## Statistical Science

- Statist. Sci.
- Volume 1, Number 4 (1986), 502-518.

### A Statistical Perspective on Ill-Posed Inverse Problems

#### Abstract

Ill-posed inverse problems arise in many branches of science and engineering. In the typical situation one is interested in recovering a whole function given a finite number of noisy measurements on functionals. Performance characteristics of an inversion algorithm are studied via the mean square error which is decomposed into bias and variability. Variability calculations are often straightforward, but useful bias measures are more difficult to obtain. An appropriate definition of what geophysicists call the Backus-Gilbert averaging kernel leads to a natural way of measuring bias characteristics. Moreover, the ideas give rise to some important experimental design criteria. It can be shown that the optimal inversion algorithms are methods of regularization procedures, but to completely specify these algorithms the signal to noise ratio must be supplied. Statistical approaches to the empirical determination of the signal to noise ratio are discussed; cross-validation and unbiased risk methods are reviewed; and some extensions, which seem particularly appropriate in the inverse problem context, are indicated. Linear and nonlinear examples from medicine, meteorology, and geophysics are used for illustration.

#### Article information

**Source**

Statist. Sci. Volume 1, Number 4 (1986), 502-518.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.ss/1177013525

**Digital Object Identifier**

doi:10.1214/ss/1177013525

**Mathematical Reviews number (MathSciNet)**

MR874480

**Zentralblatt MATH identifier**

0625.62110

**JSTOR**

links.jstor.org

**Keywords**

Averaging kernel B-splines cross-validation experimental design mean square error reservoir engineering stereology satellite meteorology

#### Citation

O'Sullivan, Finbarr. A Statistical Perspective on Ill-Posed Inverse Problems. Statist. Sci. 1 (1986), no. 4, 502--518. doi:10.1214/ss/1177013525. https://projecteuclid.org/euclid.ss/1177013525.

#### See also

- See Comment: D. M. Titterington. [A Statistical Perspective on Ill-Posed Inverse Problems]: Comment. Statist. Sci., Volume 1, Number 4 (1986), 519--521.Project Euclid: euclid.ss/1177013526
- See Comment: Grace Wahba. [A Statistical Perspective on Ill-Posed Inverse Problems]: Comment. Statist. Sci., Volume 1, Number 4 (1986), 521--522.Project Euclid: euclid.ss/1177013527
- See Comment: John A. Rice. [A Statistical Perspective on Ill-Posed Inverse Problems]: Comment. Statist. Sci., Volume 1, Number 4 (1986), 522--523.Project Euclid: euclid.ss/1177013528
- See Comment: Freeman Gilbert. [A Statistical Perspective on Ill-Posed Inverse Problems]: Comment. Statist. Sci., Volume 1, Number 4 (1986), 523--523.Project Euclid: euclid.ss/1177013529
- See Comment: Finbarr O'Sullivan. [A Statistical Perspective on Ill-Posed Inverse Problems]: Rejoinder. Statist. Sci., Volume 1, Number 4 (1986), 523--527.Project Euclid: euclid.ss/1177013530