Inverse Problems in Milan: Abstracts

We consider the classical linear scattering problem for a medium with compact support, where the constitutive material properties are described by two function-valued coefficients, namely one in the principal part of the operator and the other in the lower-order term. The inverse scattering problem consists of recovering both coefficients from measured far-field data. The uniqueness of this inverse problem is well understood; in particular, data at two distinct frequencies are required to uniquely determine both scalar-valued coefficient functions. My presentation focuses on reconstruction methods, where we allow for jump discontinuities in the coefficients. We discuss the regularized Born inversion using disk prolate spheroidal wave functions as a basis, combined with corrections via the inverse Born series. These results are compared with reconstructions based on neural networks for the same problem.

In this talk, we will discuss an analog of the anisotropic Calderón problem for fractional Schrödinger operators on closed Riemannian manifolds of dimension two or higher. We will show that knowledge of the Cauchy data set of solutions to the fractional Schrödinger equation, given on a nonempty, open, a priori known subset of the manifold, determines both the Riemannian manifold up to isometry and the potential up to the corresponding gauge transformation, under certain geometric assumptions on the manifold and the observation set.

Many inverse problems, such as the Calderón problem related to electrical imaging, as well as related analytic/unique continuation problems, are known to be highly sensitive to noise. Such problems are called ill-posed or unstable, as opposed to being well-posed (a notion introduced by J. Hadamard in 1902). Instability is a crucial feature in the mathematical theory of these problems and it affects the performance and design of computational methods for solving them.
We discuss a general framework for studying instability in inverse problems based on smoothing/compression properties of the forward map, together with estimates for entropy and capacity numbers in relevant function spaces. The methods apply to various linear and nonlinear inverse problems as well as unique continuation problems, and they are valid for general geometries and low regularity coefficients. We will also discuss recent results explaining from this point of view the phenomenon of stability increasing with frequency.

We describe a general approach to solve inverse problems for nonlinear equations. We applied this method to solve inverse problems for Einstein field equations, nonlinear acoustic equations and nonlinear elastic equations. The main topic is that nonlinearity helps to solve inverse problems.

We consider the inverse scattering problem for time-harmonic acoustic waves in a medium with pointwise inhomogeneities. In the Foldy-Lax model, the estimation of the scatterers’ locations and intensities from far field measurements can be recast as the recovery of a discrete measure from nonlinear observations. We propose a linearize and locally optimize approach to perform this reconstruction. We first solve a convex program in the space of measures (known as the Beurling LASSO), which involves a linearization of the forward operator (the far field pattern in the Born approximation). Then, we locally minimize a second functional involving the nonlinear forward map, using the output of the first step as initialization. We provide guarantees that the output of the first step is close to the sought-after measure when the scatterers have small intensities and are sufficiently separated. We also provide numerical evidence that the second step still allows for accurate recovery in settings that are more involved.

Diffuse Optical Tomography (DOT) is a non-invasive medical imaging technique that makes use of Near-Infrared (NIR) light to recover the spatial distribution of optical coefficients in biological tissues for diagnostic purposes. Due to the intense scattering of light within tissues, the reconstruction process inherent to DOT is severely ill-posed [1]. In this work, we propose to tackle the ill-conditioning by learning a prior over the solution space using an autoencoder-type neural network. Specifically, the trained decoder part of the autoencoder is used as a generative model, that maps a latent code to estimated physical parameters given in input to the forward model [2]. The latent code is itself the result of an optimization loop which minimizes the discrepancy of the solution computed by the forward model with available observations. The structure and interpretability of the latent code is further enhanced by minimizing the rank of its covariance matrix, thereby promoting more effective utilization of its information-carrying capacity [3]. The deep learning-based prior significantly enhances reconstruction capabilities in this challenging domain, demonstrating the potential of integrating advanced neural network techniques into DOT.

