← ProjectsResearch · Software

Low-rank Riemannian optimization for matrix equations

Solving large multiterm linear matrix equations directly on the manifold of fixed-rank matrices, with new preconditioners for Riemannian CG.

Role: First author
With: D. Kressner (EPFL), L. Robol (Pisa)
Tools: MATLAB
Links: Paper (SISC)arXiv Code

The problem

Discretised PDEs and control problems often lead to large symmetric positive definite matrix equations

A_1 X B_1^\top + A_2 X B_2^\top + \dots + A_\ell X B_\ell^\top = F,

whose solution $X$ is well approximated by a low-rank matrix, especially when $F$ has low rank. For $\ell \le 2$ (Lyapunov and Sylvester equations) efficient low-rank solvers exist; for $\ell > 2$ most methods combine a classical iteration with repeated low-rank truncation.

Approach

We approximate $X$ directly on the manifold $\mathcal{M}_r$ of rank- $r$ matrices, minimising the energy functional with Riemannian conjugate gradient. The key difficulty is preconditioning a first-order Riemannian method; we propose

a change of metric in the ambient space,
preconditioning of the Riemannian gradient, and
a variant of ADI acting on the tangent space,

combined with a strategy that adapts the rank during the iteration.

Outcome

The method is competitive on examples representative of typical applications. This work grew out of my M.Sc. thesis at EPFL and was published in the SIAM Journal on Scientific Computing (2025).