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
Consider an example where we want to obtain information about the wave speed structure \(g\) inside the earth from the observed seismic wave-field \(\Lambda_g\) at the boundary of the earth. In the following 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. The wave travels through geodesics 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 with a seismograph.
The mathematical formulation of this problem is described by the wave equation in a compact domain \(\Omega\) as follows: \begin{equation*} \begin{array}{rclcr} (\partial_t^2 + \triangle_g )u &= &0 & \mbox{in}& (0,T)\times \Omega,\\ u(0,x) = \partial_t u(0,x) &=& 0 & \mbox{for}& x\in \Omega,\\ u(t,x) &=& f(t,x)& \mbox{on} & (0,T)\times \partial \Omega, \end{array} \end{equation*} where \(\triangle_g\) is the Laplace-Beltrami operator for the metric \(g\).
The information of the seismic wave field is encoded in the hyperbolic Dirichlet to Neumann map \begin{equation*} \Lambda_g: f \to \left.\frac{\partial u}{\partial \nu}\right|_{\partial \Omega} \end{equation*} where \(\nu\) is the outer unit co-normal to \(\partial \Omega\). The forward problem of finding \(\Lambda_g\) from \(g\) is a well-posed problem under reasonable regularity assumptions over \(g\). That is the forward operator \(F\) \begin{equation*} F(g) = \Lambda_g \end{equation*} is well defined. The inverse problems consist in study properties of the inverse operator \(F^{-1}\).
In general the central questions that we want to address on inverse problems are:
- Existence: Given observed measurements (e.g., \(\Lambda_g\)) for the system is there any unknown parameter(s) (e.g., \(g\)) that actually yields this observations?
- Uniqueness: Can we determine uniquely the unknown parameter(s) by an observed measurements?
- Stability: How are the errors in the measurements amplified in the resulting unknown parameter(s)?
- Reconstruction: Is there a computationally efficient formula or procedure to recover the unknown parameters from the data?
Research interests
I am interested in the mathematics surrounding inverse problems, this includes: differential geometry, microlocal analysis, partial differential equations, probability, and applied functional analysis. My research is focus on stability and reconstruction for different types of inverse problems. I use micro-local analysis to understand the inverse operators and to exploit the underlying geometry of the problem, by doing so you are able to extract information that allows stable reconstructions of solutions under certain regularity conditions. Here there is a summary of some of my work: