Inverse Problems and Control Theory. 

2025-05-09
15:00hrs.

Nicolás Barnafi. Universidad Católica de Chile
The inverse elasticity problem, its extension to porous media and applications in biomedicine
Auditorio del Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile
Abstract:

The inverse elasticity problem can be simply stated as: given a deformed configuration and the forces that act on it, find an initial stress-free configuration such that when the given forces are applied to it, one recovers the given deformed configuration. Surprisingly, this problem can be framed as a (direct) elasticity one, whose mathematical properties are inherited from the original direct problem if the underlying material is sufficiently regular.

In this seminar, I will review this problem and its main mathematical properties. After this brief introduction, I will show some artifacts that appear when solving this problem, such as self-intersections and geometrically incompatible solutions. The talk will finish with an extension of this system to poroelastic materials, where I will show that the strong form of the equations does not allow for a weak formulation, and this requires some special treatment. All models will be shown to work in realistic heart geometries.

https://eventos.cmm.uchile.cl/seminarioipct/

2025-04-10
16:00hrs.

Rodrigo Lecaros Lira. Universidad Técnica Federico Santa María
Estabilidad Lipschitz en un problema inverso para un operador parabólico semi-discreto
Auditorio Ninoslav Bralic, Facultad de Matemática, Universidad Católica de Chile.
Abstract:
En esta presentación, se abordará un problema inverso para una ecuación parabólica semi-discreta, donde el objetivo es identificar el término en el lado derecho de la ecuación (la fuente) a partir de mediciones de la solución en un tiempo intermedio y dentro de un subdominio espacial. A partir de este resultado, se puede establecer una estimación de estabilidad para la función potencial cuando esta depende únicamente del espacio.
Se presentará una nueva estimación de Carleman en el contexto semi-discreto, cuya dependencia del parámetro está determinada por el tamaño de la malla. Además, se analizará cómo los términos discretos que surgen en la desigualdad de Carleman introducen un término de error asociado a la condición inicial de la solución en el sistema semi-discreto.
https://eventos.cmm.uchile.cl/seminarioipct/

2025-01-16
16:00hrs.

Eric Bonnetier. Institut Fourier, Université Grenoble-Alpes
Uniform estimates for small volume asymptotics
Sala de Seminarios Felipe Álvarez, Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile.
Abstract:
We revisit the problem of  studying the impact of a perturbation of the coefficients of an elliptic PDE on a set of small size. We show that the asymptotic structure of the perturbed solution can be described in terms of the spectrum of the Poincaré variational operators defined by the perturbations. This approach turns out to be useful in obtaining estimates which are uniform in the coefficient contrast.
https://eventos.cmm.uchile.cl/seminarioipct/

2024-12-17
15:00hrs.

Grégoire Allaire. Centre de Mathématiques Appliquées, École Polytechnique, France
Optimization of accessibility and application to supports in additive manufacturing
Sala A-016, Universidad Técnica Federico Santa María, campus Santiago San Joaquín.
Abstract:
In this talk I will discuss a geometric constraint, called accessibility constraint, for shape and topology optimization of structures built by additive manufacturing. The motivation comes from the use of sacrificial supports to maintain a structure, submitted to intense thermal residual stresses during its building process. Once the building stage is finished, the supports are of no more use and should be removed. However, such a removal can be very difficult or even impossible if the supports are hidden deep inside the complex geometry of the structure. A rule of thumb for evaluating the ease of support removal is to ask that the contact zone between the structure and its supports can be accessed from the exterior by a straight line which does not cross another part of the structure. It represents the possibility to cut the head of the supports attached to the structure with some cutting tool. We propose a new mathematical approach to evaluate such an accessibility constraint, which is based on distance functions, solutions of eikonal equations. The main advantage is the possibility of computing its shape derivative by the adjoint method, well-knonw in control theory. We numerically demonstrate in 2D and 3D that, in the context of the level-set method for topology optimization, our algorithm can optimize simultaneously the mechanical performance of a structure and the accessibility of its building supports, guaranteeing its additive manufacturing. This is a joint work with M. Bihr, B. Bogosel and M. Godoy.

https://eventos.cmm.uchile.cl/seminarioipct/

2024-11-08
15:00 - 16:00hrs.

