11th Applied Inverse Problems Conference
September 4 - 8, 2023 | Göttingen, Germany
Conference Agenda
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
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Session Overview | |
| Location: VG2.105 |
| Date: Monday, 04/Sept/2023 | |
| 1:30pm - 3:30pm | MS39: Statistical inverse problems: regularization, learning and guarantees Location: VG2.105 Session Chair: Kim Knudsen Session Chair: Abhishake Abhishake |
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On the Regularized Functional Regression The Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austria
Functional Regression (FR) involves data consisting of a sample of functions taken from some population. Most work in FR is based on a variant of the functional linear model first introduced by Ramsay and Dalzell in 1991. A more general form of polynomial functional regression has been introduced only quite recently by Yao and Müller (2010), with quadratic functional regression as the most prominent case. A crucial issue in constructing FR models is the need to combine information both across and within observed functions, which Ramsay and Silverman (1997) called replication and regularization, respectively. In this talk we are going to present a general approach for the analysis of regularized polynomial functional regression of
arbitrary order and indicate the possibility for using here a technique that has been recently developed in the context of supervised learning. Moreover, we are going to describe of how multiple penalty regularization can be used in the context of FR and demonstrate an advantage of such use. Finally, we briefly discuss the application of FR in stenosis detection.
Joint research with S. Pereverzyev Jr. (Uni. Med. Innsbruck), A. Pilipenko
(IMATH, Kiev) and V.Yu. Semenov (DELTA SPE, Kiev) supported by the consortium
of Horizon-2020 project AMMODIT and the Austrian National Science Foundation (FWF).
Inverse learning in Hilbert scales LUT University Lappeenranta, Finland
We study the linear ill-posed inverse problem with noisy data in the statistical learning setting. Approximate reconstructions from random noisy data are sought with general regularization schemes in Hilbert scale. We discuss the rates of convergence for the regularized solution under the prior assumptions and a certain link condition. We express the error in terms of certain distance functions. For regression functions with smoothness given in terms of source conditions, the error bound can then be explicitly established.
Stability and Generalization for Stochastic Gradient Methods SUNY Albany, United States of America
Stochastic gradient methods (SGMs) have become the workhorse of machine learning (ML) due to their incremental nature with a computationally cheap update. In this talk, I will first discuss the close interaction between statistical generalization and computational optimization for SGMs in the framework of statistical learning theory (SLT). The core concept for this study is algorithmic stability which characterizes how the output of an ML algorithm changes upon a small perturbation of the training data. Our theoretical studies have led to new insights into understanding the generalization of overparameterized neural networks trained by SGD. Then, I will describe how this interaction framework can be used to derive lower bounds for the convergence of existing methods in the task of maximizing the AUC score which further inspires a new direction for designing efficient AUC optimization algorithms.
Causality and Consistency in Bayesian Inference Paradigms University of Copenhagen, Denmark
Bayesian inference paradigms are regarded as powerful tools for solution of inverse problems. However, Bayesian formulations suffer from a number of difficulties that are often overlooked.
The well known, but mostly neglected, difficulty is connected to the use of conditional probability densities. Borel, and later Kolmogorov's (1933/1956), found that the traditional definition of probability densities is incomplete: In different parameterizations it leads to different, conditional probability measures. This inconsistency is generally neglected in the scientific literature, and therefore threatens the objectivity of Bayesian inversion, Bayes Factor computations, and trans-dimensional inversion. We will show that this problem is much more serious than usually assumed.
Additional inconsistencies in Bayesian inference are found in the so-called hierarchical methods where so-called hyper-parameters are used as variables to control the uncertainties. We will see that these methods violate causality, and analyze how this challenges the validity of Bayesian computations.
