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Giovanni S. Alberti

Publications and source records attributed to Giovanni S. Alberti.

52 records · Page 3Linked to original sources
arXiv2018

Calderón's Inverse Problem with a Finite Number of Measurements

We prove that an $L^\infty$ potential in the Schrödinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace $\mathcal W$. As a corollary, we obtain a similar result for Calderón's inverse conductivity problem. Lipschitz stability estimates and a globally convergent nonlinear reconstruction algorithm for both inverse problems are also presented. These are the first results on global uniqueness, stability and reconstruction for nonlinear inverse boundary value problems with finitely many measurements. We also discuss a few relevant examples of finite dimensional subspaces $\mathcal W$, including bandlimited and piecewise constant potentials, and explicitly compute the number of required measurements as a function of $\dim \mathcal W$.

math.AP↗
arXiv2017

Critical Points for Elliptic Equations with Prescribed Boundary Conditions

This paper concerns the existence of critical points for solutions to second order elliptic equations of the form $\nabla\cdot σ(x)\nabla u=0$ posed on a bounded domain $X$ with prescribed boundary conditions. In spatial dimension $n=2$, it is known that the number of critical points (where $\nabla u=0$) is related to the number of oscillations of the boundary condition independently of the (positive) coefficient $σ$. We show that the situation is different in dimension $n\geq3$. More precisely, we obtain that for any fixed (Dirichlet or Neumann) boundary condition for $u$ on $\partial X$, there exists an open set of smooth coefficients $σ(x)$ such that $\nabla u$ vanishes at least at one point in $X$. By using estimates related to the Laplacian with mixed boundary conditions, the result is first obtained for a piecewise constant conductivity with infinite contrast, a problem of independent interest. A second step shows that the topology of the vector field $\nabla u$ on a subdomain is not modified for appropriate bounded, sufficiently high-contrast, smooth coefficients $σ(x)$. These results find applications in the class of hybrid inverse problems, where optimal stability estimates for parameter reconstruction are obtained in the absence of critical points. Our results show that for any (finite number of) prescribed boundary conditions, there are coefficients $σ(x)$ for which the stability of the reconstructions will inevitably degrade.

math.AP↗
arXiv2016

Mathematical Analysis of Ultrafast Ultrasound Imaging

This paper provides a mathematical analysis of ultrafast ultrasound imaging. This newly emerging modality for biomedical imaging uses plane waves instead of focused waves in order to achieve very high frame rates. We derive the point spread function of the system in the Born approximation for wave propagation and study its properties. We consider dynamic data for blood flow imaging, and introduce a suitable random model for blood cells. We show that a singular value decomposition method can successfully remove the clutter signal by using the different spatial coherence of tissue and blood signals, thereby providing high-resolution images of blood vessels, even in cases when the clutter and blood speeds are comparable in magnitude. Several numerical simulations are presented to illustrate and validate the approach.

math.NA↗
arXiv2016

The Linearized Inverse Problem in Multifrequency Electrical Impedance Tomography

This paper provides an analysis of the linearized inverse problem in multifrequency electrical impedance tomography. We consider an isotropic conductivity distribution with a finite number of unknown inclusions with different frequency dependence, as is often seen in biological tissues. We discuss reconstruction methods for both fully known and partially known spectral profiles, and demonstrate in the latter case the successful employment of difference imaging. We also study the reconstruction with an imperfectly known boundary, and show that the multifrequency approach can eliminate modeling errors and recover almost all inclusions. In addition, we develop an efficient group sparse recovery algorithm for the robust solution of related linear inverse problems. Several numerical simulations are presented to illustrate and validate the approach.

math.NA↗
arXiv2016

Recent Progress in Shearlet Theory: Systematic Construction of Shearlet Dilation Groups, Characterization of Wavefront Sets, and New Embeddings

The class of generalized shearlet dilation groups has recently been developed to allow the unified treatment of various shearlet groups and associated shearlet transforms that had previously been studied on a case-by-case basis. We consider several aspects of these groups: First, their systematic construction from associative algebras, secondly, their suitability for the characterization of wavefront sets, and finally, the question of constructing embeddings into the symplectic group in a way that intertwines the quasi-regular representation with the metaplectic one. For all questions, it is possible to treat the full class of generalized shearlet groups in a comprehensive and unified way, thus generalizing known results to an infinity of new cases. Our presentation emphasizes the interplay between the algebraic structure underlying the construction of the shearlet dilation groups, the geometric properties of the dual action, and the analytic properties of the associated shearlet transforms.