We present iterative methods which compute solutions to ill-posed problems that additionally consist of multiple components. Each of these contains certain information about the reconstruction. Typically, the ability to treat different parts of an approximation individually results in more robust methods that yield higher quality results. We consider two different ways of decomposition. First, we give a brief overview of structural decompositions. Those aim for components that represent different types of features, such as smooth, piecewise constant, or oscillatory parts. Secondly, hierarchical decomposition means expressing reconstructed solutions as sums of components, where each component contains information at a different level of detail. We focus on the multiscale hierarchical decomposition method (MHDM) for the blind image deblurring problem.

We consider the inverse acoustic obstacle scattering problem in the plane, given the far field of a single time-harmonic incident plane wave. Kusiak and Sylvester have used this information in 2005 to determine an approximation of the convex hull of the scatterers, and few years later we have tried to distinguish multiple scatterers [1]. The present work is similar in spirit, with the essential difference being that we now only use the low-frequent (instead of the high-frequent) part of the far field.
We employ the z-transform of the corresponding Fourier coefficients, and use a modification of the AAA algorithm [2] to compute (complex) rational approximations of this function. It is shown that the polar angles of the poles of this rational function can be used to generate data for the Radon transform of the support of the scatterers. We present numerical examples to indicate the potential and limitations of this approach.

Sobolev Spaces are useful for the formulation of Inverse Problems as well as the incorporation of, e.g., smoothness properties or statistics of their solution. As a result, the Sobolev embedding operator and its adjoint are common components in both iterative and variational regularization methods for the computation of a solution. However, while the embedding operator itself is trivial, its adjoint is typically not, and the study of its properties and different representations is of importance both theoretically and practically. Hence, in this talk we will present different characterizations of the adjoint embedding operators and their use in standard Tomography, Atmospheric Tomography and Photoacoustic Tomography.

In this work, we investigate the problem of vertical infiltration in a homogeneous, bounded soil profile. The governing evolution equation is formulated in a one-dimensional setting with boundary conditions that model either flooding or rainfall. To solve the direct problem—computing soil moisture given specific soil properties—we employ the unified transform method (also known as the Fokas method). An exact analytical solution is derived and shown to align perfectly with well-known approximate solutions. Furthermore, we address an inverse control problem: determining a boundary control function at the bottom of the profile that drives the solution to zero at a prescribed time. Numerical examples are presented to demonstrate the effectiveness and applicability of the proposed approach.

Variational regularization is one of the most popular strategies for obtaining stable solutions of ill-posed problems. While the topic of error estimates in the event of additive noise distorsion has been extensively researched, the literature on such quantitative analysis in the case of non-additive noise is quite scarce. In this work, we derive higher order error estimates for variational regularization of ill-posed problems perturbed by non-additive noise. The results are obtained by means of a novel source condition inspired by the Fenchel-dual of the regularized problem. While our focus is on regularization having the Kullback-Leibler divergence as data fidelity, we also point out how our approach can be extended to the framework of more general convex data fidelities.

Quantum optics is the quantum theory of the interaction of light and matter. In this talk, I will describe a real-space formulation of quantum electrodynamics. The goal is to understand the propagation of nonclassical states of light in systems consisting of many atoms. Applications to inverse problems for nonlocal PDEs will be described.

It is always good to have a friend, and even better if they know things that you don’t! In a career of fluid mechanics and industrial problem solving, I have continually arrived at problems where some form of optimization or inverse problem becomes evident, either as the key issue or otherwise as an interesting sideshow. These have formed the basis for many enjoyable collaborations with Otmar. Problems range from additive manufacturing, through image processing, to the leakage of oil and gas wells and stability of suspensions. I will give a rapid tour of the hits and misses!

A common ansatz in inverse problems for PDEs is that the sought solutions are piecewise constant, modelling situations like localized inclusions of different material properties within an otherwise homogeneous medium. In this situation, variational regularization with a total variation penalty balances being compatible with piecewise constant minimizers with retaining convexity of the regularizer. However, its lack of differentiability means that most numerical methods require some level of smoothing, so that such piecewise constant structures can be observed only approximately and/or at very fine resolutions.
In this talk, we instead focus on fully-corrective generalized conditional gradient methods alternating between updating an active set of subsets of the spatial domain, and an iterate given by a conic combination of the associated characteristic functions. The computation of a new candidate set is performed by solving prescribed mean curvature problems which, after discretization on a triangulation used for FEM simulations, can be realized using graph cut algorithms. We will discuss some convergence results in terms of energy decay rates and mesh refinement, and demonstrate their effectiveness in numerical examples on large meshes.

