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Matteo Santacesaria

Publications and source records attributed to Matteo Santacesaria.

46 records · Page 3Linked to original sources
arXiv2015

Positive-energy D-bar method for acoustic tomography: a computational study

A new computational method for reconstructing a potential from the Dirichlet-to-Neumann map at positive energy is developed. The method is based on D-bar techniques and it works in absence of exceptional points -- in particular, if the potential is small enough compared to the energy. Numerical tests reveal exceptional points for perturbed, radial potentials. Reconstructions for several potentials are computed using simulated Dirichlet-to-Neumann maps with and without added noise. The new reconstruction method is shown to work well for energy values between $10^{-5}$ and $5$, smaller values giving better results.

math.NA↗
arXiv2014

On an inverse problem for anisotropic conductivity in the plane

Let $\hat Ω\subset \mathbb R^2$ be a bounded domain with smooth boundary and $\hat σ$ a smooth anisotropic conductivity on $\hat Ω$. Starting from the Dirichlet-to-Neumann operator $Λ_{\hat σ}$ on $\partial \hat Ω$, we give an explicit procedure to find a unique domain $Ω$, an isotropic conductivity $σ$ on $Ω$ and the boundary values of a quasiconformal diffeomorphism $F:\hat Ω\to Ω$ which transforms $\hat σ$ into $σ$.

math.AP↗
arXiv2013

A Holder-logarithmic stability estimate for an inverse problem in two dimensions

The problem of the recovery of a real-valued potential in the two-dimensional Schrodinger equation at positive energy from the Dirichlet-to-Neumann map is considered. It is know that this problem is severely ill-posed and the reconstruction of the potential is only logarithmic stable in general. In this paper a new stability estimate is proved, which is explicitly dependent on the regularity of the potentials and on the energy. Its main feature is an efficient increasing stability phenomenon at sufficiently high energies: in some sense, the stability rapidly changes from logarithmic type to Holder type. The paper develops also several estimates for a non-local Riemann-Hilbert problem which could be of independent interest.

math.AP↗
arXiv2012

Stability estimates for an inverse problem for the Schrödinger equation at negative energy in two dimensions

We study the inverse problem of determining a real-valued potential in the two-dimensional Schrödinger equation at negative energy from the Dirichlet-to-Neumann map. It is known that the problem is ill-posed and a stability estimate of logarithmic type holds. In this paper we prove three new stability estimates. The main feature of the first one is that the stability increases exponentially with respect to the smoothness of the potential, in a sense to be made precise. The others show how the first estimate depends on the energy, for low and high energies (in modulus). In particular it is found that for high energies the stability estimate changes, in some sense, from logarithmic type to Lipschitz type: in this sense the ill-posedness of the problem decreases when increasing the energy (in modulus).

math.AP↗
arXiv2012

New global stability estimates for the Calderón problem in two dimensions

We prove a new global stability estimate for the Gel'fand-Calderón inverse problem on a two-dimensional bounded domain or, more precisely, the inverse boundary value problem for the equation $-Δψ+ v\, ψ= 0$ on $D$, where $v$ is a smooth real-valued potential of conductivity type defined on a bounded planar domain $D$. The principal feature of this estimate is that it shows that the more a potential is smooth, the more its reconstruction is stable, and the stability varies exponentially with respect to the smoothness (in a sense to be made precise). As a corollary we obtain a similar estimate for the Calderón problem for the electrical impedance tomography.

math.AP↗
arXiv2011

Monochromatic reconstruction algorithms for two-dimensional multi-channel inverse problems

We consider two inverse problems for the multi-channel two-dimensional Schrödinger equation at fixed positive energy, i.e. the equation $-Δψ+ V(x)ψ= E ψ$ at fixed positive $E$, where $V$ is a matrix-valued potential. The first is the Gel'fand inverse problem on a bounded domain $D$ at fixed energy and the second is the inverse fixed-energy scattering problem on the whole plane $\R^2$. We present in this paper two algorithms which give efficient approximate solutions to these problems: in particular, in both cases we show that the potential $V$ is reconstructed with Lipschitz stability by these algorithms up to $O(E^{-(m-2)/2})$ in the uniform norm as $E \to +\infty$, under the assumptions that $V$ is $m$-times differentiable in $L^1$, for $m \geq 3$, and has sufficient boundary decay.

math.AP↗
arXiv2011

Global uniqueness and reconstruction for the multi-channel Gel'fand-Calderón inverse problem in two dimensions

We study the multi-channel Gel'fand-Calderón inverse problem in two dimensions, i.e. the inverse boundary value problem for the equation $-Δψ+ v(x) ψ= 0$, $x\in D$, where $v$ is a smooth matrix-valued potential defined on a bounded planar domain $D$. We give an exact global reconstruction method for finding $v$ from the associated Dirichlet-to-Neumann operator. This also yields a global uniqueness results: if two smooth matrix-valued potentials defined on a bounded planar domain have the same Dirichlet-to-Neumann operator then they coincide.

math.AP↗
arXiv2011

Gel'fand-Calderón's inverse problem for anisotropic conductivities on bordered surfaces in $\mathbb{R}^3$

Let $X$ be a smooth bordered surface in $\real^3$ with smooth boundary and $\hat σ$ a smooth anisotropic conductivity on $X$. If the genus of $X$ is given, then starting from the Dirichlet-to-Neumann operator $Λ_{\hat σ}$ on $\partial X$, we give an explicit procedure to find a unique Riemann surface $Y$ (up to a biholomorphism), an isotropic conductivity $σ$ on $Y$ and the boundary values of a quasiconformal diffeomorphism $F: X \to Y$ which transforms $\hat σ$ into $σ$. As a corollary we obtain the following uniqueness result: if $σ_1, σ_2$ are two smooth anisotropic conductivities on $X$ with $Λ_{σ_1}= Λ_{σ_2}$, then there exists a smooth diffeomorphism $Φ: \bar X \to \bar X$ which transforms $σ_1$ into $σ_2$.

math-ph↗