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| 4:00pm - 6:00pm | MS35: Edge-preserving uncertainty quantification for imaging Location: VG2.105 Session Chair: Amal Mohammed A Alghamdi Session Chair: Jakob Sauer Jørgensen |
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Efficient Bayesian computation for low-photon imaging problems 1Heriot-Watt University, United Kingdom; 2University of Edinburgh, United Kingdom
This talk presents a new and highly efficient MCMC methodology to perform Bayesian inference in low-photon imaging problems, with particular attention to situations involving observation noise processes that deviate significantly from Gaussian noise, such as binomial, geometric and low-intensity Poisson noise. These problems are challenging for many reasons. From an inferential viewpoint, low photon numbers lead to severe identifiability issues, poor stability and high uncertainty about the solution. Moreover, low-photon models often exhibit poor regularity properties that make efficient Bayesian computation difficult; e.g., hard non-negativity constraints, non-smooth priors, and log-likelihood terms with exploding gradients. More precisely, the lack of suitable regularity properties hinders the use of state-of-the-art Monte Carlo methods based on numerical approximations of the Langevin stochastic differential equation (SDE) or other similar dynamics, as both the continuous-time process and its numerical approximations behave poorly. We address this difficulty by proposing an MCMC methodology based on a reflected and regularised Langevin SDE, which is shown to be well-posed and exponentially ergodic under mild and easily verifiable conditions. This then allows us to derive four reflected proximal Langevin MCMC algorithms to perform Bayesian computation in low-photon imaging problems. The proposed approach is illustrated with a range of experiments related to image deblurring, denoising, and inpainting under binomial, geometric and Poisson noise.
Advancements of $\alpha$-stable priors for Bayesian inverse problems 1Heriot Watt University, United Kingdom; 2LUT, Finland
In this talk, we will summarize the recent advacements made for non-Gaussian
process priors for statistical inversion. This will be primarily focused on $\alpha$-stable
distributions which provide a natural generalization of a family of distributions, such as the normal and Cauchy. We discuss recently proposed priors which include various Cauchy priors, hierarchical and neural-network based $\alpha$-stable priors. The focus will be computational where we demonstrate their gains on a range of examples for fully Bayesian and MAP-based estimation. We also provide some theoretical insights which include error bounds.
Edge preserving Random Tree Besov Priors 1Delft University of Technology, Netherlands; 2University of Helsinki, Finland
Gaussian process priors are often used in practice due to their fast computational properties. The smoothness of the resulting estimates, however, is not well suited for modelling functions with sharp changes. We propose a new prior that has same kind of good edge-preserving properties than total variation or Mumford-Shah but correspond to a well-defined infinite dimensional random variable. This is done by introducing a new random variable $T$ that takes values in the space of ‘trees’, and which is chosen so that the realisations have jumps only on a small set.
CUQIpy - Computational Uncertainty Quantification for Inverse problems in Python Technical University of Denmark (DTU), Denmark
In this talk we present CUQIpy (pronounced ”cookie pie”) - a new computational modelling environment in Python that uses uncertainty quantification (UQ) to access and quantify the uncertainties in solutions to inverse problems. The overall goal of the software package is to allow both expert and non-expert (without deep knowledge of statistics and UQ) users to perform UQ related analysis of their inverse problem while focusing on the modelling aspects. To achieve this goal the package utilizes state-of-the-art tools and methods in statistics and scientific computing specifically tuned to the ill-posed and often large-scale nature of inverse problems to make UQ feasible. We showcase the software on problems relevant to imaging science such as computed tomography and partial differential equation-based inverse problems. CUQIpy is developed as part of the CUQI project at the Technical University of Denmark and is available at https://github.com/CUQI-DTU/CUQIpy.
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| Date: Tuesday, 05/Sept/2023 | |
| 1:30pm - 3:30pm | MS38 1: Inverse eigenvalue problems in astrophysics Location: VG2.105 Session Chair: Charlotte Gehan Session Chair: Damien Fournier |
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No planet is an island: what we can learn from how Saturn interacts with its surroundings Canadian Institute for Theoretical Astrophysics (CITA), Canada
Direct observations provide limited information about the deep internal structures and basic properties of gaseous planets, even in our own Solar system. However, more can be learned from how planets interact with their surroundings. I will introduce research focused on interpreting the satellite Cassini's observations of Saturn's gravitational interactions with its rings and satellite moons. Observing these interactions yields information about Saturn's internal oscillation modes and tidally excited waves, the successful inversion of which may provide our best hope for constraining the planet's deep internal structure and rotation state.
Inversion methods in asteroseismology LESIA, France
In this talk, I will review the different kernel-based inversion techniques that have been used in asteroseismology. In particular, I will describe regularised least-squares (RLS) as well as optimally localised averages (OLA) type inversions. These have been applied to rotation and structural profiles as well as to integrated quantities such as the total kinetic rotation energy, the mean density, the acoustic radius, and various evolutionary phase and convective region indicators. I will also briefly show how inverse techniques can lead to more subtle constraints such as inequalities on rotational splittings if one makes certain assumptions on the rotation profile.