In recent years, it has been more and more common to try to approach the regularised solution of inverse problems by data-driven methods combining classical variational regularisation with techniques from machine learning. Examples of such approaches include methods for the learning of optimal (weighted) regularisation parameters or methods for the learning of regularisation terms to be used in generalised Tikhonov regularisation. However, as of now there exist only relatively few results demonstrating that the resulting methods actually are regularising operators in the sense of inverse problems. Even more, the existing results often are focussed on the question whether a learned regulariser acts as a classically convergent regularisation method. That is, whether the reconstructions converge to the true solution as the noise level for a given data point converges to zero and additional parameters are chosen in a suitable manner. However, they do not address the important question of how, or if, improved access to more accurate data will also lead to improved reconstructions. In this talk we will introduce a framework for data-driven regularisation methods that allows for the discuss of this question by reformulating the inverse problem at hand. Most importantly, we will treat the training data as additional input to the inverse problem, which then allows us to interpret incompleteness or inaccuracy of the training data as additional noise.

In this work, we consider the problem of reconstructing a planar vector field from partial knowledge of its zeroth and first moment ray transforms. Different from existing works the data is known on a subset of lines, namely the ones intersecting a given arc. We present a reconstruction method which recovers the vector field in the convex hull of the arc.

There are many ways to measure the height of a wave. One of these ways is to deduce the height of the wave from the measure of pressure. Nevertheless, in some cases, current methods could be inaccurate. In order to get a more precise process, we are interested in a particular configuration, with small periodic travelling waves with a constant vorticity flow (coming from an underlying current).
We study the distribution of the hydrodynamic pressure. The hydrodynamic pressure is the sum of the hydrostatic pressure, caused by gravity and atmospheric pressure, and the dynamic pressure, caused by the water waves.
We show that a complex and different distribution appears when there is a reversal flow, i.e. when the underlying current is faster than the velocity wave.

Proving convergence of regularization methods for nonlinear inverse problems requires certain structural assumptions on the forward operator. Most of them turn out to rely on a decomposition of the inverse problem into an ill-posed linear and a well-posed nonlinear one, in one way or the other, as well as generalizations thereof. After revisiting some of these conditions, this talk will dwell on their verification in some coefficient identification problems for PDEs by means of appropriate liftings.

Several problems in imaging are based on recovering coefficients of a PDE, resulting in a non-linear inverse problem that is typically solved by an iterative algorithm with the gradient obtained by an adjoint state method. When the forward problem is time-varying this corresponds to the method of time-reversal which convolves a forward and time-reversed field with the derivative of the spatial operator (sometimes called the imaging condition). Applications include full-waveform imaging (FWI) in Ultrasound Computed Tomography, PhotoAcoustic Tomography (PAT) and time-resolved Diffuse Optical Tomography (tDOT). Within Learned Physics approaches time reversal corresponds to the Neural ODE method for learning the time-derivative of an ODE parameterised by a neural network. By combining the trained network with symbolic regression an interpretable model can be discovered. In this talk I will discuss application of these methods for solving some forward and inverse problems in imaging.

Photoacoustic tomography (PAT) is a hybrid imaging modality that combines the unique optical contrast with the high spatial resolution of ultrasound. In the inverse problem of PAT, the initial pressure distribution, induced by an externally introduced light pulse, is estimated from measured photoacoustic waves generated by this pressure increase. Generally, when solving the inverse problem, the speed of sound within the target is assumed to be known. However, this assumption is not valid in general, and approaches for dealing with an unknown, and possibly spatially distributed, speed of sound are being studied. In this work, we study simultaneous estimation of initial pressure and speed of sound in PAT. As such, this is a highly ill-posed problem. We propose an approach where multiple initial distributions are generated in the imaged target. Then, the speed of sound is estimated simultaneously with different initial pressure distributions. The approach is evaluated with numerical simulations.

