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Alexandre Jollivet

Publications and source records attributed to Alexandre Jollivet.

18 recordsLinked to original sources
arXiv2021

Boundary Control for Transport Equations

This paper considers two types of boundary control problems for linear transport equations. The first one shows that transport solutions on a subdomain of a domain X can be controlled exactly from incoming boundary conditions for X under appropriate convexity assumptions. This is in contrast with the only approximate control one typically obtains for elliptic equations by an application of a unique continuation property, a property which we prove does not hold for transport equations. We also consider the control of an outgoing solution from incoming conditions, a transport notion similar to the Dirichlet-to-Neumann map for elliptic equations. We show that for well-chosen coefficients in the transport equation, this control may not be possible. In such situations and by (Fredholm) duality, we obtain the existence of non-trivial incoming conditions that are compatible with vanishing outgoing conditions.

math.AP↗
arXiv2020

Convexity properties of the difference over the real axis between the Steklov zeta functions of a smooth planar domain with $2π$ perimeter and of the unit disk

We consider the zeta function $ζ_Ω$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $Ω$ bounded by a smooth closed curve of perimeter $2π$. We prove that $ζ_Ω''(0)\ge ζ_{\mathbb{D}}''(0)$ with equality if and only if $Ω$ is a disk where $\mathbb{D}$ denotes the closed unit disk. We also provide an elementary proof that for a fixed real $s$ satisfying $s\le-1$ the estimate $ζ_Ω''(s)\ge ζ_{\mathbb{D}}''(s)$ holds with equality if and only if $Ω$ is a disk. We then bring examples of domains $Ω$ close to the unit disk where this estimate fails to be extended to the interval $(0,2)$. Other computations related to previous works are also detailed in the remaining part of the text.

math.AP↗
arXiv2020

An estimate for the Steklov zeta function of a planar domain derived from a first variation formula

We consider the Steklov zeta function $ζ$ $Ω$ of a smooth bounded simply connected planar domain $Ω$ $\subset$ R 2 of perimeter 2$π$. We provide a first variation formula for $ζ$ $Ω$ under a smooth deformation of the domain. On the base of the formula, we prove that, for every s $\in$ (--1, 0) $\cup$ (0, 1), the difference $ζ$ $Ω$ (s) -- 2$ζ$ R (s) is non-negative and is equal to zero if and only if $Ω$ is a round disk ($ζ$ R is the classical Riemann zeta function). Our approach gives also an alternative proof of the inequality $ζ$ $Ω$ (s) -- 2$ζ$ R (s) $\ge$ 0 for s $\in$ (--$\infty$, --1] $\cup$ (1, $\infty$); the latter fact was proved in our previous paper [2018] in a different way. We also provide an alternative proof of the equality $ζ$ $Ω$ (0) = 2$ζ$ R (0) obtained by Edward and Wu [1991].

math.AP↗
arXiv2017

Generalized stability estimates in inverse transport theory

Inverse transport theory concerns the reconstruction of the absorption and scattering coefficients in a transport equation from knowledge of the albedo operator, which models all possible boundary measurements. Uniqueness and stability results are well known and are typically obtained for errors of the albedo operator measured in the $L^1$ sense. We claim that such error estimates are not always very informative. For instance, arbitrarily small blurring and misalignment of detectors result in $O(1)$ errors of the albedo operator and hence in $O(1)$ error predictions on the reconstruction of the coefficients, which are not useful. This paper revisit such stability estimates by introducing a more forgiving metric on the measurements errors, namely the $1-$Wasserstein distances, which penalize blurring or misalignment by an amount proportional to the width of the blurring kernel or to the amount of misalignment. We obtain new stability estimates in this setting. We also consider the effect of errors, still measured in the $1-$Wasserstein distance, on the generation of the probing source. This models blurring and misalignment in the design of (laser) probes and allow us to consider a discretized sources. Under appropriate assumptions on the coefficients, we quantify the effect of such errors on the reconstructions.

