New approaches to determination of constrained lumping schemes for a reaction system in the whole composition space

Two new approaches to the determination of constrained lumping schemes are presented. They are based on the property that the lumping schemes validated in the whole composition Y_n-space of y are only determined by the invariance of the subspace spanned by the row vectors of lumping matrix M with respect to the transpose of the Jacobian matrix J^T(y) for the kinetic equations. It is proved that, when a part of a lumping matrix M_G is given, each row of the part of the lumping matrix to be determined, M_D, is certain linear combinations of a set of eigenvectors of a special symmetric matrix. This symmetric matrix is related to M^T_G and A_kM^T_G, where A_k are the basis matrices of J^T (y). It is shown that the approximate lumping matrices containing M_G with different row number n̂(n̂ < n) and global minimum errors can be determined by an optimization method. Using the concept of the minimal invariant subspace of a constant matrix over a given subspace one can directly obtain the lumping matrices containing M_G with different n̂. The accuracy of these lumping matrices are shown to be satisfactory in sample calculations.