We propose and analyze a stochastic-gradient-descent (SGD) type method for solving systems of nonlinear ill-posed equations [2]. The method considered here extends the SGD type iteration introduced in [1] for solving linear ill-posed systems. A distinctive feature of our method resides in the adaptive choice of the stepsize, which promotes a relaxed orthogonal projection of the current iterate onto a conveniently chosen convex set. This characteristic distinguishes our method from other SGD type methods in the literature (where the stepsize is typically chosen a priori) and accounts for the faster convergence observed in the numerical experiments conducted in this manuscript. Our convergence analysis includes: monotonicity and mean square convergence of the iteration error (exact data case), stability and semi-convergence (noisy data case). In the later case, our method is coupled with an a priori stopping rule. Numerical experiments are presented for two large scale nonlinear inverse problems in machine learning (both with real data):

1. we address, using neural networks, the big data problem of CO-concentration prediction considered in [1];
2. we tackle the classification problem for the MNIST database (http://yann.lecun.com/exdb/mnist/).

This talk is concerned with the inverse spectral problem of recovering the vector field B and the electric potential q from some asymptotic knowledge of the boundary spectral data of the bi-harmonic operator ${\mathscr{H}}_{B,q}=(-\mathrm{\Delta }{)}^2-2iB\cdot \nabla -i\text{div}B+q$. More precisely, in a bounded smooth domain of ${ℝ}^n$, with $n\ge 2$, we prove the stability estimate of the first order perturbation coefficients $(B,q)$ from some asymptotic knowledge of a subset of the Dirichlet eigenvalues and Neumann traces of the associated eigenfunctions of ${\mathscr{H}}_{B,q}$. This can be viewed as an extension of the famous one-dimensional Borg-Levinson theorem.

The Calderón problem consists in recovering an unknown coefficient of a partial differential equation from boundary measurements of its solution. These measurements give rise to a highly nonlinear forward operator. Consequently, the development of reconstruction methods for this inverse problem is challenging, as they usually suffer from the problem of local convergence. To circumvent this issue, we propose an alternative approach based on lifting and convex relaxation techniques, that have been successfully developed for solving finite-dimensional quadratic inverse problems. This leads to a convex optimization problem whose solution coincides with the sought-after coefficient, provided that a non-degenerate source condition holds. We demonstrate the validity of our approach on a toy model where the solution of the partial differential equation is known everywhere in the domain. In this simplified setting, we verify that the non-degenerate source condition holds under certain assumptions on the unknown coefficient. We leave the investigation of its validity in the Calderón setting for future works.

Inverse problems are at the core of image reconstruction in computed tomography, where the goal is to recover a high-quality image from indirect, incomplete, and often noisy measurements. Classical model-based methods, such as variational regularization techniques, provide interpretability and stability guarantees, but can be computationally demanding. In recent years, deep learning has shown strong potential in accelerating and enhancing these reconstructions, yet purely data-driven approaches often struggle with generalization and lack theoretical transparency. In this talk, I introduce hybrid methods that combine the strengths of both worlds: deep neural networks are embedded into model-based iterative schemes to guide the reconstruction process and improve convergence speed.

We will present a concise overview of Sergio Vessella’s scientific journey, emphasizing his most significant results and accomplishments.

Detecting ischemic regions is crucial for preventing lethal ventricular ischemic tachycardia. This is typically done by recording the heart’s electrical activity using either noninvasive or minimally invasive methods, such as body surface or intracardiac measurements. Mathematical and numerical models of cardiac electrophysiology can provide insight into how electrical measurements can be used to detect ischemia. The goal is to combine boundary measurements of potentials with a mathematical model of the heart’s electrical activity to identify the position, shape, and size of ischemia and/or infarctions. Ischemic regions can be modeled as electrical insulators using the monodomain model, which is a semilinear reaction-diffusion system that describes cardiac electrical activity comprehensively. In this talk, I will focus on the case of an insulated heart without coupling to the torso. I will first review some results related to reconstructing cavities for the stationary model, and then present some results obtained recently in the case of the time-dependent nonlinear monodomain model for different types of nonlinearities. Finally, I will go back to the low conducting case ischemia and present a problem, arising from the application, that reduces to a nonlinear version of EIT.

