The laboratory of Inverse Problems in Applied Sciences aims to conduct high-level applications of inverse problems in Atmospheric Sciences, Solid Earth, Medical Imaging and Astronomy arising from these strategic areas in Chile. Most of these applications involve the direct and inverse modeling of complex partial differential equations. The research is intended to be interdisciplinary, in collaboration with national and international researchers in mathematics and applied sciences, searching for frontier and original developments. Laboratory activities will also a) promote the development of regional and international collaboration network in these areas b) make knowledge transfer by means of interdisciplinary training of young researchers and students and c) continuously identify, organize, and diffuse information and opportunities about inverse problems research in these topics through its web page.

### Research areas:

IP in atmospheric sciences: Inverse problems in atmospheric sciences are of main importance in Chile and other countries in South America, since it is linked to air-quality problems in megacities that with high population density as is the case of Santiago, and to industrial and natural environmental atmospheric impacts as is the case of copper smelters or volcanic activity in Chile. We will continue the fruitful collaboration between CMM and Laura Gallardo, from the Atmospheric Sciences section of the Geophysical Department, and use the already built regional network linked to other air-quality groups in Sao Paulo, Lima, Buenos Aires, Bogotá and Medellín and the Chilean Weather Service DMC. In this new period, we will also include other researchers with expertise in numerical analysis interested in this type of subjects. The three main topics to develop are the following:

##### a) Inverse problems for radiative transfer equation.

Since 2011, lidar (light detection and ranging) measurements are being collected at the Chilean Weather Service DMC. We are interested in the implementation and development of lidar signal inversion techniques in collaboration with physicist from UDEC in order to infer optical properties of aerosols from DMC lidar data.

##### b) New methods for inverse source problems and applications to air-quality.

There is an important gap between theoretical results for inverse source problems for parabolic equations and the standard least-square approach used in practice in atmospheric sciences in order to afford inverse source problems. It seems to exist place for new methods, but there are practical difficulties due to the big size of the corresponding discretized systems. We will analyze methods based on exact controls and other recently proposed methodologies such as back and forth nudging and the enclosure methods applied to some real data in city scale and regional scale scenarios.

##### c) Analysis of air-quality observational systems using information tools.

Covariance error matrices obtained as a subproduct of data assimilation analysis can be used for the analysis of air-quality observational systems including all kind of data (in-situ, satellite) and the design of air-quality monitoring network. This kind of tools has been applied in the context of weather forecast but they are new in the context of air-quality forecast. We plan to develop these variational information tools applied to the air-quality monitoring network in Santiago city, and to compare the results with other standard statistical information tools used in practice for these type of problems and to provide knowledge to national and regional air-quality services.

#### IP in solid earth:

Inverse problems in Solid Earth linked to earthquake studies and seismic prospection are of primary importance in Chile, geographically situated more than 4000 km along the so called ring of fire. This is and applied research in collaboration with Eduardo Contreras, from the solid Earth section of the Geophysics Department, U. de Chile, with expertise in seismic tomography, inverse gravimetry and tectonics. The problems we plan to study are the following:

##### a) Parameter identification in tectonics.

These are parameter identification problems for models of the Nazca plate subduction under the Chilean trench. The main problem is the numerical modeling of plate equations with variable thickness under loading, sediments and subduction boundary conditions and the identification of the effective thickness near the subduction region or under seamount regions using only partial bathymetry and/or seismic measurements. These types of studies are relevant for better understanding earthquake phenomenology and lithosphere aging. We plan to model plate equations with variable thickness using finite elements and solve some of these parameter identification problems for real data in some selected regions of the Chilean trench using MonteCarlo methods.

##### b) Inverse problem in geodesy.

This is a relatively new area in geophysics of increasing interest due to the large amount of available GPS data. The basic inverse problem consists in recovering incremental stresses into the Earth from direct or remote observations near the ground surface. Our aim is to model and solve some simple three-dimensional inverse geodetic problems by considering an elastostatic model and real data.

##### c) Inverse problems and compressive sensing.

This is a prospective joint work with Marcos Díaz and Jorge Soto, from the Electrical Engineering Department, U. de Chile and some researchers in optimization from the UTFSM external CMM group. We are interested in solving L1 norm optimization problems appearing when applying compressive sensing ideas to build some data acquisition instrumentation in seismology.

#### IP in medical imaging:

These are applications in collaboration with Matías Courdurier from Pontificia Universidad Católica, the international associate member Gunther Uhlmann from UW-UCI and Maya de Buhan from Paris 5.

##### a) Attenuated ray transform and SPECT.

We are interested in developing practical algorithms for identifying a source of radiation for single photon emission computed tomography (SPECT) assuming you can have separate measurements of the ballistic and scattering parts of the radiation intensity on the boundary. Matías Courdurier has connections with the Center of Biomedical Images of the Pontificia Universidad Católica, where this inverse problem was initially proposed.

##### b) Elastography and viscoelasticity.

Theoretical studies on inverse problems for hyperbolic equations with viscoelastic or memory effects are of primary interest for elastography and thermo-acoustic and photo-acoustic tomography. We plan to study viscoelastic effects for inverse problems for the wave equation in the context of hybrid inverse problems and for some hyperbolic systems arising in elasticity and plate equations.

#### IP related with astronomy:

##### a) modeling and design of semiconductor devices used in radio astronomy.

This is a joint research with Marcos Díaz and other researchers from the astronomical instrumentation group of the Electric Engineering Department in U. de Chile. The aim is the numerical modeling of drift diffusion equations using finite differences, finite volumes and finite elements and to provide design tools through inverse techniques for some prototypes of photo-diodes with improved attenuation, designed to be used in the front-end of some equipment in radio astronomy (ALMA).

##### b) Other inverse problem.

We plan to identify other applied inverse problem in astronomy under G. Uhlmann’s advice.

Director: Axel Osses

Researchers: A. Osses, L. Gallardo

International associate researchers: G. Uhlmann (U. Washington & U. California Irvine)

Collaborators: M. Courdurier (Mathematics Department, PUC), E. Contreras (Geophysics Department, U. Chile), M. Díaz (Electric Engineering Department, U. Chile), G. García (Mathematical Department, USACH), A. Mercado, E. Hernández, E. Cerpa (Mathematical Department, UTFSM), B. Palacios, J. Escribano.

Postdoc: Nicole Spillane

Engineers: Alfredo Torrico Palacios , Adolfo Rene Henriquez Saa , Francisco Romero

Students: Nicolás Molina (MsC), Rodolfo Núñez Uribe (MsC), Sélim Cornet (ECP, France)