math.AP↗
arXiv2016

Steklov zeta-invariants and a compactness theorem for isospectral families of planar domains

The inverse problem of recovering a smooth simply connected multisheet planar domain from its Steklov spectrum is equivalent to the problem of determination, up to a gauge transform, of a smooth positive function $a$ on the unit circle from the spectrum of the operator $aΛ$, where $Λ$ is the Dirichlet-to-Neumann operator of the unit disk. Zeta-invariants are defined by $Z_m(a)={\rm Tr}[(aΛ)^{2m}-(aD)^{2m}]$ for every smooth function $a$. In the case of a positive $a$, zeta-invariants are determined by the Steklov spectrum. We obtain some estimate from below for $Z_m(a)$ in the case of a real function $a$. On using the estimate, we prove the compactness of a Steklov isospectral family of planar domains in the $C^\infty$-topology. We also describe all real functions $a$ satisfying $Z_m(a)=0$.

math.SP↗
arXiv2015

An inequality for the zeta function of a planar domain

We consider the zeta function $ζ\_Ω$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $Ω$ bounded by a smooth closed curve.We prove non-negativeness and growth properties for $ζ\_Ω(s)-2\big({L(\partial Ω)\over 2π}\big)^sζ\_R(s)\ (s\leq-1)$, where $L(\partial Ω)$ is the length of the boundary curve and $ζ\_R$ stands for the classical Riemann zeta function.Two analogs of these results are also provided.

math-ph↗
arXiv2013

On inverse scattering at high energies for the multidimensional relativistic Newton equation in a long range electromagnetic field

We define scattering data for the relativistic Newton equation in an electric field $-\nabla V\in C^1(\R^n,\R^n)$, $n\ge 2$, and in a magnetic field $B\in C^1(\R^n,A_n(\R))$ that decay at infinity like $r^{-α-1}$ for some $α\in (0,1]$, where $A_n(\R)$ is the space of $n\times n$ antisymmetric matrices. We provide estimates on the scattering solutions and on the scattering data and we prove, in particular, that the scattering data at high energies uniquely determine the short range part of $(\nabla V,B)$ up to the knowledge of the long range tail of $(\nabla V,B)$. The Born approximation at fixed energy of the scattering data is also considered. We then change the definition of the scattering data to study their behavior in other asymptotic regimes. This work generalizes [Jollivet, 2007] where a short range electromagnetic field was considered.

math-ph↗
arXiv2013

Inverse scattering at high energies for the multidimensional Newton equation in a long range potential

We define scattering data for the Newton equation in a potential $V\in C^2(\R^n,\R)$, $n\ge2$, that decays at infinity like $r^{-α}$ for some $α\in (0,1]$. We provide estimates on the scattering solutions and scattering data and we prove, in particular, that the scattering data at high energies uniquely determine the short range part of the potential up to the knowledge of the long range tail of the potential. The Born approximation at fixed energy of the scattering data is also considered. We then change the definition of the scattering data to study inverse scattering in other asymptotic regimes. These results were obtained by developing the inverse scattering approach of [Novikov, 1999].

math-ph↗
arXiv2012

Inverse scattering at fixed energy for the multidimensional Newton equation in short range radial potentials

We consider the inverse scattering problem at fixed and sufficiently large energy for the nonrelativistic and relativistic Newton equation in $\R^n$, $n \ge 2$, with a smooth and short range electromagnetic field $(V,B)$. Using results of [Firsov, 1953] or [Keller-Kay-Shmoys, 1956] we obtain a uniqueness result when $B$ is assumed to be zero in a neighborhood of infinity and $V$ is assumed to be spherically symmetric in a neighborhood of infinity.

