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":

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

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

3. [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 y-space (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)

 

Pseudo-inverse operator

We can introduce the pseudo-inverse operator

     (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):

References

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