Problemas inversos: desde la teoría a las aplicaciones
Organizadores: Adriano Dr Cezaro (adrianocezaro@furg.br), Alberto Mercado (alberto.mercado@usm.cl), Juan Pablo Agnelli (agnelli@famaf.unc.edu.ar)

Lunes 13
15:00  15:45A Splitting Strategy for the Calibration of JumpDiffusion Models
Vinicus V. Albani (Universidade Federal de Santa Catarina, Brasil)
We present a splitting strategy to identify simultaneously the localvolatility surface and the jumpsize distribution from quoted European prices. The underlying is a jumpdiffusion driven asset with time and price dependent volatility. Our approach uses a forward partialintegrodifferential equation for the option prices to produce a parametertosolution map. The corresponding inverse problem is then solved by Tikhonovtype regularization combined with a splitting strategy.
15:45  16:30Bayesian Approach Helmholtz Inverse Problem
Lilí Guadarrama (Centro de Investigación en Matemáticas, México)
This work present an implementation for solving the Helmholtz inverse problem via Finite Element Methods and Bayesian inference theory. The solution of the inverse problem is a posterior distribution based on a number observation data sets, the forward problem governing the state solution, and a statistical prior distribution which describes the behavior of the uncertainty linked to the parameters.
16:45  17:30Lipschitz Stability for the Backward Heat Equation and its application to Lightsheet Fluorescence Microscopy
Matias Courdurier (Pontificia Universidad Católica, Chile)
In this talk we will present a result about Lipschitz stability for the reconstruction of a compactly supported initial temperature for the heat equation in \(\mathbb{R}^n\), from measurements along a positive time interval and over an open set containing its support. We apply these results to deduce a Lipschitz stability inequality for the backwards heat equation problem in \(\mathbb{R}\) with measurements on a curve contained in \( \mathbb{R}\times[0,T]\), leading to stability estimates for an inverse problem arising in 2D Fluorescence Microscopy.
This is joint work with Pablo Arratia, Evelyn Cueva, Axel Osses and Benjamin Palacios.

Martes 14
15:00  15:45Anatomical atlas of the upper part of the human head for electroencephalography and bioimpedance applications
Fernando Moura da Silva (Universidade Federal do ABC, Brasil y University of Helsinki, Finlandia)
Electrophysiology is the branch of physiology that investigates the electrical properties of biological tissues. Volume conductor problems in cerebral electrophysiology and bioimpedance do not have analytical solutions for nontrivial geometries and require a 3D model of the head and its electrical properties for solving the associated PDEs numerically.
Ideally, the model should be made with patientspecific information. In clinical practice, this is not always the case and an average head model is often used. Also, the electrical properties of the tissues might not be completely known due to natural variability.
A 4D (3D+T) statistical anatomical atlas of the electrical properties of the upper part of the human head for cerebral electrophysiology and bioimpedance applications is presented. The atlas is constructed based on MRI images of human individuals and comprises the electrical properties of the main internal structures and can be adjusted for specific electrical frequencies. The atlas also comprises a timevarying model of arterial brain circulation, based on the solution of the NavierStokes equations.
The atlas is an important tool for in silico studies on cerebral circulation and electrophysiology that require statistically consistent data, e.g., machine learning, sensitivity analyses, and as a benchmark to test inverse problem solvers. The atlas can also be used as statistical prior information for inverse problems in electrophysiology.
15:45  16:30Partial data Photoacoustic Tomography in unbounded domains
Benjamín Palacios (Pontificia Universidad Católica de Chile, Chile)
Photoacoustic Tomography is a promising hybrid medical imaging modality that is able to generate highresolution and highcontrast images, by exploiting the coupling of electromagnetic pulses (in the visible region) and ultrasound waves via de photoacoustic effect.
In this talk I will introduce the modality and focus on the ultrasound propagation part which is mathematically modeled as an inverse initial source problem for the wave equation. We will pose this problem in a setting consisting of an unbounded domain with boundary, with observation taking place in a subset of it. We will address the question of uniqueness, stability and reconstruction of the initial source of acoustic waves.
16:45  17:30Synergistic multispectral CT reconstruction with directional total variation
Evelyn Cueva (Escuela Politécnica Nacional, Ecuador)
This work considers synergistic multispectral CT reconstruction where information from all available energy channels is combined to improve the reconstruction of each individual channel. We propose to fuse these available data (represented by a single sinogram) to obtain a polyenergetic image that keeps structural information shared by the energy channels with an increased signaltonoise ratio. This new image is used as prior information during a channelbychannel minimization process through the directional total variation. We analyze the use of directional total variation within variational regularization and iterative regularization. Our numerical results on simulated and experimental data show improvements in terms of image quality and in computational speed.
17:30  18:15Joint curvaturedriven diffusion and weighted anisotropic total variation regularization for local inpainting
Ruben D. Spies (Instituto de Matemática Aplicada del Litoral, Argentina)
An image inpainting problem consists of restoring an image from a (possibly noisy) observation, in which data from one or more regions is missing. Several inpainting models to perform this task have been developed, and although some of them perform reasonably well in certain types of images, quite a few issues are yet to be sorted out. For instance, if the original image is expected to be smooth, inpainting can be performed with reasonably good results by modeling the solution as the result of a diffusion process using the heat equation. For nonsmooth images, however, such an approach is far from being satisfactory. On the other hand, Total Variation (TV) inpainting models based on high order PDE diffusion equations can be used whenever edge restoration is a priority. More recently, the introduction of spatially variant conductivity coefficients on these models, such as in the case of CurvatureDriven Diffusions (CDD), has allowed inpainted images with well defined edges and enhanced object connectivity. The CDD approach, nonetheless, is not suitable wherever the image is smooth, as it tends to produce piecewise constant solutions. Based upon this, we propose using CDD to gather apriori information used at a second step in a weighted anisotropic mixed Tikhonov plus total variation model that allows for both edge preservation and object connectivity while precluding the staircasing effect that all pure TVbased methods entail. Comparisons between the results of the implemented models will be illustrated by several computed examples, along with performance measures.