Internal structure of giant planets from gravity data IRAP, CNRS, France
The Juno and Cassini spacecrafts have measured the gravity fields of Jupiter and Saturn with exquisite precision. The gravity field can then be projected onto the Legendre polynomials to obtain the gravitational moments, signatures of the density distribution in the planet as a function of radius and angle. In the past few years, a lot of effort has thus been dedicated to create precise methods to calculate gravitational moments from synthetic models and optimise the retrieval of internal structure by comparing with Juno and Cassini data.
In this talk, I will detail how we tackled this inverse problem in the case of Jupiter and Saturn. I will quickly introduce the concentric Maclaurin spheroid method used to calculate gravitational moments, before detailing the recovered internal structures. I will expose the dominant influence of winds on the gravity field and how planetary oscillations can constrain further the recovered density profiles. These results have strong implication for the formation and evolution of the giant planets and solar system in general.
Accurate asteroseismic surface rotation rates for evolved red giants 1Heidelberg Institute for Theoretical Studies, Germany; 2Max Planck Institute for Astrophysics, Germany; 3Stellar Astrophysics Centre, Aarhus University, Denmark; 4Center for Astronomy (ZAH/LSW), Heidelberg University, Germany; 5Department of Astronomy, Yale University, USA
The understanding of the internal stellar rotation and its evolution are important ingredients for the construction of accurate stellar models. We use asteroseismology, the study of global stellar oscillations, to probe the interior rotation of stars, particularly that of red giants. Large systematic errors previously hindered the accurate determination of near-surface rotation rates in evolved red giants e.g. [1]. We have developed a method of effectively eliminating these systematic errors by introducing an extension to a currently used rotational inversion method for red-giant stars [2].
We demonstrate the ability of the new inversion technique to compute accurate envelope rotation rates of stars along the red giant branch (RGB). Furthermore, we show the resulting improvement of our new method compared to other seismic inversion methods. Subsequently, we aim at quantifying systematic uncertainties in asteroseismic rotational inversions occurring due to inaccurate stellar modelling (Ahlborn et al. in prep). More accurate surface rotation rates for evolved red giants will be an important probe to understand the loss of angular momentum in red-giant cores, and an important milestone to improve the theory of rotation in stellar models.
[1] F. Ahlborn, E. P. Bellinger, S. Hekker, S. Basu and G. C. Angelou. Asteroseismic sensitivity to internal rotation along the red-giant branch, Astronomy and Astrophysics, 639:A98, 2020. https://doi.org/10.1051/0004-6361/201936947
[2]
F. Ahlborn, E. P. Bellinger, S. Hekker, S. Basu and D. Mokrytska. Improved asteroseismic inversions for red-giant surface rotation rates, Astronomy and Astrophysics, 668:A98, 2022. https://doi.org/10.1051/0004-6361/202142510
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| 4:00pm - 6:00pm | MS38 2: Inverse eigenvalue problems in astrophysics Location: VG2.105 Session Chair: Charlotte Gehan Session Chair: Damien Fournier |
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Mode identification in rapidly-rotating stars: paving the way to inverse methods Universidad de Granada, Spain
Asteroseismology has opened a window on the internal physics of thousands of stars, by relating pulsations of stars to their internal physics. Mode identification, namely the process of associating a measured oscillation frequency to the corresponding mode geometry and properties, is the preliminary step of the seismic analysis. In upper main-sequence stars, that often rotate rapidly, this identification is challenging and largely incomplete, as modes assume complex geometries and frequencies shift under the combined influence of the Coriolis force and centrifugal flattening.
In this contribution, I will describe the various classes of mode geometries that emerge in rapidly rotating stars and their differences with slow rotators. After discussing how their frequencies and periods relate with structural quantities, allowing us to derive constraints on the stellar evolution, I will discuss the approaches developed towards inversion methods.
Progress in Asteroseismology: Where We Stand and Where We'll Go Max Planck Institute for Astrophysics, Germany
Over the past decade, asteroseismic inversion techniques have emerged as crucial tools to help identify the missing physics in our understanding of stellar evolution. In this talk, I will provide a comprehensive overview of the recent progress in asteroseismology and showcase the major advancements in the field, with a focus on novel methods for probing stellar structure and evolution. I will present new inferences into various different types of pulsating stars and improvements in our ability to infer internal stellar dynamics. I will also share our latest research on non-linear inversion methods and the application of inversions to massive stars. Finally, I will discuss the future of asteroseismology, including the expected yield of several forthcoming missions. This talk aims to highlight the remarkable progress in asteroseismology and stimulate discussions on future avenues for continued advancement.
