The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
A new iterative Monte Carlo approach for inverse matrix problem
Journal of Computational and Applied Mathematics
Scientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey
Monte Carlo Methods for Applied Scientists
Monte Carlo Methods for Applied Scientists
Improved Monte Carlo linear solvers through non-diagonal splitting
ICCSA'03 Proceedings of the 2003 international conference on Computational science and its applications: PartIII
Stochastic tomography and its applications in 3D imaging of mixing fluids
ACM Transactions on Graphics (TOG) - SIGGRAPH 2012 Conference Proceedings
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Monte Carlo (MC) linear solvers can be considered stochastic realizations of deterministic stationary iterative processes. That is, they estimate the result of a stationary iterative technique for solving linear systems. There are typically two sources of errors: (i) those from the underlying deterministic iterative process and (ii) those from the MC process that performs the estimation. Much progress has been made in reducing the stochastic errors of the MC process. However, MC linear solvers suffer from the drawback that, due to efficiency considerations, they are usually stochastic realizations of the Jacobi method (a diagonal splitting), which has poor convergence properties. This has limited the application of MC linear solvers. The main goal of this paper is to show that efficient MC implementations of non-diagonal splittings too are feasible, by constructing efficient implementations for one such splitting. As a secondary objective, we also derive conditions under which this scheme can perform better than MC Jacobi, and demonstrate this experimentally. The significance of this work lies in proposing an approach that can lead to efficient MC implementations of a wider variety of deterministic iterative processes.