Solving the inverse problem of an inhomogeneous Fredholm integral equation by numerical method

Fredholm equations involve definite integrals with fixed upper and lower limits. An inhomogeneous Fredholm equation of the first kind has the form

$g(t)=\int_{a}^{b}K(t,s)f(s)ds$                                                                                     (1)

Here f(s) is the unknown function to be solved for, while g(t) is known.

Fredholm equations of the first kind are often extremely ill-conditioned. Applying the kernel to a function is generally a smoothing operation, which often actually loses information, and there is no way to get it back in an inverse operation. Specialized methods have been developed for such equations, which are often called inverse problems. In general, some prior knowledge of the nature of the solution must be used, in one way or another, to restore the lost information.

To allow the equation to be solved numerically we need to discretize the integral using some quadrature rules. Here we use the Gaussian-Legendre quadrature, yielding a linear system of equations for the unknown function f(s) as:

$g_{i}=\sum_{j=1}^{N}\omega _{j}K(t_{i},s_{j})f(s_{j})$                                                                                   (2)

where (t_i, g_i) is the i-th experimental data set, N is the number of integration points, w_j and s_j are the quadrature weights and points, respectively.

Most often we do not have enough data points to solve Eqn. 2 directly. Even if we have enough data points, it is still impossible to do so because of experimental uncertainty and/or measurement noises. Instead we seek the f(s) that satisfy:

$minimize: A=\sum_{i}^{ }\left [ g_{i}-\sum_{j=1}^{N}w_{j}K(t_{i},s_{j})f(s_{j}) \right ]^{2}$                                           (3)

However, the mere Eqn. 3 is not enough. When A is minimized, though the agreement is very good, the solution could become unstable, and in some way unrealistic because of the degenerate natures of the equations. We need to introduce another factor B, which is supposed to give a solution that is “smooth” or stable, and meanwhile has nothing to do with the measured data. In fact, the central idea of inverse theory is to find

$\textup{minimize: } A+\lambda B$                                                                                          (4)

for various values of 0 < λ < ∞, and then to settle on a “best” value of λ by certain criteria.

How to determine the smoothing factor B? Generally, if the priori belief is that a linear function is a good approximation to f(s), we can take

                        $B=\sum_{j=1}^{N-2}\left [ -f(s_{j})+2f(s_{j+1})-f(s_{j+2}) \right ]^{2}\propto \int \left [ f{(s)}'' \right ]^{2}ds$                          (5)

If the priori belief is that a quadratic function is a good approximation, then we can take

                        $B=\sum_{j=1}^{N-3}\left [ -f_{j}+3f_{j+1}-3f_{j+2}+f_{j+3} \right ]^{2}\propto \int \left [ f{{}'}'' \right ]^{2}dx$                               (6)

To solve the optimization problem, we can use CVX, a package for specifying and solving convex programs that was developed by Michael Grant and Stephen Boyd from Stanford University. CVX is software based on MATLAB, and can be downloaded from the website: http://cvxr.com/cvx/download/.

The implementation of CVX in MATLAB is very simple. Just the following code segment will do the job.

An example of the solution of the inverse problem of inhomogeneous Fredholm integral equation of the first kind is to find the thermal conductivity accumulation function α based on experimental measurements of thermal conductivities of nanostructured materials.

                $\frac{k_{nano}}{k_{bulk}}=\frac{L_{c}}{\Lambda _{max}+L_{c}}+\int_{0}^{\Lambda _{max}}\alpha (\Lambda )\frac{L_{c}}{\left ( \Lambda +L_{c} \right )^{2}}d\Lambda$

where L_c is the characteristic length of nanostructure materials, $\Lambda$ is the mean free path of phonons, $\alpha (\Lambda )$ is the thermal conductivity accumulation function, which is defined as the fraction of thermal conductivity contributed by phonons with mean free paths shorter than $\Lambda$ .