In this paper, we study an anisotropic Calderón’s problem of recovering potentials by the boundary measurements. We consider the Laplace-Beltrami operator with any Riemannian metric which is a small perturbation of the Euclidean one. We prove that the Dirichlet-to-Neumann map associated with the Schrödinger operator determines the potential with a Lipschitz stability estimate in a negative Sobolev norm under a sign condition. In other words, assuming a priori sign condition, we are able to prove two new results in Calderón’s problem: small anisotropic media without structure assumption and a Lipschitz stability estimate of reconstructing potentials.

For the reconstruction in linear inverse ill-posed problems, I consider a Tikhonov regularization given by a suitable Hilbert space norm penalization. The regularization coefficient is linked, although in a rather implicit way, to the scale of the reconstructed unknown, that is, to the amount of details recovered.
I consider a multiscale procedure introduced in imaging by Tadmor, Nezzar and Vese and extended to inverse problems by Modin, Nachman and myself. I show that the multiscale procedure provides a more stable reconstruction at a finer scale. Numerical examples confirm the theoretical results.

About 40 years of experience in the development of the boundary control method (BCM) can be reduced to a couple of basic ideas and theses.

1. By the general system theory, the observer, which investigates a system S from its input/output correspondence (Inverse Data) R is, in principle, not able to recover the system itself. The only relevant and properly formalized understanding of to recover a system is to create its copy (model) S', whose input/output map R' coincides with the original one, i.e., provides R' = R.
2. The main source of the copies are the model theorems of functional analysis, which provide ‘material’ from which the copy can be constructed. In the algebraic version of the BCM for 2d electric impedance tomography, the copy of the surface under reconstruction is constructed from the Gelfand spectrum of the holomorphic function algebra, with the spectrum being extracted from the Dirichlet-to-Neumann map (as ID). In the time-domain IPs, by the ID the observer constructs a canonical functional model (the so-called wave model) of the operator which governs the evolution of the system S.

We will give a survey about the great scientific achievements of Otmar Scherzer from his diploma thesis up to now and the immediate future.

We consider the inverse problem of detecting in an isotropic homogeneous elastic body a polyhedral inclusion made by a different homogeneous Lamé material, from boundary measurements of tractions and displacements. Under suitable a priori information, we prove a quantitative Lipschitz stability estimate from the local Dirichlet to Neumann map.

Electrical Impedance Tomography (EIT) is a non-invasive imaging technique whose mathematical foundation is closely linked to the Calderón problem: specifically, the reconstruction of a conductivity coefficient from (partial) knowledge of the associated Neumann-to-Dirichlet map. In this talk, I will focus on the complete electrode model of EIT, under the additional assumption that the unkwown conductivity is piecewise linear on a fixed triangulation. In this setting, similarly to the results obtained in the seminal paper [1], one can establish the Lipschitz stability of the conductivity with respect to the measurements. I will present a reconstruction method that promotes sparsity of the solution in an appropriate representation, which shares many similarity with the compressed sensing approach, although it is generalized to a nonlinear framework. In a supervised learning setting, I further enhance the method by incorporating a data-driven support estimator based on a trained graph neural network. The resulting approach, which we refer to as Oracle-Net, shows encouraging numerical performance and is amenable to rigorous theoretical analysis. In particular, by employing classical tools such as the source condition and the tangential cone condition, we prove that the method defines a convergent regularization scheme for the EIT problem. I will present a reconstruction method that promotes sparsity of the solution in an appropriate representation. While it shares conceptual similarities with compressed sensing, it is formulated in a nonlinear framework. In a supervised learning setting, I further enhance the method by incorporating a data-driven support estimator based on a trained graph neural network.