Miércoles 15
15:00  15:45Particle Source Estimation via Analytical Formulations of the Adjoint Flux
Liliane Basso Barichello (Universidade Federal do Rio Grande do Sul, Brasil)
Numerous applications of research interest may benefit from the study and development of particle sources estimation methods, such as nondestructive material identification and nuclear safety. In such problems, we are concerned with estimating the spatial distribution and intensity of internal sources of neutral particles. In this context, we use the adjoint to the transport operator to derive a linear model that relates the absorption rate measurements of a series of internal particle detectors with the coefficients of the source expansion on a given basis. We consider both onedimensional energydependent and twodimensional (\(XY\)geometry) monoenergetic transport problems. We use the adjoint version of the Analytical Discrete Ordinates (ADO) method to write spatially explicit solutions for the adjoint flux for onedimensional problems. We consider a nodal version of the adjoint ADO method to writing similar explicit expressions for the \(x\) and \(y\) averaged adjoint fluxes for the twodimensional case. In both cases, we use explicit expressions to derive closedform formulas to the absorption rate. Then, we apply the iterated Tikhonov algorithm to estimate the sources from noisy measurements and a Bayesian approach through the MetropolisHastings algorithm. We successfully estimated onedimensional polynomial and piecewise constant localized energy dependent sources and twodimensional piecewise constant localized sources from noise measurements. In either case, the use of closedform expressions allowed a relevant reduction in computational time.
15:45  16:30Inverse scattering for the Jacobi system using input data containing transmission eigenvalues
Abdon Choque Rivero (Universidad Michoacana de San Nicolás de Hidalgo, México)
The Jacobi system with the Dirichlet boundary condition is considered on a halfline lattice with realvalued coefficients including weight factors. The inverse problem of recovery of the coefficients is analyzed by using certain input data sets containing the transmission eigenvalues.
This is joint work with T. Aktosun of University of Texas at Arlington and V. G. Papanicolaou of National Technical University of Athens.
16:45  17:30Identificación del coeficiente de velocidad local en un modelo bidireccional para la propagación de ondas en la superficie de un canal con fondo variable
Juan Carlos Muñoz Grajales (Universidad del Valle, Colombia)
Un problema inverso es aquel donde se requiere determinar el valor de algunos parámetros de un modelo a partir de datos observados. El estudio de este tipo de problemas es de mucha importancia y aparece en diversas ramas de la ciencia y de las Matemáticas. En esta charla consideramos el problema inverso de identificación del coeficiente de velocidad lineal local en un sistema de tipo Boussinesq linealizado de tipo dispersivo deducido por Quintero y Muñoz (2004), a partir de las mediciones \(V_T(x), N_T(x)\) en un tiempo final \(T\), de la velocidad de las partículas en una profundidad fija y la amplitud de la onda, que se propaga en la superficie de un canal raso con fondo variable.
El problema inverso es reformulado como un problema de optimización con restricciones, donde la funcional objetivo es de tipo usado en métodos de mínimos cuadrados junto con una regularización del tipo introducido por Tikhonov (1963). El proceso de minimización del funcional Tikhonov adoptado se realiza utilizando el método iterativo quasiNewton LBFGSB, introducido por Byrd et al. (1995). Se presentan simulaciones numéricas para mostrar la convergencia y robustez del método numérico propuesto, a\'un en el caso en que el coeficiente posee discontinuidades, o existe ruido en la medici\'on de los datos de entrada \(V_T(x), N_T(x)\).
Investigación realizada con el apoyo de la Universidad del Valle mediante proyecto C.I. 71235.
Referencias:
Tikhonov, A.N. (1963): Solution of incorrectly formulated problems and the regularization method. Soviet Math. Dokl. 4, 10351038.
Byrd, R.H., Lu, P., Nocedal, J. (1995): A limited memory algorithm for Bound Constrained Optimization. SIAM J. Sci. Stat. Comp. 16 (5): 11901208.
Quintero, J.R. , Muñoz, J.C. (2004): Existence and uniqueness for a Boussinesq system with a disordered forcing. Methods Appl. Anal., 11 (1): 015032.