# Analysis of Monte Carlo accelerated iterative methods for sparse linear systems: Analysis of Monte Carlo accelerated iterative methods for sparse linear systems

## Abstract

Here, we consider hybrid deterministic-stochastic iterative algorithms for the solution of large, sparse linear systems. Starting from a convergent splitting of the coefficient matrix, we analyze various types of Monte Carlo acceleration schemes applied to the original preconditioned Richardson (stationary) iteration. We expect that these methods will have considerable potential for resiliency to faults when implemented on massively parallel machines. We also establish sufficient conditions for the convergence of the hybrid schemes, and we investigate different types of preconditioners including sparse approximate inverses. Numerical experiments on linear systems arising from the discretization of partial differential equations are presented.

- Authors:

- Department of Mathematics and Computer Science, Emory University, Atlanta 30322 GA USA
- Oak Ridge National Laboratory, 1 Bethel Valley Rd. Oak Ridge 37831 TN USA

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1346688

- Alternate Identifier(s):
- OSTI ID: 1400612

- Grant/Contract Number:
- AC05-00OR22725; ERKJ247

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Numerical Linear Algebra with Applications

- Additional Journal Information:
- Journal Volume: 24; Journal Issue: 3; Journal ID: ISSN 1070-5325

- Publisher:
- Wiley

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; iterative methods; Monte Carlo methods; preconditioning; resilience; Richardson iteration; sparse approximation inverse; sparse linear systems

### Citation Formats

```
Benzi, Michele, Evans, Thomas M., Hamilton, Steven P., Lupo Pasini, Massimiliano, and Slattery, Stuart R.
```*Analysis of Monte Carlo accelerated iterative methods for sparse linear systems: Analysis of Monte Carlo accelerated iterative methods for sparse linear systems*. United States: N. p., 2017.
Web. doi:10.1002/nla.2088.

```
Benzi, Michele, Evans, Thomas M., Hamilton, Steven P., Lupo Pasini, Massimiliano, & Slattery, Stuart R.
```*Analysis of Monte Carlo accelerated iterative methods for sparse linear systems: Analysis of Monte Carlo accelerated iterative methods for sparse linear systems*. United States. doi:10.1002/nla.2088.

```
Benzi, Michele, Evans, Thomas M., Hamilton, Steven P., Lupo Pasini, Massimiliano, and Slattery, Stuart R. Sun .
"Analysis of Monte Carlo accelerated iterative methods for sparse linear systems: Analysis of Monte Carlo accelerated iterative methods for sparse linear systems". United States.
doi:10.1002/nla.2088. https://www.osti.gov/servlets/purl/1346688.
```

```
@article{osti_1346688,
```

title = {Analysis of Monte Carlo accelerated iterative methods for sparse linear systems: Analysis of Monte Carlo accelerated iterative methods for sparse linear systems},

author = {Benzi, Michele and Evans, Thomas M. and Hamilton, Steven P. and Lupo Pasini, Massimiliano and Slattery, Stuart R.},

abstractNote = {Here, we consider hybrid deterministic-stochastic iterative algorithms for the solution of large, sparse linear systems. Starting from a convergent splitting of the coefficient matrix, we analyze various types of Monte Carlo acceleration schemes applied to the original preconditioned Richardson (stationary) iteration. We expect that these methods will have considerable potential for resiliency to faults when implemented on massively parallel machines. We also establish sufficient conditions for the convergence of the hybrid schemes, and we investigate different types of preconditioners including sparse approximate inverses. Numerical experiments on linear systems arising from the discretization of partial differential equations are presented.},

doi = {10.1002/nla.2088},

journal = {Numerical Linear Algebra with Applications},

number = 3,

volume = 24,

place = {United States},

year = {Sun Mar 05 00:00:00 EST 2017},

month = {Sun Mar 05 00:00:00 EST 2017}

}