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Giovanni S. Alberti

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math.AP math.FA math.RT eess.IV Machine Learning math.GR math.NA math.PR

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preprint2026arXiv

Diffusion Graph Posterior Sampling for Nonlinear Inverse Problems with Application to Electrical Impedance Tomography

Deep generative models have emerged as state-of-the-art for solving inverse problems, but applying them to inverse problems for PDEs, like electrical impedance tomography (EIT) remains challenging. Because physical domains are naturally discretized as unstructured meshes rather than regular grids, standard convolutional architectures are often inadequate. In this paper, we propose a novel framework that extends diffusion posterior sampling (DPS) to graph-structured data. We develop an unconditional score-based diffusion model directly on a 2D triangular mesh to learn an accurate prior over the physical solution space. Furthermore, we introduce a regularized variant, RDPS, which incorporates explicit regularization terms, such as total variation and generalized Tikhonov, to complement the implicit diffusion prior and mitigate severe ill-posedness. Extensive experiments on synthetic and real 2D EIT datasets demonstrate that RDPS produces stable, physically plausible reconstructions. Our approach generalizes well to out-of-distribution inclusion geometries, is highly robust to measurement noise, and outperforms current state-of-the-art solvers (e.g., GPnP-BM3D, DP-SGS) in reconstruction accuracy and artifact reduction.

preprint2020arXiv

Calderón's Inverse Problem with a Finite Number of Measurements II: Independent Data

We prove a local Lipschitz stability estimate for Gel'fand-Calderón's inverse problem for the Schrödinger equation. The main novelty is that only a finite number of boundary input data is available, and those are independent of the unknown potential, provided it belongs to a known finite-dimensional subspace of $L^\infty$ . A similar result for Calderón's problem is obtained as a corollary. This improves upon two previous results of the authors on several aspects, namely the number of measurements and the stability with respect to mismodeling errors. A new iterative reconstruction scheme based on the stability result is also presented, for which we prove exponential convergence in the number of iterations and stability with respect to noise in the data and to mismodeling errors.

preprint2020arXiv

On the randomised stability constant for inverse problems

In this paper we introduce the randomised stability constant for abstract inverse problems, as a generalisation of the randomised observability constant, which was studied in the context of observability inequalities for the linear wave equation. We study the main properties of the randomised stability constant and discuss the implications for the practical inversion, which are not straightforward.

preprint2020arXiv

Radon Transform: Dual Pairs and Irreducible Representations

We illustrate the general point of view developed in [SIAM J. Math. Anal., 51(6), 4356-4381] that can be described as a variation of Helgason's theory of dual $G$ -homogeneous pairs $(X,Ξ)$ and which allows us to prove intertwining properties and inversion formulae of many existing Radon transforms. Here we analyze in detail one of the important aspects in the theory of dual pairs, namely the injectivity of the map label-to-manifold $ξ\to\hatξ$ and we prove that it is a necessary condition for the irreducibility of the quasi-regular representation of $G$ on $L^2(Ξ)$ . We further explain how the theory in [SIAM J. Math. Anal., 51(6), 4356-4381] applies to the classical Radon and X-ray transforms in $\mathbb R^3$ .

preprint2016arXiv

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.

preprint2015arXiv

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.

preprint2015arXiv

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.

preprint2015arXiv

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.

preprint2013arXiv

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.

preprint2012arXiv

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.

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Giovanni S. Alberti

Research overview

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10 publications

Diffusion Graph Posterior Sampling for Nonlinear Inverse Problems with Application to Electrical Impedance Tomography

Calderón's Inverse Problem with a Finite Number of Measurements II: Independent Data

On the randomised stability constant for inverse problems

Radon Transform: Dual Pairs and Irreducible Representations

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

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

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

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

On Multiple Frequency Power Density Measurements

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

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Giovanni S. Alberti

Research overview

Is this your work?

10 publications

Diffusion Graph Posterior Sampling for Nonlinear Inverse Problems with Application to Electrical Impedance Tomography

Calderón's Inverse Problem with a Finite Number of Measurements II: Independent Data

On the randomised stability constant for inverse problems

Radon Transform: Dual Pairs and Irreducible Representations

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

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

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

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

On Multiple Frequency Power Density Measurements

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

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Building this map preview

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