Workshop on Computational methods for Inverse and Ill-posed problems

Workshop on Computational methods for Inverse and Ill-posed problems


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Invited speakers

Abstracts

Simon Arridge: Tomography with Sound and Light
Several different techniques exist for indirectly recovering the optical absorption and/or scattering coefficients of biological objects, and from there to inferring concentrations of chromophores of interest, from observations of transmitted and reflected light at multiple wavelengths; these include diffuse optical tomography, fluorescence optical tomography, and bioluminescence tomography. These modalities exhibit a tradeoff between greater contrast against lower resolution due to increased scattering. Acoustic waves also have a long tradition in imaging with both qualitative and quantitative interpretations. These concepts are combined in photo-acoustic tomography (PAT) which generates contrast with optical photons and develops resolution using ultrasound. In this talk I review some recent progress in these areas including the acceleration of PAT using Compressed Sensing and Machine Learning techniques.

Joint Work with P.Beard, M.Betcke, B.Cox, A. Hauptman, N.Huynh, Z.Kereta, F.Lucka, A. Pulkkinen, T.Tarvainen
Bastian Harrach: Monotonicity and Convexity in inverse coefficient problems
Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill posed inverse problem, for which unique reconstructability results, stability and resolution estimates and global convergence of numerical methods are very hard to achieve. In this talk we will review some recent results on Loewner Monotonicity and Convexity that may help in overcoming these issues.
Rob Stevenson: Ultra-weak least squares discretizations for Unique Continuation and Cauchy problems
We present conditional stability estimates for unique continuation and Cauchy problems associated to the Poisson equation in ultra-weak variational form. Numerical approximations are obtained as minima of regularized least squares functionals. The arising dual norms are replaced by discretized dual norms, which leads to a mixed formulation in terms of trial- and test-spaces.
For stable pairs of such spaces, and a proper choice of the regularization parameter, the $L_2$-error on a subdomain in the obtained numerical approximation can be bounded by the best possible fractional power of the sum of the data error and the error of best approximation. Compared to the use of a standard variational formulation, the latter two errors are measured in weaker norms. To avoid the use of $C^1$-finite element test spaces, nonconforming finite element test spaces can be applied as well. They either lead to the qualitatively same error bound, or in a simplified version, to such an error bound modulo an additional data oscillation term. Numerical results illustrate our theoretical findings.
Lauri Oksanen: Optimality of stabilized finite element methods for elliptic unique continuation
We consider finite element approximation in the context of the ill-posed elliptic unique continuation problem and introduce a notion of optimal error estimates that includes convergence with respect to a mesh parameter and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the finite element approximation space. We present a stabilized finite element method satisfying the optimal estimate and discuss a proof showing that no finite element approximation can converge at a better rate.

The talk is based on joint work with Erik Burman and Mihai Nechita.
Muriel Boulakia: Numerical approximation of the unique continuation problem enriched by a database for the Stokes equations
In this talk, we are interested in the unique continuation problem for the Stokes equations in the case where the local measurement is part of a large family of measurements. We propose a numerical resolution of this problem by introducing a FE approximation of an optimization problem regularized by the database through a term involving the POD basis associated to the database. We will present to what extent the resolution is improved both theoretically and numerically by the additional knowledge of the database. 
Daniel Lesnic : An alternating method of fundamental solutions for solving the Cauchy problem for the Brinkman equations
Cauchy problems for the Brinkman equations governing the flow of fluids in porous media are investigated. The physical scenario corresponds to situations where part of the boundary of the fluid domain is hostile or inaccessible, whilst on the remaining friendly part of the boundary we prescribe or measure both the fluid velocity and traction. The resulting mathematical formulation leads to a linear but ill-posed inverse problem. A convergent algorithm based on solving two sub sequences of mixed direct problems is developed. The direct solver is based on the method of fundamental solutions which is a meshless boundary collocation method. Since the investigated problem is ill posed, the iterative process is stopped according to the discrepancy principle at a threshold given by the amount of noise with which the input measured data is contaminated in order to prevent the manifestation of instability. Results inverting both exact and noisy data for two and three-dimensional problems demonstrate the convergence and stability of the proposed numerical algorithm.

