A two-directional Arnoldi process and its application to parametric model order reduction

We consider a two-directional Krylov subspace Kk(A[j],b[j]), where besides the dimensionality kk of the subspace increases, the matrix A[j] and vector b[j] which induce the subspace may also augment. Specifically, we consider the case where the matrix A[j] and the vector b[j] are augmented by block triangular bordering. We present a two-directional Arnoldi process to efficiently generate a sequence of orthonormal bases Qk[j] of the Krylov subspaces. The concept of a two-directional Krylov subspace and an Arnoldi process is triggered by the need of a multiparameter moment-matching based model order reduction technique for parameterized linear dynamical systems. Numerical examples illustrate computational efficiency and flexibility of the proposed two-directional Arnoldi process.