Spectral calibration of exponential Lévy Models [1]

We investigate the problem of calibrating an exponential Lévy model based on market prices of vanilla options. We show that this inverse problem is in general severely ill-posed and we derive exact minimax rates of convergence. The estimation procedure we propose is based on the explicit inversion of the option price formula in the spectral domain and a cut-off scheme for high frequencies as regularisation.European option, jump diffusion, minimax rates, severely ill-posed, nonlinear inverse problem, spectral cut-off

A variational method for quantitative photoacoustic tomography with piecewise constant coefficients

In this article, we consider the inverse problem of determining spatially heterogeneous absorption and diffusion coefficients from a single measurement of the absorbed energy (in the steady-state diffusion approximation of light transfer). This problem, which is central in quantitative photoacoustic tomography, is in general ill-posed since it admits an infinite number of solution pairs. We show that when the coefficients are known to be piecewise constant functions, a unique solution can be obtained. For the numerical determination of the coefficients, we suggest a variational method based based on an Ambrosio-Tortorelli-approximation of a Mumford-Shah-like functional, which we implemented numerically and tested on simulated two-dimensional data

Inverse Scattering and Acousto-Optic Imaging

We propose a tomographic method to reconstruct the optical properties of a highly-scattering medium from incoherent acousto-optic measurements. The method is based on the solution to an inverse problem for the diffusion equation and makes use of the principle of interior control of boundary measurements by an external wave field.Comment: 10 page

Inverse problem of determining time-dependent leading coefficient in the time-fractional heat equation

In this paper, we investigate direct and inverse problems for the time-fractional heat equation with a time-dependent diffusion coefficient for positive operators. First, we consider the direct problem, and the unique existence of the generalized solution is established. We also deduce some regularity results. Here, our proofs are based on the eigenfunction expansion method. Second, we consider the inverse problem of determining the diffusion coefficient. The well-posedness of this inverse problem is shown by reducing the problem to an operator equation for the diffusion coefficient.Comment: arXiv admin note: text overlap with arXiv:1708.07756 by other author

Diffusion and convection of gaseous and fine particulate from a chimney

Particle dispersion from a high chimney is considered and an expression for the subsequent concentration of the particulate deposited on the ground is derived. We consider the general case wherein the effects of both diffusion and convection on the steady state ground concentration of particulate are incorporated. Two key parameters emerge from this analysis: the ratio of diffusion to convection and the nondimensionalised surface mass transfer rate. We also solve the inverse problem of recovering these two parameters given the boundary concentration profile and provide an estimate of the concentration flux above the chimney stack

Cell Detection by Functional Inverse Diffusion and Non-negative Group Sparsity−-Part I: Modeling and Inverse Problems

In this two-part paper, we present a novel framework and methodology to analyze data from certain image-based biochemical assays, e.g., ELISPOT and Fluorospot assays. In this first part, we start by presenting a physical partial differential equations (PDE) model up to image acquisition for these biochemical assays. Then, we use the PDEs' Green function to derive a novel parametrization of the acquired images. This parametrization allows us to propose a functional optimization problem to address inverse diffusion. In particular, we propose a non-negative group-sparsity regularized optimization problem with the goal of localizing and characterizing the biological cells involved in the said assays. We continue by proposing a suitable discretization scheme that enables both the generation of synthetic data and implementable algorithms to address inverse diffusion. We end Part I by providing a preliminary comparison between the results of our methodology and an expert human labeler on real data. Part II is devoted to providing an accelerated proximal gradient algorithm to solve the proposed problem and to the empirical validation of our methodology.Comment: published, 15 page