Tensor Train Multiplication

TL;DR

The paper introduces the Tensor Train Multiplication (TTM) algorithm, significantly improving the efficiency of elementwise tensor train multiplication in terms of computational complexity and memory, enabling faster tensor network simulations.

Contribution

The TTM algorithm reduces computational complexity from  to  and memory from  to , representing a substantial advancement over conventional methods.

Findings

Demonstrates improved runtime and memory scaling in turbulence simulations

Enables GPU-accelerated tensor network computations for fluid dynamics

Shows potential for handling larger bond dimensions efficiently

Abstract

We present the Tensor Train Multiplication (TTM) algorithm for the elementwise multiplication of two tensor trains with bond dimension $\chi$. The computational complexity and memory requirements of the TTM algorithm scale as $\chi^3$ and $\chi^2$, respectively. This represents a significant improvement compared with the conventional approach, where the computational complexity scales as $\chi^4$ and memory requirements scale as $\chi^3$.We benchmark the TTM algorithm using flows obtained from artificial turbulence generation and numerically demonstrate its improved runtime and memory scaling compared with the conventional approach. The TTM algorithm paves the way towards GPU accelerated tensor network simulations of computational fluid dynamics problems with large bond dimensions due to its dramatic improvement in memory scaling.