We present a splitting strategy to identify simultaneously the local-volatility surface and the jump-size distribution from quoted European prices. The underlying is a jump-diffusion driven asset with time and price dependent volatility. Our approach uses a forward partial-integro-differential equation for the option prices to produce a parameter-to-solution map. The corresponding inverse problem is then solved by Tikhonov-type regularization combined with a splitting strategy.

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.

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.

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 patient-specific 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 time-varying model of arterial brain circulation, based on the solution of the Navier-Stokes 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.

Photoacoustic Tomography is a promising hybrid medical imaging modality that is able to generate high-resolution and high-contrast 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.

This work considers synergistic multi-spectral 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 signal-to-noise ratio. This new image is used as prior information during a channel-by-channel 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.

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 non-smooth 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 Curvature-Driven 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 a-priori 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 TV-based methods entail. Comparisons between the results of the implemented models will be illustrated by several computed examples, along with performance measures.

Numerous applications of research interest may benefit from the study and development of particle sources estimation methods, such as non-destructive 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 one-dimensional energy-dependent and two-dimensional (\(XY\)-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 one-dimensional 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 two-dimensional case. In both cases, we use explicit expressions to derive closed-form 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 Metropolis-Hastings algorithm. We successfully estimated one-dimensional polynomial and piecewise constant localized energy dependent sources and two-dimensional piecewise constant localized sources from noise measurements. In either case, the use of closed-form expressions allowed a relevant reduction in computational time.

The Jacobi system with the Dirichlet boundary condition is considered on a half-line lattice with real-valued 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.

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 quasi-Newton L-BFGS-B, 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, 1035-1038.
Byrd, R.H., Lu, P., Nocedal, J. (1995): A limited memory algorithm for Bound Constrained Optimization. SIAM J. Sci. Stat. Comp. 16 (5): 1190-1208.
Quintero, J.R. , Muñoz, J.C. (2004): Existence and uniqueness for a Boussinesq system with a disordered forcing. Methods Appl. Anal., 11 (1): 015-032.