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In this paper we discuss the performance of solvers for low-rank linearsystems in the tensor-train format, also known as matrix-product states (MPS)in physics. We focus on today's many-core CPU systems and the interplay of theperformance and the required linear algebra operations in this setting.Specifically, we consider the tensor-train GMRES method, the modifiedalternating linear scheme (MALS) and the alternating minimal energy (AMEn).Based on the example of a simple non-symmetric system from discretizing amultidimensional convection-diffusion equation in, e.g., $50^{10}$ grid points,we illustrate the computational complexity of the three methods. This showsthat the projection to smaller sub-problems reduces the required number offloating point operations by orders of magnitude: between GMRES and MALS, andagain between MALS and AMEn. All three methods require similar underlyinglinear algebra operations (building blocks). We suggest several building blockimprovements regarding orthogonalization steps, singular value decompositions,and tensor contractions. In addition, we propose a simple genericpreconditioner in the tensor-train format based on a rank-1 approximation ofthe operator. Combining all optimizations, we obtain roughly a 5x speedup overthe reference implementation for the fastest method (AMEn) on a currentmulti-core CPU.

Performance of linear solvers in tensor-train format on current multicore architectures