Inverse problems involve for example reconstruction of an object based on physical measurements and finding a best model/parameters out of a family given observed data. Typically the corresponding "forward" problems are well-posed and can be solved straightforwardly, while the inverse problems are often ill-posed. Not to be confused with the (inverse) tag.
Typically a forward problem is a family of well-posed problems that are parametrized by some set $\mathcal{P}$. We can often write the forward problems as a nonlinear transformation $G_p$ indexed by $p\in \mathcal{P}$ such that given initial data $d$ the transformation returns some final data (or solution) $s_p$. Or, in notations: $$s_p = G_p[d]$$
The forward problem then is the problem of finding $s_p$ for a given parameter $p$ and a given initial data $d$. Some examples are:
Given a family of wave equations parametrised by the potential function $V$ $$ -\partial_t^2 u(x,t) + \partial_x^2 u(x,t) = V(x,t)u(x,t) $$ one possible forward problem would be solving for the function $u$ and its velocity $\partial_t u$ at time $t = T$ given the initial data $(u,\partial_t u)_{t = 0} = (f,g)$.
We let the parameter space $\mathcal{P}$ be the space of compactly supported smooth functions on Euclidean space $\mathbb{R}^n$, and for any initial data $d$ we let $G_p[d] = s_p$ be the X ray transform of $p$.
We let the parameter space $\mathcal{P}$ be the density distribution in the bedrock of a region of land. We let that transformation $G_p$ be the mapping that sends the initial data of the strength of controlled explosion to the final data which is the observed surface seismic wave. The operator $G_p$ can be (in theory) computed from $p$ based on accepted model of the elastodynamics of the interior of the earth.
In principle the forward problems can be solved, at least numerically, by straightforward methods.
The corresponding inverse problems are the problems of finding the best (or some) parameter $p$ such that given some initial data elicits some observed solution. Quite often the inverse problems are ill-posed: there may not be admissible parameters at all that reproduce the transformation from initial to final data (in this case the observed data are believed to contain errors, or that our a priori assumptions on what the operators are can be fallacious); or there could be multiple admissible parameters that produce the same observed results.
The inverse problem corresponding to the examples above are
- Solving for the potential $V(x,t)$ between time $t = 0$ and $t = T$ given the solution $u$ and its time derivative at those two boundary times.
- Finding the compactly support function $p$ given its X ray transform
- Solving for the density of the rock by seismic sounding.
Some often-studied inverse problems include the inverse scattering problem in partial differential equations and the inverse Sturm-Liouville problem in ordinary differential equations. Typical applications include medical imaging (X rays, CAT scans), seismic sounding, various tomography methods, machine learning, statistical analysis and more.