math.FA↗
arXiv2016

Absence of Critical Points of Solutions to the Helmholtz Equation in 3D

The focus of this paper is to show the absence of critical points for the solutions to the Helmholtz equation in a bounded domain $Ω\subset\mathbb{R}^{3}$, given by \[ \left\{ \begin{array}{l} -\rm{div}(a\,\nabla u_ω^{g})-ωqu_ω^{g}=0\quad\text{in $Ω$,}\\ u_ω^{g}=g\quad\text{on $\partialΩ$.} \end{array}\right. \] We prove that for an admissible $g$ there exists a finite set of frequencies $K$ in a given interval and an open cover $\overlineΩ=\cup_{ω\in K}Ω_ω$ such that $|\nabla u_ω^{g}(x)|>0$ for every $ω\in K$ and $x\inΩ_ω$. The set $K$ is explicitly constructed. If the spectrum of the above problem is simple, which is true for a generic domain $Ω$, the admissibility condition on $g$ is a generic property.

math.AP↗
arXiv2015

Reproducing subgroups of $Sp(2,\mathbb{R})$. Part I: algebraic classification

We classify the connected Lie subgroups of the symplectic group $Sp(2,\mathbb{R})$ whose elements are matrices in block lower triangular form. The classification is up to conjugation within $Sp(2,\mathbb{R})$. Their study is motivated by the need of a unified approach to continuous 2D signal analyses, as those provided by wavelets and shearlets.

math.GR↗
arXiv2015

Disjoint sparsity for signal separation and applications to hybrid inverse problems in medical imaging

The main focus of this work is the reconstruction of the signals $f$ and $g_{i}$, $i=1,...,N$, from the knowledge of their sums $h_{i}=f+g_{i}$, under the assumption that $f$ and the $g_{i}$'s can be sparsely represented with respect to two different dictionaries $A_{f}$ and $A_{g}$. This generalizes the well-known "morphological component analysis" to a multi-measurement setting. The main result of the paper states that $f$ and the $g_{i}$'s can be uniquely and stably reconstructed by finding sparse representations of $h_{i}$ for every $i$ with respect to the concatenated dictionary $[A_{f},A_{g}]$, provided that enough incoherent measurements $g_{i}$ are available. The incoherence is measured in terms of their mutual disjoint sparsity. This method finds applications in the reconstruction procedures of several hybrid imaging inverse problems, where internal data are measured. These measurements usually consist of the main unknown multiplied by other unknown quantities, and so the disjoint sparsity approach can be directly applied. As an example, we show how to apply the method to the reconstruction in quantitative photoacoustic tomography, also in the case when the Grüneisen parameter, the optical absorption and the diffusion coefficient are all unknown.

math.AP↗
arXiv2015

Enforcing local non-zero constraints in PDEs and applications to hybrid imaging problems

We study the boundary control of solutions of the Helmholtz and Maxwell equations to enforce local non-zero constraints. These constraints may represent the local absence of nodal or critical points, or that certain functionals depending on the solutions of the PDE do not vanish locally inside the domain. Suitable boundary conditions are classically determined by using complex geometric optics solutions. This work focuses on an alternative approach to this issue based on the use of multiple frequencies. Simple boundary conditions and a finite number of frequencies are explicitly constructed independently of the coefficients of the PDE so that the corresponding solutions satisfy the required constraints. This theory finds applications in several hybrid imaging modalities: some examples are discussed.

math.AP↗
arXiv2015

Elliptic regularity theory applied to time harmonic anisotropic Maxwell's equations with less than Lipschitz complex coefficients

The focus of this paper is the study of the regularity properties of the time harmonic Maxwell's equations with anisotropic complex coefficients, in a bounded domain with $C^{1,1}$ boundary. We assume that at least one of the material parameters is $W^{1,3+δ}$ for some $δ>0$. Using regularity theory for second order elliptic partial differential equations, we derive $W^{1,p}$ estimates and Hölder estimates for electric and magnetic fields up to the boundary. We also derive interior estimates in bi-anisotropic media.

