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

*y*[data],*F*[scientific principles relating cause and effect], and*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 statementF(x) = yis almost certainly asimplificationof reality due to the presence of so-called lurking variables:Fmay depend on many more independent variables than solelyx. 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."

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); aviobj = addframe(aviobj, getframe); pause(.1); end aviobj = close(aviobj);