## What are inverse problems?

Inverse Problems is a research area dealing with inversion of models or data. An inverse problem is a mathematical framework that is used to obtain information about a physical object or system from observed measurements. The solution to this problem is useful because it generally provides information about a physical parameter that we cannot directly observe. Thus, inverse problems are some of the most important and well-studied mathematical problems in science and mathematics. There are many different applications including, medical imaging, geophysics, computer vision, astronomy, nondestructive testing, and many others

We illustrate this by considering an example in geophysics, consider a situation where we want to obtain information about the wave speed structure ${g}$ (or in another words information about the metric ${g}$) inside the earth ${\Omega}$, from the observed seismic wave-field, denoted by ${\Lambda_g}$, at the boundary of the earth ${\partial \Omega}$ (the meaning of ${\Lambda_g}$ becomes clear when we give the precise definition in (2), but think of it as the measurements available in the surface of the earth).

In the previous picture we see the travel seismic waves ${g}$ produced by an earthquake inside the earth, knowledge of the travel seismic waves provide information about the location of oil and minerals deposits. To obtain this information the boundary of the earth is perturbed by an artificial explosion or due to a natural earthquake, this perturbation produces waves that travels through geodesics (trajectories that minimize the travel time of the wave) in the metric ${g}$ and hits the oil and mineral deposits and reflects back to the surface of the earth. We wait up to time ${T}$ until the waves have reached the boundary and we measure the wave-field using a seismograph.

The mathematical formulation of this problem is described by the wave equation in a compact domain ${\Omega}$ as follows:

$\displaystyle \begin{array}{rclcr} \partial_t^2u + \triangle_gu &= &0 & \mbox{in}& (0,T)\times M,\\ u(0,x) = \partial_t u(0,x) &=& 0 & \mbox{for}& x\in M,\\ u(t,x) &=& f(t,x)& \mbox{on} & (0,T)\times \partial M, \end{array} \ \ \ \ \ (1)$

where ${\triangle_g}$ is the Laplace-Beltrami operator for the metric ${g}$ and ${u(x,t)}$ is the displacement in some direction of the point ${x}$ at time ${t}$. The information of the seismic wave field is encoded in the hyperbolic Dirichlet to Neumann map

$\displaystyle \Lambda_g: f \rightarrow \left. \frac{\partial u}{\partial \nu}\right|_{\partial \Omega} \ \ \ \ \ (2)$

where ${\nu}$ is the outer unit co-normal to ${\partial \Omega}$. Notice that ${\Lambda_g}$ maps the initial perturbation ${f}$ (Dirichlet data) to the recovered wave-field ${\frac{\partial u}{\partial \nu}|_{\partial \Omega}}$ (Neumann data), that carries within information of the inside metric of the earth.

The forward problem of finding ${\Lambda_g}$ from ${g}$ is a well-posed problem under reasonable regularity assumptions over ${g}$. This means that if we know the underlying geometry of the problem (i.e., the metric ${g}$) them by solving for ${u}$ the Dirichlet problem (1), with Dirichlet data ${f}$, we can obtain the Neumann data ${\frac{\partial u}{\partial \nu}|_{\partial \Omega}}$. We define the forward operator ${F}$ as

$\displaystyle F(g) = \Lambda_g,$

as said before this is a well-possed operator, but it requires knowledge of the metric ${g}$ in ${\Omega}$ (that is exactly what we are interested in recover).

In our example the unknown parameter is ${g}$ while the data is ${\Lambda_g}$. The framework of inverse problems consist in trying to invert this operator and study the properties of this inversion. Basically we reason by asking: If we have enough measurements (i.e., ${\Lambda_g = (f,\frac{\partial u}{\partial \nu}|_{\partial \Omega})}$ for many ${f}$‘s), what can we say about the metric ${g}$? If we denote by ${F^{-1}}$ the inverse operator then the central questions that we want to address on inverse problems are:

• Existence: Given observed data measurements for the system is there any unknown parameters that actually yields this observations?
• Uniqueness: Can we determine uniquely the unknown parameters by the observed data measurements?
• Stability: How are the errors in the data measurements amplified in the resulting unknown parameters?
• Reconstruction: Is there a computationally efficient formula or procedure to recover the unknown parameters from the data measurements?

You can think that the forward problem is mainly concerned in the problem of prediction, in that sense that given all background knowledge of the problem (mathematical model and parameters in the PDE) and initial conditions you can predict what will happen to the solution over time or space. The inverse problem in the other hand is concerned in the problem of recovering information, in the sense that given only some background knowledge of the problem (mathematical model, but not all parameters in PDE) and some data observed in the boundary of your domain, you want to recover the missing information that carries the solution and with it important physical properties that cannot be directly observed

In general this problems are highly non-linear and ill posed and there is not a comprehensive theory that deals with such problems in this generality, so we need to focus on each particular mathematical model one at the time (vaguely speaking, similar models have similar answers). But so, can we recover the the metric and with it the oil and minerals only from measuring the boundary? Actually, there is a partial positive answer to this question that holds even for more general situations, if you are interested you can find my paper (here) that deals with the question of stability for this example.

I am a curious individual with a deep desire of making the world better for those in need.
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### 3 Responses to What are inverse problems?

1. Great post! Keep posting!

2. cmontalto says:

Hi Maikol! Great you like it, thank you so much for the feedback.

3. Carlos Montalto Gracía says:

I find it interesting this application of modern mathematical

regards