def _batch_sqrt_solve(self, rhs):
# Recall the square root of this operator is M + VDV^T.
# The Woodbury formula gives:
# (M + VDV^T)^{-1}
# = M^{-1} - M^{-1} V (D^{-1} + V^T M^{-1} V)^{-1} V^T M^{-1}
# = M^{-1} - M^{-1} V C^{-1} V^T M^{-1}
# where C is the capacitance matrix.
m = self._operator
v = self._v
cchol = self._chol_capacitance(batch_mode=True)
# The operators will use batch/singleton mode automatically. We don't
# override.
# M^{-1} rhs
minv_rhs = m.solve(rhs)
# V^T M^{-1} rhs
vt_minv_rhs = math_ops.batch_matmul(v, minv_rhs, adj_x=True)
# C^{-1} V^T M^{-1} rhs
cinv_vt_minv_rhs = linalg_ops.batch_cholesky_solve(cchol, vt_minv_rhs)
# V C^{-1} V^T M^{-1} rhs
v_cinv_vt_minv_rhs = math_ops.batch_matmul(v, cinv_vt_minv_rhs)
# M^{-1} V C^{-1} V^T M^{-1} rhs
minv_v_cinv_vt_minv_rhs = m.solve(v_cinv_vt_minv_rhs)
# M^{-1} - M^{-1} V C^{-1} V^T M^{-1}
return minv_rhs - minv_v_cinv_vt_minv_rhs
def _batch_solve(self, rhs):
return linalg_ops.batch_cholesky_solve(self._chol, rhs)
def _batch_solve(self, rhs): return linalg_ops.batch_cholesky_solve(self._chol, rhs)