Helioseismic inversions for active latitudes 1Max Planck Institute for Solar System Research, Göttingen, Germany; 2Georg-August-Universität Göttingen, Göttingen, Germany
The eleven-year solar activity cycle is known to affect the acoustic p-modes; higher activity is correlated with increase in mode frequencies and decrease in their lifetimes. This is also seen in the autocorrelation function of the integrated light. Recently, the solar cycle is also observed in travel-time measurements of p-mode wavepackets for multiple skips [1]. In this work, we first construct a forward model to explain the variation in travel-time measurements with solar activity. A simplified model is constructed by considering axisymmetric averages of the magnetic activity associated with perturbations in the near-surface wave-speed. The perturbations are constructed by longitudinally averaging synoptic magnetograms from SDO/HMI and the SOHO/MDI. The maximum correlation between observed and modeled travel-time shifts is as high as $0.92$ for some skips, much less for others. Subsequently, we setup an inverse problem to invert for the latitudinal distribution of solar activity from travel-time observations. This work is a first step towards the goal of retrieving stellar butterfly diagrams from asteroseismic observables.
[1] V. Vasilyev, L. Gizon, 2023, submitted.
Probing solar turbulent viscosity with inertial modes 1Max Planck Institute for Solar System Research, Germany; 2Institut für Astrophysik, Georg-August-Universität Göttingen
Solar inertial modes offer new possibilities to probe the solar interior down to the tachocline, and can be used to constrain such properties as the differential rotation or the spectrum of turbulent energy throughout the convection zone. Linear analysis enables us to compute the discrete eigenfrequencies of these modes [1,2]. However, because the inertial modes overlap in the frequency domain, especially for high azimuthal order $m$, this is not enough: it is necessary to model the power spectral density in the whole inertial frequency range, which can be done by modelling the stochastic source of excitation of the modes by turbulent vorticity.
In this presentation, I will show how this can be achieved in a 2D spherical setting, based on the formalism by [3]. I will then describe how this formalism can be used to relate changes in turbulent properties, with a focus on the turbulent viscosity, to their effects on the whole inertial range power spectral density (forward problem), as well as discuss the corresponding inverse problem.
[1] L. Gizon, D. Fournier, M. Albekioni. Effect of latitudinal differential rotation on solar Rossby waves: Critical layers, eigenfunctions, and momentum fluxes in the equatorial $\beta$ plane, Astron. Astrophys 642: A178, 2020.
[2] Y. Bekki, R.H. Cameron, L. Gizon. Theory of solar oscillations in the inertial frequency range: Linear modes of the convection zone. Astron. Astrophys 662: A16, 2022.
[3] J. Philidet, and L. Gizon. Interaction of solar inertial modes with turbulent convection,
A 2D model for the excitation of linearly stable modes. Astron. Astrophys 673: A124, 2023.
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| Date: Wednesday, 06/Sept/2023 | |
| 9:00am - 11:00am | CT02: Contributed talks Location: VG2.105 Session Chair: Roman Novikov |
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Lipschitz Stability of Recovering the Conductivity from Internal Current Densities 1Yau Mathematical Sciences Center, Tsinghua University, Beijing, People's Republic of China; 2Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, People's Republic of China
We investigates the inverse problem of reconstructing the electrical conductivity of an object in hybrid imaging methods. These techniques have been developed in recent years to produce clearer images than those produced by electrical impedance tomography. We focus on the inverse problem arising in the quantitative step of many hybrid imaging techniques. The problem is formulated as recovering the isotropic conductivity of an object given internal current densities generated by applying different boundary conditions to the electrostatic equation. We will present two specific examples of these techniques, current density impedance imaging and magneto-acousto-electric tomography, to illustrate the different boundary conditions that can be used. We provide a local Lipschitz stability for the general inverse problem in both full and partial data cases.
Geometric regularization in three-dimensional inverse obstacle scattering 1Georg-August-Universität Göttingen, Germany; 2Technische Universität Chemnitz, Germany
We study the classical inverse problem to determine the shape of a three-dimensional scattering obstacle from measurements of scattered waves or their far-field patterns. Previous research on this subject has mostly assumed the object to be star-shaped and imposed a Sobolev penalty on the radial function or has defined the penalty term in some other ad-hoc manner which is not invariant under coordinate transformations.
