by Nikolai V. Shokhirev
Basics of Indirect Measurements
Singular Value Decomposition
Contents
- Introduction
- Notations
- Singular Value Decomposition
- Completeness of singular basis sets
- Formal solution
- Pseudo-inverse operator
- References
Analysis of accuracy and resolution
Implementation
(1a)
A broad class of such inverse problems is instrument distortions of spectra. Indirect measurements in which g(y) and f(x) are different physical dependences are also described by the above equation. For example, g(y) is the decay of the electron spin echo amplitude, while f(x) is the distribution function of radical pairs over distances. This equation can be also written in the operator form:
(1b)
The integral operator is a mathematical model of an experiment (instrument). The operator consists of three "parts":
[c, d] is the interval of reconstruction (determination)
Now we discuss the a very efficient method of the analysis of accuracy and resolution of the solution and the influence of experimental accuracy and the interval of measurements.
and vectors
The same notation will be used for the y-space (the interval [c, d] ). The two spaces are different, but it will be clear from a context which one is currently considered.
The kernel allows the following expansion (in general infinite):
(2)
In linear algebra such expansion is called Singular Value Decomposition (SVD). Here are the singular values and v_{n} and u_{n} are the singular functions (vectors). Within each set the functions can be chosen orthogonal and normalized:
(3a)
(3b)
Here is Kroneker delta.
(4)
They act in the appropriate space in the following way: Now we can expand functions (in each space) using the above form of the identity operator: Hereare the expansion coefficients.
The integral equation (1) can be rewritten as follows
(6)
Using completeness (4) and orthogonality (5), the above equation is reduced to the following set of equations for the expansion coefficients
It can be easy resolved
and the unknown function can be reconstructed
(7)
(8)
here the sums run over all non-zero . This operator acts as an inverse operator in the sub-space formed by corresponding to non-zero : The reconstructed function can be expressed in terms of pseudo-inverse operator (or pseudo-inverse matrix for vectors):©Nikolai Shokhirev, 2001-2009