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Michel Cristofol

Publications and source records attributed to Michel Cristofol.

17 recordsLinked to original sources
arXiv2026

Inverse problems for the diffusion equation with one time observation

We examine the inverse problem of recovering the potential term of an n-dimensional heat equation from a single time solution profile. We show that it is possible to uniquely determine this potential term via a stability inequality using solution measurements taken at a fixed time, assuming the potential is known within an arbitrary subdomain. The approach is based on the Carleman estimate. Additionally, we discuss the optimality of this type of observation.

math.AP↗
arXiv2026

Inverse scattering in an asymptotically flat multilayer domain

We consider a scattering problem for a wave equation $\partial_t^2 u = \frac{1}{\sqrt{g}}\partial_i(\sqrt{g}g^{ij}\partial_j)u$ in a multilayer domain $Ω\subset {\bf R}^{n+1}_x = {\bf R}^n_y \times {\bf R}^1_{x^{n+1}}$ of the form $Ω= \mathcal K \cup Ω_1 \cup \cdots \cup Ω_N$, where $\mathcal K$ is a bounded open set and $Ω_k$ is asymptotically equal to a slab domain ${\bf R}^n \times (c_k,c_k + d_k)$ as $|y| \to \infty$. Assuming that $\partial_x^α\big(g_{ij}(x) - δ_{ij}\big) = O(|x|^{-|α| - δ_0}), \ δ_0 > 1, \forall α$, we show that $Ω$ and $g^{ij}$ are determined by one diagonal component $S_{11}(λ)$, for all energies, of the S-matrix associated with the slab $Ω_1$, provided $Ω_1$ is flat: $Ω_1 \cap \{|y| > R\} = \{|y| > R\} \times (c_1, c_1+d_1)$ for some constants $c_1, d_1, R > 0$, and the metric is Euclidean on $Ω_1\cap \{|y| > R\}$.

math.SP↗
arXiv2017

Stable reconstruction of the volatility in a regime-switching local volatility model

Prices of European call options in a regime-switching local volatility model can be computed by solving a parabolic system which generalises the classical Black and Scholes equation, giving these prices as functionals of the local volatilities. We prove Lipschitz stability for the inverse problem of determining the local volatilities from quoted call option prices for a range of strikes, if the calls are indexed by the different states of the continuous Markov chain which governs the regime switches.

math.AP↗
arXiv2017

Simultaneous determination of the drift and diffusion coefficients in stochastic differential equations

In this work, we consider a one-dimensional It{ô} diffusion process X t with possibly nonlinear drift and diffusion coefficients. We show that, when the diffusion coefficient is known, the drift coefficient is uniquely determined by an observation of the expectation of the process during a small time interval, and starting from values X 0 in a given subset of R. With the same type of observation, and given the drift coefficient, we also show that the diffusion coefficient is uniquely determined. When both coefficients are unknown, we show that they are simultaneously uniquely determined by the observation of the expectation and variance of the process, during a small time interval, and starting again from values X 0 in a given subset of R. To derive these results, we apply the Feynman-Kac theorem which leads to a linear parabolic equation with unknown coefficients in front of the first and second order terms. We then solve the corresponding inverse problem with PDE technics which are mainly based on the strong parabolic maximum principle.

math.AP↗
arXiv2016

Inverse Problem for a Curved Quantum Guide

In this paper, we consider the Dirichlet Laplacian operator -Δ on a curved quantum guide in R n (n = 2, 3) with an asymptotically straight reference curve. We give uniqueness results for the inverse problem associated to the reconstruction of the curvature by using either observations of spectral data or a boot-strapping method. keywords: Inverse Problem, Quantum Guide, Curvature

math.AP↗
arXiv2015

Optimization approach for the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions from limited observations

We consider the inverse problem of the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions of the Maxwell's system in 3D with limited boundary observations of the electric field. The theoretical stability for the problem is provided by the Carleman estimates. For the numerical computations the problem is formulated as an optimization problem and hybrid finite element/difference method is used to solve the parameter identification problem.

math.NA↗
arXiv2012

Inverse boundary value problem for the dynamical heterogeneous Maxwell system

We consider the inverse problem of determining the isotropic inhomogeneous electromagnetic coefficients of the non-stationary Maxwell equations in a bounded domain of R^3, from a finite number of boundary measurements. Our main result is a Hölder stability estimate for the inverse problem, where the measurements are exerted only in some boundary components. For it, we prove a global Carleman estimate for the heterogeneous Maxwell's system with boundary conditions.

math.AP↗
arXiv2012

Stability estimate in an inverse problem for non-autonomous Schrödinger equations

