, or
+(at your option) any later version.
+
+This program is distributed in the hope that it will be useful,
+but WITHOUT ANY WARRANTY; without even the implied warranty of
+MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 
+GNU General Public License for more details.
+
+You should have received a copy of the GNU General Public License
+along with this program. If not, see <http.//www.gnu.org/licenses/>.
+
+
+Classes:
+   nm_trans: Uses matrix multiplication to do normal mode transformations.
+   nm_rescale: Uses matrix multiplication to do ring polymer contraction
+      or expansion.
+   nm_fft: Uses fast-Fourier transforms to do normal modes transformations.
+
+Functions:
+   mk_nm_matrix: Makes a matrix to transform between the normal mode and bead
+      representations.
+   mk_rs_matrix: Makes a matrix to transform between one number of beads and
+      another. Higher normal modes in the case of an expansion are set to zero.
+"""
+
+__all__ = ['nm_trans', 'nm_rescale', 'nm_fft']
+
+import numpy as np
+from ipi.utils.messages import verbosity, info
+
+def mk_nm_matrix(nbeads):
+   """Gets the matrix that transforms from the bead representation
+   to the normal mode representation.
+
+   If we return from this function a matrix C, then we transform between the
+   bead and normal mode representation using q_nm = C . q_b, q_b = C.T . q_nm
+
+   Args:
+      nbeads: The number of beads.
+   """
+
+   b2nm = np.zeros((nbeads,nbeads))
+   b2nm[0,:] = np.sqrt(1.0)
+   for j in range(nbeads):
+      for i in range(1, nbeads/2+1):
+         b2nm[i,j] = np.sqrt(2.0)*np.cos(2*np.pi*j*i/float(nbeads))
+      for i in range(nbeads/2+1, nbeads):
+         b2nm[i,j] = np.sqrt(2.0)*np.sin(2*np.pi*j*i/float(nbeads))
+   if (nbeads%2) == 0:
+      b2nm[nbeads/2,0:nbeads:2] = 1.0
+      b2nm[nbeads/2,1:nbeads:2] = -1.0
+   return b2nm/np.sqrt(nbeads)
+
+def mk_rs_matrix(nb1, nb2):
+   """Gets the matrix that transforms a path with nb1 beads into one with
+   nb2 beads.
+
+   If we return from this function a matrix T, then we transform between the
+   system with nb1 bead and the system of nb2 beads using q_2 = T . q_1
+
+   Args:
+      nb1: The initial number of beads.
+      nb2: The final number of beads.
+   """
+
+   if (nb1 == nb2):
+      return np.identity(nb1,float)
+   elif (nb1 > nb2):
+      b1_nm = mk_nm_matrix(nb1)
+      nm_b2 = mk_nm_matrix(nb2).T
+
+      #builds the "reduction" matrix that picks the normal modes we want to keep
+      b1_b2 = np.zeros((nb2, nb1), float)
+      b1_b2[0,0] = 1.0
+      for i in range(1, nb2/2+1):
+         b1_b2[i,i] = 1.0
+         b1_b2[nb2-i, nb1-i] = 1.0
+      if (nb2 % 2 == 0):
+         #if we are contracting down to an even number of beads, then we have to
+         #pick just one of the last degenerate modes to match onto the single
+         #stiffest mode in the new path
+         b1_b2[nb2/2, nb1-nb2/2] = 0.0
+
+      rs_b1_b2 = np.dot(nm_b2, np.dot(b1_b2, b1_nm))
+      return rs_b1_b2*np.sqrt(float(nb2)/float(nb1))
+   else:
+      return mk_rs_matrix(nb2, nb1).T*(float(nb2)/float(nb1))
+
+
+class nm_trans:
+   """Helper class to perform beads <--> normal modes transformation.
+
+   Attributes:
+      _b2nm: The matrix to transform between the bead and normal mode
+         representations.
+      _nm2b: The matrix to transform between the normal mode and bead
+         representations.
+   """
+
+   def __init__(self, nbeads):
+      """Initializes nm_trans.
+
+      Args:
+         nbeads: The number of beads.
+      """
+
+      self._b2nm = mk_nm_matrix(nbeads)
+      self._nm2b = self._b2nm.T
+
+   def b2nm(self, q):
+      """Transforms a matrix to the normal mode representation.
+
+      Args:
+         q: A matrix with nbeads rows, in the bead representation.
+      """
+
+      return np.dot(self._b2nm,q)
+
+   def nm2b(self, q):
+      """Transforms a matrix to the bead representation.
+
+      Args:
+         q: A matrix with nbeads rows, in the normal mode representation.
+      """
+
+      return np.dot(self._nm2b,q)
+
+
+class nm_rescale:
+   """Helper class to rescale a ring polymer between different number of beads.
+
+   Attributes:
+      _b1tob2: The matrix to transform between a ring polymer with 'nbeads1'
+         beads and another with 'nbeads2' beads.
+      _b2tob1: The matrix to transform between a ring polymer with 'nbeads2'
+         beads and another with 'nbeads1' beads.
+   """
+
+   def __init__(self, nbeads1, nbeads2):
+      """Initializes nm_rescale.
+
+      Args:
+         nbeads1: The initial number of beads.
+         nbeads2: The rescaled number of beads.
+      """
+
+      self._b1tob2 = mk_rs_matrix(nbeads1,nbeads2)
+      self._b2tob1 = self._b1tob2.T*(float(nbeads1)/float(nbeads2))
+
+   def b1tob2(self, q):
+      """Transforms a matrix from one value of beads to another.
