Multi-way Monte Carlo Method for Linear Systems
This work addresses a bottleneck in computational linear algebra for researchers and practitioners by providing a more broadly applicable and efficient method, though it is incremental as it builds on existing Monte Carlo techniques.
The paper tackles the limitation of the Monte Carlo method for solving linear systems, which traditionally requires a strict norm condition, by proposing a multi-way Markov random walk that relaxes the requirement to a spectral radius condition, enabling it to solve more problems and work faster, as demonstrated in numerical experiments.
We study the Monte Carlo method for solving a linear system of the form $x = H x + b$. A sufficient condition for the method to work is $\| H \| < 1$, which greatly limits the usability of this method. We improve this condition by proposing a new multi-way Markov random walk, which is a generalization of the standard Markov random walk. Under our new framework we prove that the necessary and sufficient condition for our method to work is the spectral radius $ρ(H^{+}) < 1$, which is a weaker requirement than $\| H \| < 1$. In addition to solving more problems, our new method can work faster than the standard algorithm. In numerical experiments on both synthetic and real world matrices, we demonstrate the effectiveness of our new method.