### Abstract

The solution of the exterior-value problem for the fractional Laplacian can be computed by a Walk Outside Spheres algorithm. This involves sampling α-stable Levy processes on their exit from maximally inscribed balls and sampling their occupation distribution. Kyprianou, Osojnik, and Shardlow (2018) developed this algorithm, providing a complexity analysis and an implementation, for approximating the solution at a single point in the domain. This paper shows how to efficiently sample the whole field by generating an approximation in L
^{2}(D) for a domain D. The method takes advantage of a hierarchy of triangular meshes and uses the multilevel Monte Carlo method for Hilbert space-valued quantities of interest. We derive complexity bounds in terms of the fractional parameter α and demonstrate that the method gives accurate results for two problems with exact solutions. Finally, we show how to couple the method with the variable-accuracy Arnoldi iteration to compute the smallest eigenvalue of the fractional Laplacian. A criteria is derived for the variable accuracy and a comparison is given with analytical results of Dyda (2012).

Original language | English |
---|---|

Pages (from-to) | 2767-2792 |

Number of pages | 26 |

Journal | Mathematics of Computation (MCOM) |

Volume | 88 |

Issue number | 320 |

DOIs | |

Publication status | Published - 14 Mar 2019 |

### Keywords

- math.NA
- Arnoldi algorithm
- Eigenvalue problems
- Exterior-value problems
- Levy processes
- Multilevel Monte Carlo
- Numerical solution of PDEs
- Fractional Laplacian
- Walk on spheres

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics
- Algebra and Number Theory

### Cite this

**A walk outside spheres for the fractional Laplacian : fields and first eigenvalue.** / Shardlow, Tony.

Research output: Contribution to journal › Article

*Mathematics of Computation (MCOM)*, vol. 88, no. 320, pp. 2767-2792. https://doi.org/10.1090/mcom/3422

}

TY - JOUR

T1 - A walk outside spheres for the fractional Laplacian

T2 - fields and first eigenvalue

AU - Shardlow, Tony

PY - 2019/3/14

Y1 - 2019/3/14

N2 - The solution of the exterior-value problem for the fractional Laplacian can be computed by a Walk Outside Spheres algorithm. This involves sampling α-stable Levy processes on their exit from maximally inscribed balls and sampling their occupation distribution. Kyprianou, Osojnik, and Shardlow (2018) developed this algorithm, providing a complexity analysis and an implementation, for approximating the solution at a single point in the domain. This paper shows how to efficiently sample the whole field by generating an approximation in L 2(D) for a domain D. The method takes advantage of a hierarchy of triangular meshes and uses the multilevel Monte Carlo method for Hilbert space-valued quantities of interest. We derive complexity bounds in terms of the fractional parameter α and demonstrate that the method gives accurate results for two problems with exact solutions. Finally, we show how to couple the method with the variable-accuracy Arnoldi iteration to compute the smallest eigenvalue of the fractional Laplacian. A criteria is derived for the variable accuracy and a comparison is given with analytical results of Dyda (2012).

AB - The solution of the exterior-value problem for the fractional Laplacian can be computed by a Walk Outside Spheres algorithm. This involves sampling α-stable Levy processes on their exit from maximally inscribed balls and sampling their occupation distribution. Kyprianou, Osojnik, and Shardlow (2018) developed this algorithm, providing a complexity analysis and an implementation, for approximating the solution at a single point in the domain. This paper shows how to efficiently sample the whole field by generating an approximation in L 2(D) for a domain D. The method takes advantage of a hierarchy of triangular meshes and uses the multilevel Monte Carlo method for Hilbert space-valued quantities of interest. We derive complexity bounds in terms of the fractional parameter α and demonstrate that the method gives accurate results for two problems with exact solutions. Finally, we show how to couple the method with the variable-accuracy Arnoldi iteration to compute the smallest eigenvalue of the fractional Laplacian. A criteria is derived for the variable accuracy and a comparison is given with analytical results of Dyda (2012).

KW - math.NA

KW - Arnoldi algorithm

KW - Eigenvalue problems

KW - Exterior-value problems

KW - Levy processes

KW - Multilevel Monte Carlo

KW - Numerical solution of PDEs

KW - Fractional Laplacian

KW - Walk on spheres

UR - http://www.scopus.com/inward/record.url?scp=85073220126&partnerID=8YFLogxK

U2 - 10.1090/mcom/3422

DO - 10.1090/mcom/3422

M3 - Article

VL - 88

SP - 2767

EP - 2792

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 1088-6842

IS - 320

ER -