+
+      Args:
+         q: A matrix with nbeads1 rows, in the bead representation.
+      """
+
+      return np.dot(self._b1tob2,q)
+
+   def b2tob1(self, q):
+      """Transforms a matrix from one value of beads to another.
+
+      Args:
+         q: A matrix with nbeads2 rows, in the bead representation.
+      """
+
+      return np.dot(self._b2tob1,q)
+
+
+
+class nm_fft:
+   """Helper class to perform beads <--> normal modes transformation
+      using Fast Fourier transforms.
+
+   Attributes:
+      fft: The fast-Fourier transform function to transform between the
+         bead and normal mode representations.
+      ifft: The inverse fast-Fourier transform function to transform
+         between the normal mode and bead representations.
+      qdummy: A matrix to hold a copy of the bead positions to transform
+         them to the normal mode representation.
+      qnmdummy: A matrix to hold a copy of the normal modes to transform
+         them to the bead representation.
+      nbeads: The number of beads.
+      natoms: The number of atoms.
+   """
+
+   def __init__(self, nbeads, natoms):
+      """Initializes nm_trans.
+
+      Args:
+         nbeads: The number of beads.
+         natoms: The number of atoms.
+      """
+
+      self.nbeads = nbeads
+      self.natoms = natoms
+      try:
+         import pyfftw
+         info("Import of PyFFTW successful", verbosity.medium)
+         self.qdummy = pyfftw.n_byte_align_empty((nbeads, 3*natoms), 16, 'float32')
+         self.qnmdummy = pyfftw.n_byte_align_empty((nbeads//2+1, 3*natoms), 16, 'complex64')
+         self.fft = pyfftw.FFTW(self.qdummy, self.qnmdummy, axes=(0,), direction='FFTW_FORWARD')
+         self.ifft = pyfftw.FFTW(self.qnmdummy, self.qdummy, axes=(0,), direction='FFTW_BACKWARD')
+      except ImportError: #Uses standard numpy fft library if nothing better
+                          #is available
+         info("Import of PyFFTW unsuccessful, using NumPy library instead", verbosity.medium)
+         self.qdummy = np.zeros((nbeads,3*natoms), dtype='float32')
+         self.qnmdummy = np.zeros((nbeads//2+1,3*natoms), dtype='complex64')
+         def dummy_fft(self):
+            self.qnmdummy = np.fft.rfft(self.qdummy, axis=0)
+         def dummy_ifft(self):
+            self.qdummy = np.fft.irfft(self.qnmdummy, n=self.nbeads, axis=0)
+         self.fft = lambda: dummy_fft(self)
+         self.ifft = lambda: dummy_ifft(self)
+
+   def b2nm(self, q):
+      """Transforms a matrix to the normal mode representation.
+
+      Args:
+         q: A matrix with nbeads rows and 3*natoms columns,
+            in the bead representation.
+      """
+
+      if self.nbeads == 1:
+         return q
+      self.qdummy[:] = q
+      self.fft()
+      if self.nbeads == 2:
+         return self.qnmdummy.real/np.sqrt(self.nbeads)
+
+      nmodes = self.nbeads/2
+
+      self.qnmdummy /= np.sqrt(self.nbeads)
+      qnm = np.zeros(q.shape)
+      qnm[0,:] = self.qnmdummy[0,:].real
+
+      if self.nbeads % 2 == 0:
+         self.qnmdummy[1:-1,:] *= np.sqrt(2)
+         (qnm[1:nmodes,:], qnm[self.nbeads:nmodes:-1,:]) = (self.qnmdummy[1:-1,:].real, self.qnmdummy[1:-1,:].imag)
+         qnm[nmodes,:] = self.qnmdummy[nmodes,:].real
+      else:
+         self.qnmdummy[1:,:] *= np.sqrt(2)
+         (qnm[1:nmodes+1,:], qnm[self.nbeads:nmodes:-1,:]) = (self.qnmdummy[1:,:].real, self.qnmdummy[1:,:].imag)
+
+      return qnm
+
+   def nm2b(self, qnm):
+      """Transforms a matrix to the bead representation.
+
+      Args:
+         qnm: A matrix with nbeads rows and 3*natoms columns,
+            in the normal mode representation.
+      """
+
+      if self.nbeads == 1:
+         return qnm
+      if self.nbeads == 2:
+         self.qnmdummy[:] = qnm
+         self.ifft()
+         return self.qdummy*np.sqrt(self.nbeads)
+
+      nmodes = self.nbeads/2
+      odd = self.nbeads - 2*nmodes  # 0 if even, 1 if odd
+
+      qnm_complex = np.zeros((nmodes+1, len(qnm[0,:])), complex)
+      qnm_complex[0,:] = qnm[0,:]
+      if not odd:
+         (qnm_complex[1:-1,:].real, qnm_complex[1:-1,:].imag) = (qnm[1:nmodes,:], qnm[self.nbeads:nmodes:-1,:])
+         qnm_complex[1:-1,:] /= np.sqrt(2)
+         qnm_complex[nmodes,:] = qnm[nmodes,:]
+      else:
+         (qnm_complex[1:,:].real, qnm_complex[1:,:].imag) = (qnm[1:nmodes+1,:], qnm[self.nbeads:nmodes:-1,:])
+         qnm_complex[1:,:] /= np.sqrt(2)
+
+      self.qnmdummy[:] = qnm_complex
+      self.ifft()
+      return self.qdummy*np.sqrt(self.nbeads)