The talk is based on joint work with Andreas Karageorghis.
Thorsten Hohage : Quantitative helioseismic holography
Helioseismology is the study of the interior of the Sun given observations of oscillations of the solar surface. Due to the lack of controlled sources, passive imaging techniques are typically used, and correlations of observed oscillations are considered as input data of the inverse problem.
Numerical reconstructions are challenging since such correlation data are extremely noisy and high dimensional. They depend of 5 variables (one frequency variable and 2x2=4 surface variables), and it is computationally unfeasibly to fully compute and store them in a preprocessing step. Classically, quantitative reconstructions in local helioseismology are achieved by computing a moderate number of linear functionals of the correlation data which can, e.g., be interpreted as travel times of certain types of waves, and then solving inverse problems with such reduced data. However, it can be shown that these procedures waste lots of the information contained in the correlation data. As an alternative, helioseismic holography has been used successfully since more than two decades to compute qualitative images of, e.g., the far side of the Sun, without the need of computing correlation data as preprocessing step.
We present a new mathematical interpretation of (certain versions of) helioseismic holography, and based on this interpretation we propose an iterative method which is provably convergent. In particular, it provides not only qualitative, but also quantitatively correct reconstructions of the solar interior and the far side. It takes advantage of the full information content of correlation data without the need to compute these data in a preprocessing step, and as a consequence, it achieves significantly improved resolution compared to traditional approaches.

Joint work with Björn Müller, Damien Fournier, and Laurent Gizon.
Florian Faucher : Numerical inverse wave problem in anisotropic elastic media
We consider the quantitative inverse problem for the reconstruction of physical properties of a medium are from wave measurements. We consider linear viscoelasticity to model the propagation of waves in solid materials, taking into account attenuation and anisotropy of the media. In this work, we employ the time-harmonic wave equations to facilitate working with different attenuation models, which are frequency-dependent. For numerical discretization, we use the hybridizable discontinuous Galerkin (HDG) method which employs static condensation to reduce the computational cost. We perform quantitative inversion by following a minimization algorithm where the elastic properties are iteratively updated. We carry out reconstructions with attenuation model uncertainty, and highlight the importance of considering anisotropy in the model. We provide synthetic experiments for ultrasound imaging of breast samples.
Janosch Preuss : Unique continuation for the wave equation: The stability landscape
We consider a unique continuation problem subject to the wave equation: Given measurements of the solution in a subset of the space-time cylinder but no initial data, we aim to extend the solution as far as possible beyond the data set - ideally to the entire cylinder. It is well-known that this maximal extension is indeed possible in a Lipschitz-stable manner, provided that additional measurements of the solution on the lateral boundary of the cylinder are available (and certain geometric conditions are fulfilled). In this setting, many innovative numerical methods have been proposed in recent years to compute the extension numerically.
However, the question what happens if no measurements on the boundary are available seems to be far less understood. In this talk we provide analytical and numerical results which show that in this case the solution can be extended in a Hölder-stable manner to a certain proper(!) subset of the space-time domain. Trying to extend further leads to degeneration of the stability until a logarithmic regime is reached and the accuracy of the corresponding extension becomes too poor to suit practical purposes. We present a possible remedy for this problem: If one can explicitly construct a small finite dimensional space in which the trace of the solution on the boundary can be approximated with high-accuracy, then the solution can be recovered with Lipschitz stability everywhere. This is an extremely powerful yet dangerous technique as its success hinges entirely on a proper choice of this finite dimensional space.