Matías Courdurier. Facultad de Matemáticas, Universidad Católica de Chile.
Reconstructing elastic strain fields from its Longitudinal Ray Transform
Auditorio del Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile
Abstract:
In the problem of Bragg-edge elastic strain tomography, measurements are obtained from energy resolved neutron transmission imaging, which provides information about the Longitudinal Ray Transform (LRT) of the elastic strain field. The goal is to recover the elastic strain field by inverting its LRT.

The inversion of the ray transform for tensor fields is a well studied problem [1]. It is known that only the solenoidal part of symmetric tensor fields can be recovered from their LRT, there are inversion formulas available that reconstruct such solenoidal component and there are stability estimates for such reconstruction.

Nonetheless, by taking into account that elastic strain fields additionally satisfy an equilibrium equation, it is possible to also recover the potential part of the elastic strain, for simply connected object [3], or under some extra requirements [2].

In this talk I will present the problem of inverting the LRT for elastic strain tomography that arises from energy resolved neutron transmission imaging,  I will recall the classic results about the inversion of the ray transform for tensor fields, and I will present the contributions of [2,3] dealing with the reconstruction of the potential part of the elastic strain field using the equilibrium equation that is satisfied.

[1] V. A. Sharafutdinov, Integral geometry of tensor fields, Vol. 1, Walter de Gruyter, 2012.

 

[2] C M Wensrich, S Holman, M Courdurier, W R B Lionheart, A P Polyakova and I E Svetov, Direct inversion of the Longitudinal ray transform for 2D residual elastic strain fields, Inverse Problems, Volume 40, Number 7, 2024.

 

[3] C M Wensrich, S Holman, W R B Lionheart, M Courdurier, A P Polyakova, I E Svetov, T Doubikin, General reconstruction of elastic strain fields from their Longitudinal Ray Transform, arXiv:2408.10250, 2024.
https://eventos.cmm.uchile.cl/seminarioipct/seminarios/

2024-10-17
16:30hrs.

Sylvain Ervedoza. Institut de Mathématiques de Bordeaux, Université de Bordeaux and Cnrs
On the reachable space for the heat equation
Auditorio Ninoslav Bralic, Facultad de Matemática, Universidad Católica de Chile.
Abstract:
The goal of this talk is to explain how perturbative arguments can be applied to derive a sharp description of the reachable space for heat equations having lower order terms. The main result I will present is the following one. Let us consider an abstract system y’ = Ay + Bu, where A is an operator generating a C⁰ semigroup (exp(tA))_t≥0 on a Hilbert space X, and B is a control operator, for instance a linear operator from an Hilbert space U to X, and let us assume that this system is null-controllable in X in any positive time. Then, setting R the reachable set of the system (that is all the states that can be achieved by y solution of y’ = Ay + Bu, y(0) = 0), the restriction of (exp(tA))_t≥0 to R forms a C⁰ semigroup on R. Accordingly, the system y’ = Ay + Bu is exactly controllable on R, and one can then perform classical perturbative arguments to handle lower order terms, as I will explain on a few examples. This talk is based on a joint work with Kévin Le Balc’h (INRIA Paris) and Marius Tucsnak (Bordeaux). If time allows, I will also explain the strategy we develop in a recent work with Adrien Tendani-Soler (Bordeaux) to get a more refined description of the reachable space in the case of a ball controlled from its entire boundary, following the recent approach by Alexander Strohmaier and Alden Waters.
https://eventos.cmm.uchile.cl/seminarioipct/

2024-09-06
16:00hrs.

Eduardo Cerpa. Instituto de Ingeniería Matemática y Computacional, Facultad de Matemáticas, Pontificia Universidad Católica de Chile
Singular perturbation method for stability of infinite-dimensional systems
Sala de Seminarios Felipe Álvarez, Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile.
Abstract:
Coupled systems appear everywhere in complex models and in some cases there are different time scales involved. The coupling and the scales make this kind of system very difficult to study from theoretical and computational viewpoints. One hopes that some particular properties of the system could be studied through simpler uncoupled systems. This is what the singular perturbation method (SPM) does concerning stability properties. The SPM approach has been introduced for ordinary differential equations and can also be applied for partial differential equations but in the latter case there are no general theorems and stability properties have to be obtained for each particular system. In this talk we introduce the SPM for infinite-dimensional systems and obtain stability results. We will consider parabolic, hyperbolic and dispersive equations appearing in coupled systems in some cases also involving ordinary differential equations.