# Mathematical Inverse Problems

Joseph Myers, editor

Inverse Problems are problems of identifying unknown causes from known effects. The cause may be an embedded anomaly within a mission-critical semiconductor device, or an underground tunnel containing anti-personnel munitions. The effects may be measurements of the electric current flowing through the semiconductor, or gravitational data from locations surrounding the underground tunnel.

Solving an inverse problem means solving a scientific research problem using applied mathematical analysis. The added ingredient of applied mathematical analysis is the key that makes the study of inverse problems successful. An expert in the field of inverse problems has a deep understanding of concepts in physics, ideas in mathematical theory, and techniques in numerical analysis.

In mathematics a problem is considered as a function which computes data from independent variables. An inverse problem is to determine the independent variables from the data—in essence, to compute the inverse function. The information given for solving an inverse problem includes the functional relationship between the data and the independent variables. Thus, research problems are connected to mathematics and thereby simplified to an analytical and computational procedure of finding x given

1. y [data],
2. F [scientific principles relating cause and effect], and
3. F(x) = y [scientific law that assumes the data y are an effect of the cause(s) x].

Most often, the functional relationship is a system of partial differential equations, duly constituted according to natural and physical laws.

Note. The preceding statement F(x) = y is almost certainly a simplification of reality due to the presence of so-called lurking variables: F may depend on many more independent variables than solely x. One must constantly keep in mind this intrinsic limitation in order to report honest results and to be an effective researcher—saving oneself from quite a few impractical "wild goose chases."

### Conductivity Problem Example

Simulation of time-dependent interior voltage profile of doped (k = 10) semiconductor device with sinusoidal imposed boundary voltage

The above movie was produced with MatLab 7.8.0 (R2009a) by the following commands:

```cp = conductivityProblem();
% create a conductivity problem solution class using the computer
% software package for matlab written by Joseph K. Myers (c) 2009.

cp.n = 50; % set discretization size
cp.k = 10; % ratio of exterior/interior conductivity coefficients
% for a P-N type semiconductor device
cp.D = .5; % define the doped region D by the radial Fourier
% coefficients of its boundary, i.e., the P-N junction.

% Note that D can be a vector of arbitrary finite positive length,
% and the conductivity problem solution class will deal with
% all the resulting complexity automatically (within the limits
% of the mesh/grid size and the computer hardware).
% Naturally, the sequence defined by elements of D must satisfy
% some type of convergence criterion (at least |D| < 1) in order
% to be representative of some physically attainable domain
% configuration.

cp.beforeSolving;
% create Chebyshev interpolation points and optimized distribution
% of source points.

cp.solver = @solveTransmission;
% define the solver for transmission-type problems

p = 30; % the number of points used in plotting solutions
% will be p^2.

hold off
jframe = 1;
aviobj = avifile('voltage-time.avi', 'fps', 4, 'compression', 'Cinepak');
for d=0:.1:10;

cp.f_g0 = @(z) .5 + sin(pi*real(z) - d)/2;
% impose boundary voltage

cp.solve;
% solve direct conductivity problem

cp.plots(p, 1);
% create a plot of voltage profile in figure 1.

axis([-1 1 -1 1 0 1]*1.5);