The talk is based on joint work with Erik Burman, Lauri Oksanen and Ziyao Zhao.
Guillaume Delay : Solving a unique continuation problem for some transient equations by means of a high-order space-time method
We discretize a unique continuation problem subject to a transient equation and present the associated numerical analysis. This unique continuation problem consists in reconstructing the solution of the equation in a target space-time subdomain given its (noised) value in a subset of the computational domain. Both initial and boundary data can be unknown. In the sequel, both the heat and the wave equations will be considered.
The corresponding continuous problem is ill-posed and does not necessarily have a standard (Lipschitz) stability estimate. Instead, we can have a conditional (Hölder) stability estimate, so that an a priori estimate on the solution is needed to bound its norm in the target subdomain.
The considered discretization method uses discontinuous high-order polynomials in space and in time. Polynomial variables attached to the faces of the mesh are also considered. The discrete solution is sought as a saddle-point of a Lagrangian functional. Moreover, some stabilization terms are considered to regularize the ill-posed problem.
We establish optimal a priori error bounds for a weak (residual) norm. This first result is proved by means of standard arguments such as inf-sup stability and consistency of the discretization. The conditional stability estimate of the continuous problem is then used to obtain a priori error bounds in energy norm. These error bounds optimally account for the ill-posedness of the continuous problem. Some numerical experiments are presented to check the actual efficiency of the method.

This is joint work with Erik Burman and Alexandre Ern.
Laurent Bourgeois : Tsunami identification from surface data: a Tikhonov-Morozov mixed formulation
Considering a simple linear model of ocean, we wish to identify the bottom deformation due to a tsunami from the observation of the perturbed free surface. In this preliminary work, we restrict to a time harmonic situation. This ill-posed problem is addressed with the help of a mixed formulation of the Tikhonov regularization, the Morozov principle being used to compute the regularization parameter through a duality approach. One step of this Morozov process requires to compute a lifting function which is associated with some noisy Neumann data and convert the noise amplitude of the surface data to a noise amplitude of the volume lifting function. We compare a deterministic and a probabilistic conversion, the second one improving the quality of the identification.
Arnaud Münch : An inverse problem for a semi-linear wave equation: remarks and constructive approaches
We consider the reconstruction of the solution of a semi-linear wave equation from partial boundary measurements. Assuming a logarithmic growth at infinity of the nonlinearity, we discuss a constructive approach based on a simple fixed point method within the usual Carleman functional setting. Numerical experiments illustrates the theoretical part. This work employs some arguments developed recently for the corresponding controllability problem and is joint with Sue Claret (Clermont-Ferrand).
Jérémi Dardé : Finite element discretization of the iterated quasi-reversibility method
We're interested in the iterated quasi-reversibility method applied to the data completion problem for the Laplacian. In particular, we focus on the finite element discretization of the method. We obtain a convergence estimate for the method, with explicit dependence on the various parameters, i.e. the number of iterations, the regularization parameter, the discretization parameter and the amplitude of the noise on the data, making it possible to choose the parameters according to the noise to guarantee convergence.

This is a joint work with Jérémi Heleine.
Mingfei Lu : Inf-sup stability and optimal convergence of the quasi-reversibility method for unique continuation subject to Poisson's equation
We develop a framework for the discretization of a mixed formulation of quasi-reversibility solutions to ill-posed problems with respect to Poisson's equations. By carefully choosing test and trial spaces a formulation that is stable in a certain residual norm is obtained. Numerical stability and optimal convergence are established based on the conditional stability property of the problem. We also provide a guideline for feasible pairs of finite element spaces that satisfy suitable stability and consistency assumptions. Numerical experiments are provided to illustrate the theoretical results.

Preliminary schedule

Thursday, 07.11.2024
10:30-11:00 Registration and coffee (Room 502)
11:00-11:15 Welcome
11:15-11:50 Simon Arridge
11:50-12:25 Bastian Harrach
12:25-13:45 Lunch
13:45-14:20 Rob Stevenson
14:20-14:55 Lauri Oksanen
14:55-15:30 Muriel Boulakia
15:30-16:00 Coffee
16:00-16:35 Daniel Lesnic
16:35-17:10 Thorsten Hohage
17:10-17:45 Florian Faucher
Friday, 08.11.2024
9:00-9:35 Janosch Preuss
9:35-10:10 Jérémi Dardé
10:10-10:45 Laurent Bourgeois
10:45-11:15 Coffee
11:15-11:50 Arnaud Münch
11:50-12:25 Guillaume Delay
12:25-13:00 Mingfei Lu
13:00-13:45 Closing and Lunch