### Abstract

Unfolding problems often arise in the context of statistical data analysis. Such problematics occur when the probability distribution of a physical quantity is to be measured, but it is randomized (smeared) by some well-understood process, such as a nonideal detector response or a well-described physical phenomenon. In such case it is said that the original probability distribution of interest is folded by a known response function. The reconstruction of the original probability distribution from the measured one is called unfolding. That technically involves evaluation of the nonbounded inverse of an integral operator over the space of L1 functions, which is known to be an ill-posed problem. For the pertinent regularized operator inversion, we propose a linear iterative formula and provide proof of convergence in a probability theory context. Furthermore, we provide formulae for error estimates at finite iteration stopping order which are of utmost importance in practical applications: the approximation error, the propagated statistical error, and the propagated systematic error can be quantified. The arguments are based on the Riesz-Thorin theorem mapping the original L1 problem to L2 space, and subsequent application of ordinary L2 spectral theory of operators. A library implementation in C of the algorithm along with corresponding error propagation is also provided. A numerical example also illustrates the method in operation.

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

Pages (from-to) | 1345-1371 |

Number of pages | 27 |

Journal | SIAM-ASA Journal on Uncertainty Quantification |

Volume | 4 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2016 |

### Fingerprint

### Keywords

- Convergence
- Error propagation
- Functional analysis
- Probability theory
- Riesz- Thorin theorem
- Statistics
- Unfolding

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Applied Mathematics
- Discrete Mathematics and Combinatorics
- Modelling and Simulation
- Statistics and Probability

### Cite this

**Convergence and error propagation results on a linear iterative unfolding method.** / László, A.

Research output: Contribution to journal › Article

*SIAM-ASA Journal on Uncertainty Quantification*, vol. 4, no. 1, pp. 1345-1371. https://doi.org/10.1137/15M1035744

}

TY - JOUR

T1 - Convergence and error propagation results on a linear iterative unfolding method

AU - László, A.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - Unfolding problems often arise in the context of statistical data analysis. Such problematics occur when the probability distribution of a physical quantity is to be measured, but it is randomized (smeared) by some well-understood process, such as a nonideal detector response or a well-described physical phenomenon. In such case it is said that the original probability distribution of interest is folded by a known response function. The reconstruction of the original probability distribution from the measured one is called unfolding. That technically involves evaluation of the nonbounded inverse of an integral operator over the space of L1 functions, which is known to be an ill-posed problem. For the pertinent regularized operator inversion, we propose a linear iterative formula and provide proof of convergence in a probability theory context. Furthermore, we provide formulae for error estimates at finite iteration stopping order which are of utmost importance in practical applications: the approximation error, the propagated statistical error, and the propagated systematic error can be quantified. The arguments are based on the Riesz-Thorin theorem mapping the original L1 problem to L2 space, and subsequent application of ordinary L2 spectral theory of operators. A library implementation in C of the algorithm along with corresponding error propagation is also provided. A numerical example also illustrates the method in operation.

AB - Unfolding problems often arise in the context of statistical data analysis. Such problematics occur when the probability distribution of a physical quantity is to be measured, but it is randomized (smeared) by some well-understood process, such as a nonideal detector response or a well-described physical phenomenon. In such case it is said that the original probability distribution of interest is folded by a known response function. The reconstruction of the original probability distribution from the measured one is called unfolding. That technically involves evaluation of the nonbounded inverse of an integral operator over the space of L1 functions, which is known to be an ill-posed problem. For the pertinent regularized operator inversion, we propose a linear iterative formula and provide proof of convergence in a probability theory context. Furthermore, we provide formulae for error estimates at finite iteration stopping order which are of utmost importance in practical applications: the approximation error, the propagated statistical error, and the propagated systematic error can be quantified. The arguments are based on the Riesz-Thorin theorem mapping the original L1 problem to L2 space, and subsequent application of ordinary L2 spectral theory of operators. A library implementation in C of the algorithm along with corresponding error propagation is also provided. A numerical example also illustrates the method in operation.

KW - Convergence

KW - Error propagation

KW - Functional analysis

KW - Probability theory

KW - Riesz- Thorin theorem

KW - Statistics

KW - Unfolding

UR - http://www.scopus.com/inward/record.url?scp=85033562578&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85033562578&partnerID=8YFLogxK

U2 - 10.1137/15M1035744

DO - 10.1137/15M1035744

M3 - Article

VL - 4

SP - 1345

EP - 1371

JO - SIAM-ASA Journal on Uncertainty Quantification

JF - SIAM-ASA Journal on Uncertainty Quantification

SN - 2166-2525

IS - 1

ER -