math-ph↗
arXiv2009

Inverse Transport Theory of Photoacoustics

We consider the reconstruction of optical parameters in a domain of interest from photoacoustic data. Photoacoustic tomography (PAT) radiates high frequency electromagnetic waves into the domain and measures acoustic signals emitted by the resulting thermal expansion. Acoustic signals are then used to construct the deposited thermal energy map. The latter depends on the constitutive optical parameters in a nontrivial manner. In this paper, we develop and use an inverse transport theory with internal measurements to extract information on the optical coefficients from knowledge of the deposited thermal energy map. We consider the multi-measurement setting in which many electromagnetic radiation patterns are used to probe the domain of interest. By developing an expansion of the measurement operator into singular components, we show that the spatial variations of the intrinsic attenuation and the scattering coefficients may be reconstructed. We also reconstruct coefficients describing anisotropic scattering of photons, such as the anisotropy coefficient $g(x)$ in a Henyey-Greenstein phase function model. Finally, we derive stability estimates for the reconstructions.

math-ph↗
arXiv2009

Time-dependent angularly averaged inverse transport

This paper concerns the reconstruction of the absorption and scattering parameters in a time-dependent linear transport equation from knowledge of angularly averaged measurements performed at the boundary of a domain of interest. We show that the absorption coefficient and the spatial component of the scattering coefficient are uniquely determined by such measurements. We obtain stability results on the reconstruction of the absorption and scattering parameters with respect to the measured albedo operator. The stability results are obtained by a precise decomposition of the measurements into components with different singular behavior in the time domain.

math-ph↗
arXiv2008

Stability for time-dependent inverse transport

This paper concerns the reconstruction of the absorption and scattering parameters in a time-dependent linear transport equation from full knowledge of the albedo operator at the boundary of a bounded domain of interest. We present optimal stability results on the reconstruction of the absorption and scattering parameters for a given error in the measured albedo operator.

math-ph↗
arXiv2008

Stability estimates in stationary inverse transport

We study the stability of the reconstruction of the scattering and absorption coefficients in a stationary linear transport equation from knowledge of the full albedo operator in dimension $n\geq3$. The albedo operator is defined as the mapping from the incoming boundary conditions to the outgoing transport solution at the boundary of a compact and convex domain. The uniqueness of the reconstruction was proved in [M. Choulli-P. Stefanov, 1996 and 1999] and partial stability estimates were obtained in [J.-N. Wang, 1999] for spatially independent scattering coefficients. We generalize these results and prove an $L^1$-stability estimate for spatially dependent scattering coefficients.

math-ph↗
arXiv2007

On inverse scattering at high energies for the multidimensional Newton equation in electromagnetic field

We consider the multidimensional (nonrelativistic) Newton equation in a static electromagnetic field $$\ddot x = F(x,\dot x), F(x,\dot x)=-\nabla V(x)+B(x)\dot x, \dot x={dx\over dt}, x\in C^2(\R,\R^n),\eqno{(*)}$$ where $V \in C^2(\R^n,\R),$ $B(x)$ is the $n\times n$ real antisymmetric matrix with elements $B_{i,k}(x)$, $B_{i,k}\in C^1(\R^n,\R)$ (and $B$ satisfies the closure condition), and $|\pa^{j_1}_xV(x)| +|\pa^{j_2}_xB_{i,k}(x)| \le β_{|j_1|} (1+|x|)^{-(α+|j_1|)}$ for $x\in \R^n,$ $1\le|j_1|\le 2,$ $0\le|j_2|\le 1$, $|j_2|=|j_1|-1$, $i,k=1... n$ and some $α> 1$. We give estimates and asymptotics for scattering solutions and scattering data for the equation $(*)$ for the case of small angle scattering. We show that at high energies the velocity valued component of the scattering operator uniquely determines the X-ray transforms $P\nabla V$ and $PB_{i,k}$ (on sufficiently rich sets of straight lines). Applying results on inversion of the X-ray transform $P$ we obtain that for $n\ge 2$ the velocity valued component of the scattering operator at high energies uniquely determines $(\nabla V,B)$. We also consider the problem of recovering $(\nabla V,B)$ from our high energies asymptotics found for the configuration valued component of the scattering operator. Results of the present work were obtained by developing the inverse scattering approach of [R. Novikov, 1999] for $(*)$ with $B\equiv 0$ and of [Jollivet, 2005] for the relativistic version of $(*)$. We emphasize that there is an interesting difference in asymptotics for scattering solutions and scattering data for $(*)$ on the one hand and for its relativistic version on the other.