In this talk, we focus on reconstructing unknown cofficients for second order partial differential equations from the knowledge of their solutions or the corresponding boundary conditions (which we called the boundary measurements).
In 2020, Giovanni S. Alberti and Yves Capdeboscq considered this type of problem in [1] for the linear elliptic equation. They showed that if parameters are smooth enough, thanks to a direct calculation, this type of inverse problem will link to a so-called non-vanishing Jacobian problem (namely, find a group of solutions whose Jacobian has maximal rank in the domain). Solutions which also solved such non-vanishing Jacobian problem will make contribution to reconstructing parameters. In dimension d ≥ 2, under suitable uniform regularity assumptions, the family of such 2d solutions is both open and dense. The approach is based on the combination of the Runge approximation property and the Whitney projection argument.
In application, a large group of piecewise coefficients are considered especially when reconstructing conductivity in composite media. This will lead us to a piecewise regular coefficients case. Also, when time variable is involved, we hope to reconstruct parameters for evolution equation (for example, the heat equation). Finally, we consider the most interesting electromagnetism reconstruction problem. Our goal is to reconstruct conductivity, permittivity and the magnetic permeability for the Maxwell system if charge density is provided. I will present how such reconstruction problems are related to some constraints problems and introduce a new approximation method combining a small dilation discussion, an extended Runge property and the extension method in time. I will also present a new Whitney argument to solve the discontinuity problem between different inner media when it comes to the piecewise regular coefficients case.

A central challenge in applying Bayesian inference to inverse problems is that in many cases—especially those governed by partial differential equations (PDEs)—the unknowns to be estimated are functions that lie in a suitable function space, typically an infinite-dimensional Hilbert space. It is therefore crucial to design Bayesian inference algorithms that are both theoretically sound and computationally effective in arbitrarily high dimensions. One way to achieve this is by lifting these problems to an infinite-dimensional space and designing inference methods directly in that setting. This approach, sometimes referred to as apply-algorithm-then-discretize, allows for the development of methods that are inherently discretization-invariant, as both the Bayes formula and the algorithms are properly defined on Hilbert spaces.
The infinite-dimensional formulation of Bayesian inference is now receiving renewed attention with the rise of deep learning methods for posterior sampling. One popular class of such methods that still lacks a complete theoretical understanding in this context is score-based generative models (SGMs), which generate samples from complex distributions by first learning the (Stein) score—the gradient of the log-density—and then using it in various sampling algorithms. Since their introduction, SGMs have been applied successfully to Bayesian inverse problems, either by learning the score conditioned on data or by using the score of the prior—the so-called unconditional score model—within Langevin-type samplers. However, these approaches typically assume that the posterior is supported in a finite-dimensional space, leaving the challenges of infinite dimensions to heuristics and ad hoc solutions.
In this talk, we address this gap and present a theoretical framework and methods for using SGMs in Bayesian inference for inverse problems formulated directly in infinite dimensions.

In this talk we present a novel methodology for estimating the intrinsic dimension (ID) of high-dimensional datasets by leveraging singular Riemannian metrics induced by Variational Autoencoders (VAEs). Under the manifold hypothesis, the data is assumed to lie on a smooth, low-dimensional manifold embedded in a high-dimensional Euclidean space. The core idea is to use the pullback metric via the VAE decoder to recover the local geometry of the manifold and estimate its dimension through the numerical rank of the resulting (possibly degenerate) metric tensor.
The proposed method is applied to the solution of inverse problems in biomedical imaging, where observations are modeled as outputs of a forward operator corrupted by noise. To address the ill-posedness and high dimensionality of such problems, we constrain the solution to lie on a learned manifold. The intrinsic dimension, estimated via the proposed technique, informs the construction of an atlas of local charts using mixtures of invertible VAEs, ensuring both accurate manifold parameterization and tractable inference.
Additionally, we investigate the effects of network pruning on the learned manifold structure and reconstruction quality, demonstrating that significant model compression is feasible up to a critical threshold beyond which manifold geometry and reconstruction fidelity degrade.

We discuss both the forward as well as an inverse problem for viscoelastic models of dislocations that represent aseismic, creeping faults. We study a nonlocal linear slip rate-traction model and a nonlinear local slip rate-slip friction model. The inverse problem consists in determining the geometry of the dislocation surface as well as the slip vector from surface displacement measurements. This is joint work with PhD student Arum Lee, and extends prior results.