We consider the inverse problem of determining the time dependent magnetic field of the Schrödinger equation in a bounded open subset of $R^n$, with $n \geq 1$, from a finite number of Neumann data, when the boundary measurement is taken on an appropriate open subset of the boundary. We prove the Lispchitz stability of the magnetic potential in the Coulomb gauge class by $n$ times changing initial value suitably.

math.AP↗
arXiv2011

Uniqueness from pointwise observations in a multi-parameter inverse problem

In this paper, we prove a uniqueness result in the inverse problem of determining several non-constant coefficients of one-dimensional reaction-diffusion equations. Such reaction-diffusion equations include the classical model of Kolmogorov, Petrovsky and Piskunov as well as more sophisticated models from biology. When the reaction term contains an unknown polynomial part of degree $N,$ with non-constant coefficients $μ_k(x),$ our result gives a sufficient condition for the uniqueness of the determination of this polynomial part. This sufficient condition only involves pointwise measurements of the solution $u$ of the reaction-diffusion equation and of its spatial derivative $\partial u / \partial x$ at a single point $x_0,$ during a time interval $(0,ε).$ In addition to this uniqueness result, we give several counter-examples to uniqueness, which emphasize the optimality of our assumptions. Finally, in the particular cases N=2 and $N=3,$ we show that such pointwise measurements can allow an efficient numerical determination of the unknown polynomial reaction term.

math.AP↗
arXiv2010

On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation

This paper is devoted to the analysis of some uniqueness properties of a classical reaction-diffusion equation of Fisher-KPP type, coming from population dynamics in heterogeneous environments. We work in a one-dimensional interval $(a,b)$ and we assume a nonlinear term of the form $u \, (μ(x)-γu)$ where $μ$ belongs to a fixed subset of $C^{0}([a,b])$. We prove that the knowledge of $u$ at $t=0$ and of $u$, $u_x$ at a single point $x_0$ and for small times $t\in (0,\varepsilon)$ is sufficient to completely determine the couple $(u(t,x),μ(x))$ provided $γ$ is known. Additionally, if $u_{xx}(t,x_0)$ is also measured for $t\in (0,\varepsilon)$, the triplet $(u(t,x),μ(x),γ)$ is also completely determined. Those analytical results are completed with numerical simulations which show that, in practice, measurements of $u$ and $u_x$ at a single point $x_0$ (and for $t\in (0,\varepsilon)$) are sufficient to obtain a good approximation of the coefficient $μ(x).$ These numerical simulations also show that the measurement of the derivative $u_x$ is essential in order to accurately determine $μ(x)$.

math.AP↗
arXiv2009

Biological invasions: deriving the regions at risk from partial measurements

We consider the problem of forecasting the regions at higher risk for newly introduced invasive species. Favourable and unfavourable regions may indeed not be known a priori, especially for exotic species whose hosts in native range and newly-colonised areas can be different. Assuming that the species is modelled by a logistic-like reaction-diffusion equation, we prove that the spatial arrangement of the favourable and unfavourable regions can theoretically be determined using only partial measurements of the population density: 1) a local "spatio-temporal" measurement, during a short time period and, 2) a "spatial" measurement in the whole region susceptible to colonisation. We then present a stochastic algorithm which is proved analytically, and then on several numerical examples, to be effective in deriving these regions.

math.AP↗
arXiv2008

Inverse problem for a parabolic system with two components by measurements of one component

We consider a $2\times 2$ system of parabolic equations with first and zeroth coupling and establish a Carleman estimate by extra data of only one component without data of initial values. Then we apply the Carleman estimate to inverse problems of determining some or all of the coefficients by observations in an arbitrary subdomain over a time interval of only one component and data of two components at a fixed positive time $θ$ over the whole spatial domain. The main results are Lipschitz stability estimates for the inverse problems. For the Lipschitz stability, we have to assume some non-degeneracy condition at $θ$ for the two components and for it, we can approximately control the two components of the $2 \times 2$ system by inputs to only one component. Such approximate controllability is proved also by our new Carleman estimate. Finally we establish a Carleman estimate for a $3\times 3$ system for parabolic equations with coupling of zeroth-order terms by one component to show the corresponding approximate controllability with a control to one component.

math.AP↗
arXiv2007

Inverse Problem for the Schrödinger Operator in an Unbounded Strip

We consider the operator $H:= i \partial_t + \nabla \cdot (c \nabla)$ in an unbounded strip $Ω$ in $\mathbb{R}^2$, where $c(x,y) \in \mathcal{C}^3(\barΩ)$. We prove adapted a global Carleman estimate and an energy estimate for this operator. Using these estimates, we give a stability result for the diffusion coefficient $c(x,y)$.

math.AP↗