math-ph↗
arXiv2007

On inverse problems in electromagnetic field in classical mechanics at fixed energy

In this paper, we consider inverse scattering and inverse boundary value problems at sufficiently large and fixed energy for the multidimensional relativistic and nonrelativistic Newton equations in a static external electromagnetic field $(V,B)$, $V\in C^2,$ $B\in C^1$ in classical mechanics. Developing the approach going back to Gerver-Nadirashvili 1983's work on an inverse problem of mechanics, we obtain, in particular, theorems of uniqueness.

math-ph↗
arXiv2006

On inverse scattering in electromagnetic field in classical relativistic mechanics at high energies

We consider the multidimensional Newton-Einstein equation in static electromagnetic field $$\eqalign{\dot p = F(x,\dot x), F(x,\dot x)=-\nabla V(x)+{1\over c}B(x)\dot x,\cr p={\dot x \over \sqrt{1-{|\dot x|^2 \over c^2}}}, \dot p={dp\over dt}, \dot x={dx\over dt}, x\in C^1(\R,\R^d),}\eqno{(*)}$$ where $V \in C^2(\R^d,\R),$ $B(x)$ is the $d\times d$ real antisymmetric matrix with elements $B\_{i,k}(x)={\pa\over \pa x\_i}\A\_k(x)-{\pa\over \pa x\_k}\A\_i(x)$, and $|\pa^j\_x\A\_i(x)|+|\pa^j\_x V(x)| \le β\_{|j|}(1+|x|)^{-(α+|j|)}$ for $x\in \R^d,$ $|j| \le 2,$ $i=1..d$ and some $α> 1$. We give estimates and asymptotics for scattering solutions and scattering data for the equation $(*)$ for the case of small angle scattering. We show that at high energies the velocity valued component of the scattering operator uniquely determines the X-ray transforms $P\nabla V$ and $PB\_{i,k}$ for $i,k=1..d,$ $i\neq k.$ Applying results on inversion of the X-ray transform $P$ we obtain that for $d\ge 2$ the velocity valued component of the scattering operator at high energies uniquely determines $(V,B)$. In addition we show that our high energy asymptotics found for the configuration valued component of the scattering operator doesn't determine uniquely $V$ when $d\ge 2$ and $B$ when $d=2$ but that it uniquely determines $B$ when $d\ge 3.$

math-ph↗
arXiv2006

On inverse scattering for the multidimensional relativistic Newton equation at high energies

Consider the Newton equation in the relativistic case (that is the Newton-Einstein equation) $$\eqalign{\dot p = F(x),& F(x)=-\nabla V(x),\cr p={\dot x \over \sqrt{1-{|\dot x|^2 \over c^2}}},& \dot p={dp\over dt}, \dot x={dx\over dt}, x\in C^1(\R,\R^d),}\eqno{(*)}$$ $${\rm where\}V \in C^2(\R^d,\R), |\pa^j\_x V(x)| \le β\_{|j|}(1+|x|)^{-(α+|j|)}$$ for $|j| \le 2$ and some $α> 1$. We give estimates and asymptotics for scattering solutions and scattering data for the equation $(*)$ for the case of small angle scattering. We show that at high energies the velocity valued component of the scattering operator uniquely determines the X-ray transform $PF.$ Applying results on inversion of the X-ray transform $P$ we obtain that for $d\ge 2$ the velocity valued component of the scattering operator at high energies uniquely determines $F$. In addition we show that our high energy asymptotics found for the configuration valued component of the scattering operator doesn't determine uniquely $F$. The results of the present work were obtained in the process of generalizing some results of Novikov [R.G. Novikov, Small angle scattering and X-ray transform in classical mechanics, Ark. Mat. 37, pp. 141-169 (1999)] to the relativistic case.

math-ph↗