# Multilevel sequential Monte Carlo samplers

## Abstract

Here, we study the approximation of expectations w.r.t. probability distributions associated to the solution of partial differential equations (PDEs); this scenario appears routinely in Bayesian inverse problems. In practice, one often has to solve the associated PDE numerically, using, for instance finite element methods and leading to a discretisation bias, with the step-size level h _{L}. In addition, the expectation cannot be computed analytically and one often resorts to Monte Carlo methods. In the context of this problem, it is known that the introduction of the multilevel Monte Carlo (MLMC) method can reduce the amount of computational effort to estimate expectations, for a given level of error. This is achieved via a telescoping identity associated to a Monte Carlo approximation of a sequence of probability distributions with discretisation levels $${\infty}$$ >h _{0}>h _{1 }...>h _{L}. In many practical problems of interest, one cannot achieve an i.i.d. sampling of the associated sequence of probability distributions. A sequential Monte Carlo (SMC) version of the MLMC method is introduced to deal with this problem. In conclusion, it is shown that under appropriate assumptions, the attractive property of a reduction of the amount of computational effort to estimate expectations, for a given level of error, can be maintained within the SMC context.

- Authors:

- Univ. College London, London (United Kingdom). Dept. of Statistical Science
- National Univ. of Singapore (Singapore). Dept. of Statistics & Applied Probability
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computer Science and Mathematics Division
- King Abdullah Univ. of Science and Technology, Thuwal (Saudi Arabia)

- Publication Date:

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

- Sponsoring Org.:
- USDOE Laboratory Directed Research and Development (LDRD) Program

- OSTI Identifier:
- 1302922

- Grant/Contract Number:
- AC05-00OR22725; R-155-000-143-112

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Stochastic Processes and Their Applications

- Additional Journal Information:
- Journal Name: Stochastic Processes and Their Applications; Journal ID: ISSN 0304-4149

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; 77 NANOSCIENCE AND NANOTECHNOLOGY; multilevel Monte Carlo; sequential Monte Carlo; Bayesian inverse problems

### Citation Formats

```
Beskos, Alexandros, Jasra, Ajay, Law, Kody, Tempone, Raul, and Zhou, Yan.
```*Multilevel sequential Monte Carlo samplers*. United States: N. p., 2016.
Web. doi:10.1016/j.spa.2016.08.004.

```
Beskos, Alexandros, Jasra, Ajay, Law, Kody, Tempone, Raul, & Zhou, Yan.
```*Multilevel sequential Monte Carlo samplers*. United States. doi:10.1016/j.spa.2016.08.004.

```
Beskos, Alexandros, Jasra, Ajay, Law, Kody, Tempone, Raul, and Zhou, Yan. Wed .
"Multilevel sequential Monte Carlo samplers". United States.
doi:10.1016/j.spa.2016.08.004. https://www.osti.gov/servlets/purl/1302922.
```

```
@article{osti_1302922,
```

title = {Multilevel sequential Monte Carlo samplers},

author = {Beskos, Alexandros and Jasra, Ajay and Law, Kody and Tempone, Raul and Zhou, Yan},

abstractNote = {Here, we study the approximation of expectations w.r.t. probability distributions associated to the solution of partial differential equations (PDEs); this scenario appears routinely in Bayesian inverse problems. In practice, one often has to solve the associated PDE numerically, using, for instance finite element methods and leading to a discretisation bias, with the step-size level hL. In addition, the expectation cannot be computed analytically and one often resorts to Monte Carlo methods. In the context of this problem, it is known that the introduction of the multilevel Monte Carlo (MLMC) method can reduce the amount of computational effort to estimate expectations, for a given level of error. This is achieved via a telescoping identity associated to a Monte Carlo approximation of a sequence of probability distributions with discretisation levels ${\infty}$ >h0>h1 ...>hL. In many practical problems of interest, one cannot achieve an i.i.d. sampling of the associated sequence of probability distributions. A sequential Monte Carlo (SMC) version of the MLMC method is introduced to deal with this problem. In conclusion, it is shown that under appropriate assumptions, the attractive property of a reduction of the amount of computational effort to estimate expectations, for a given level of error, can be maintained within the SMC context.},

doi = {10.1016/j.spa.2016.08.004},

journal = {Stochastic Processes and Their Applications},

number = ,

volume = ,

place = {United States},

year = {Wed Aug 24 00:00:00 EDT 2016},

month = {Wed Aug 24 00:00:00 EDT 2016}

}

*Citation information provided by*

Web of Science

Web of Science