math.AP↗
arXiv2015

On multiple frequency power density measurements II. The full Maxwell's equations

We shall give conditions on the illuminations $φ_{i}$ such that the solutions to Maxwell's equations \[ \left\{ \begin{array}{l} {\rm curl} E^{i}=iωμH^{i}\qquad\text{in }Ω,\\ {\rm curl} H^{i}=-i(ω\varepsilon+iσ)E^{i}\qquad\text{in }Ω,\\ E^{i}\timesν=φ_{i}\timesν\qquad\text{on }\partialΩ, \end{array}\right. \] satisfy certain non-zero qualitative properties inside the domain $Ω$, provided that a finite number of frequencies $ω$ are chosen in a fixed range. The illuminations are explicitly constructed. This theory finds applications in several hybrid imaging problems, where unknown parameters have to be imaged from internal measurements. Some of these examples are discussed. This paper naturally extends a previous work of the author [Inverse Problems 29 (2013) 115007], where the Helmholtz equation was studied.

math.AP↗
arXiv2015

On local non-zero constraints in PDE with analytic coefficients

We consider the Helmholtz equation with real analytic coefficients on a bounded domain $Ω\subset\mathbb{R}^{d}$. We take $d+1$ prescribed boundary conditions $f^{i}$ and frequencies $ω$ in a fixed interval $[a,b]$. We consider a constraint on the solutions $u_ω^{i}$ of the form $ζ(u_ω^{1},\ldots,u_ω^{d+1},\nabla u_ω^{1},\ldots,\nabla u_ω^{d+1})\neq0$, where $ζ$ is analytic, which is satisfied in $Ω$ when $ω=0$. We show that for any $Ω^{\prime}\SubsetΩ$ and almost any $d+1$ frequencies $ω_{k}$ in $[a,b]$, there exist $d+1$ subdomains $Ω_{k}$ such that $Ω^{\prime}\subset\cup_{k}Ω_{k}$ and $ζ(u_{ω_{k}}^{1},\ldots,u_{ω_{k}}^{d+1},\nabla u_{ω_{k}}^{1},\ldots,\nabla u_{ω_{k}}^{d+1})\neq0$ in $Ω_{k}$. This question comes from hybrid imaging inverse problems. The method used is not specific to the Helmholtz model and can be applied to other frequency dependent problems.

math.AP↗
arXiv2014

Multi-frequency acousto-electromagnetic tomography

This paper focuses on the acousto-electromagnetic tomography, a recently introduced hybrid imaging technique. In a previous work, the reconstruction of the electric permittivity of the medium from internal data was achieved under the Born approximation assumption. In this work, we tackle the general problem by a Landweber iteration algorithm. The convergence of such scheme is guaranteed with the use of a multiple frequency approach, that ensures uniqueness and stability for the corresponding linearized inverse problem. Numerical simulations are presented.

math.AP↗
arXiv2013

On Multiple Frequency Power Density Measurements

We shall give a priori conditions on the illuminations $ϕ_i$ such that the solutions to the Helmholtz equation $-div(a \nabla u^i)-k q u^i=0$ in Ω, $u^i=ϕ_i$ on $\partialΩ$, and their gradients satisfy certain non-zero and linear independence properties inside the domain Ω, provided that a finite number of frequencies k are chosen in a fixed range. These conditions are independent of the coefficients, in contrast to the illuminations classically constructed by means of complex geometric optics solutions. This theory finds applications in several hybrid problems, where unknown parameters have to be imaged from internal power density measurements. As an example, we discuss the microwave imaging by ultrasound deformation technique, for which we prove new reconstruction formulae.

math.AP↗
arXiv2012

Reproducing subgroups of Sp(2,R). Part II: admissible vectors

In part I we introduced the class ${\mathcal E}_2$ of Lie subgroups of $Sp(2,\R)$ and obtained a classification up to conjugation (Theorem 1.1). Here, we determine for which of these groups the restriction of the metaplectic representation gives rise to a reproducing formula. In all the positive cases we characterize the admissible vectors with a generalized Calderón equation. They include products of 1D-wavelets, directional wavelets, shearlets, and many new examples.

math.RT↗