Speaker
Description
Large-scale multiterm linear matrix equations of the form
$$
A_1 X B_1 + \cdots + A_\ell X B_\ell = C,
$$ arise as the algebraic formulation in various application problems, such as discretized multivariable PDEs, stochastic, parameterized, or space-time PDEs and inverse problems. While effective methods exist for two-term equations ($\ell=2$), limited options are available for $\ell > 2$. Thus, efficiently solving multiterm matrix equations remains an open problem in numerical linear algebra.
In this talk, we present a new iterative scheme called the Subspace-Conjugate Gradient (Ss-cg) method for the efficient solution of large‑scale multiterm linear matrix equations. This method relies on the matrix‑oriented CG scheme but better uses the underlying (low-rank) matrix structure.
By imposing a peculiar orthogonality condition, the CG scalar coefficients for the iterative solution and the descent direction are replaced by low-dimensional matrices in Ss-CG. We employ truncation strategies to maintain the computed matrix iterates of low rank since limiting memory consumption becomes essential, especially when the number of terms $\ell$ is large. To this end, an additional ad-hoc randomized range-finding strategy is developed to further speed up computations.
The features of the Ss-CG method lead to remarkable computational gains, as demonstrated by several computational experiments. In particular, we compare Ss-CG with existing algorithms for solving Sylvester and Lyapunov multiterm equations to highlight its potential.
