### Abstract

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

Pages (from-to) | 381-399 |

Number of pages | 19 |

Journal | Inverse Problems in Science and Engineering |

Volume | 18 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2010 |

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### Keywords

- alternating method
- second kind boundary integral equation
- Nyström method
- Laplace equation
- Cauchy problem

### Cite this

*Inverse Problems in Science and Engineering*,

*18*(3), 381-399. https://doi.org/10.1080/17415971003624322

}

*Inverse Problems in Science and Engineering*, vol. 18, no. 3, pp. 381-399. https://doi.org/10.1080/17415971003624322

**Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques.** / Helsing, Johan; Johansson, B. Tomas.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques

AU - Helsing, Johan

AU - Johansson, B. Tomas

PY - 2010

Y1 - 2010

N2 - We consider the problem of stable determination of a harmonic function from knowledge of the solution and its normal derivative on a part of the boundary of the (bounded) solution domain. The alternating method is a procedure to generate an approximation to the harmonic function from such Cauchy data and we investigate a numerical implementation of this procedure based on Fredholm integral equations and Nyström discretization schemes, which makes it possible to perform a large number of iterations (millions) with minor computational cost (seconds) and high accuracy. Moreover, the original problem is rewritten as a fixed point equation on the boundary, and various other direct regularization techniques are discussed to solve that equation. We also discuss how knowledge of the smoothness of the data can be used to further improve the accuracy. Numerical examples are presented showing that accurate approximations of both the solution and its normal derivative can be obtained with much less computational time than in previous works.

AB - We consider the problem of stable determination of a harmonic function from knowledge of the solution and its normal derivative on a part of the boundary of the (bounded) solution domain. The alternating method is a procedure to generate an approximation to the harmonic function from such Cauchy data and we investigate a numerical implementation of this procedure based on Fredholm integral equations and Nyström discretization schemes, which makes it possible to perform a large number of iterations (millions) with minor computational cost (seconds) and high accuracy. Moreover, the original problem is rewritten as a fixed point equation on the boundary, and various other direct regularization techniques are discussed to solve that equation. We also discuss how knowledge of the smoothness of the data can be used to further improve the accuracy. Numerical examples are presented showing that accurate approximations of both the solution and its normal derivative can be obtained with much less computational time than in previous works.

KW - alternating method

KW - second kind boundary integral equation

KW - Nyström method

KW - Laplace equation

KW - Cauchy problem

UR - http://www.tandfonline.com/10.1080/17415971003624322

U2 - 10.1080/17415971003624322

DO - 10.1080/17415971003624322

M3 - Article

VL - 18

SP - 381

EP - 399

JO - Inverse Problems in Science and Engineering

JF - Inverse Problems in Science and Engineering

SN - 1741-5977

IS - 3

ER -