Indirect and Remote measurements
Singular Value Decomposition
Nikolai Shokhirev
Introduction
In many experiments the unknown functions
f(x) is connected with the measured signal
g(y) by an integral equation of the first kind:
(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":

The kernel K(y, x), which is the response of the instrument
to the pulse (function) input
:

[c, d] is the interval of reconstruction
(determination)

[a, b] is the interval of measurement (scanning interval)
The properties of integral operators of the first kind were in
general described in the introductory tutorial (
Basics of Indirect Measurements).
The solution of integral equation brakes into two problems:

Reconstruction of g(x) and

Analysis of accuracy and resolution
Usually the main efforts are concentrated on the first problem, extracting of the unknown function.
However the solution devaluates without estimation of accuracy.
I think that the second part is more important for ill posed problems.
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.
Notations
We will use the notation suitable both continuous
functions
and
vectors
The same notation will be used for the yspace (the interval [c, d] ). The
two spaces are different, but it will be clear from a context which one is
currently considered.
Singular Value Decomposition
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.
Completeness of singular basis sets
The two sets of basis functions (or vectors) {
v_{n}}and {
u_{n}}are
not necessarily complete (in each space). However we always can expand SVD with
necessary additional functions and
= 0.
Using the complete sets we can decompose the identity operators:
(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:
Here
are the expansion coefficients.
Formal solution
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)
Pseudoinverse operator
We can introduce the pseudoinverse operator
(8)
here the sums run over all
nonzero .
This operator acts as an inverse operator in the subspace formed by
corresponding to nonzero
:
The reconstructed function can be expressed in terms of
pseudoinverse operator (or pseudoinverse matrix for vectors):
References
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