For the case of curves in $\mathbb{R}^2$, reference [1] suggests to use the bending energy as regularisation functional and proposes Tikhonov regularization and regularized Newton methods on a shape manifold. The case of surfaces in $\mathbb{R}^3$ is considerably more demanding. First, a suitable space (manifold) of shapes is not obvious. The second problem is to find a
stabilizing functional for generalised Tikhonov regularisation which on the
one hand should be bending-sensitive and on the other hand prevent the surface from self-intersections during the reconstruction.
The tangent-point energy is a parametrization-invariant and repulsive surface energy that is constructed as the double integral over a power of the tangent point radius with respect to two points on the surface, i.e. the smallest radius of a sphere being tangent to the first point and intersecting the other. The finiteness of this energy also provides $C^{1,\alpha}$ Hölder regularity of the surfaces.[2] Using this energy as the stabilising functional, we choose general surfaces of Sobolev-Slobodeckij reguality, which are naturally connected to this energy.
The proposed approach works for surfaces of arbitrary (known) topology.
In numerical examples we demonstrate that the flexibility of our approach in handling rather general shapes.
[1] J Eckhardt, R Hiptmair, T Hohage, H Schumacher, M Wardetzky. Elastic energy regularization for inverse obstacle scattering problems. 2019
[2] P. Strzlecki, H. von der Mosel. Tangent-point repulsive potentials for a class of smooth $m$-dimensional sets in $\mathbb{R}^n$. Part 1: Smoothing and self-avoidance effects. 2011
Phase retrieval and phaseless inverse scattering with background information 1Univ. Gottingen, Germany; 2CMAP, Ecole Polytechnique, France
We consider the problem of finding a compactly supported potential in the multidimensional Schrodinger equation from its differential scattering cross section (squared modulus of the scattering amplitude) at fixed energy. In the Born approximation this problem simplifies to the phase retrieval problem of reconstructing the potential from the absolute value of its Fourier transform on a ball. To compensate for the missing phase information we use the method of a priori known background scatterers. In particular, we propose an iterative scheme for finding the potential from measurements of a single differential scattering cross section corresponding to the sum of the unknown potential and a known background potential, which is sufficiently disjoint. If this condition is relaxed, then we give similar results for finding the potential from additional monochromatic measurements of the differential scattering cross section of the unknown potential without the background potential. The performance of the proposed algorithms is demonstrated in numerical examples. This talk is based on the work
Hohage, Novikov, Sivkin, preprint 2022, hal-03806616
Convergence analysis of optimization-by-continuation algorithms Université libre de Bruxelles, Belgium
We discuss several iterative optimization algorithms for the minimization of a cost function consisting of a linear combination of up to three convex terms with at least one differentiable and a second one prox-simple. Such optimization problems frequently occur in the numerical solution of inverse problems (data misfit term plus penalty or constraint term).
We present several new results on the convergence of proximal-gradient-like algorithms in the context of a optimization-by-continuation strategy. The algorithms special feature lies in their ability to approximate, in a single iteration run, the minimizers of the cost function for many different values of the parameters determining the relative weight of the three terms in the cost function (penalty parameters). As a special case, one recovers a generalization of the primal-dual algorithm of Chambolle and Pock.
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| Date: Thursday, 07/Sept/2023 | |
| 1:30pm - 3:30pm | CT05: Contributed talks Location: VG2.105 Session Chair: Tram Nguyen |
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Quaternary image decomposition with cross-correlation-based multi-parameter selection University of Bologna, Italy
Separating different features in images is a challenging problem, especially in the separation of the textural component when the image is noisy. In the last two decades many papers were published on image decomposition,
addressing modeling and algorithmic aspects and presenting the use of image decomposition in cartooning, texture separation, denoising, soft shadow/spot light removal and structure retrieval.
Given the desired properties of the image components, all the
valuable contributions to this problem rely on a variational-based formulation
which minimizes the sum of different energy norms: total variation semi-norm,
$L^1$-norm, G-norm,
approximation of the G-norm by the $div(L^p)$-norm
and by the $H^{-1}$-norm,
homogeneous Besov space,
to model the oscillatory component of an image. The intrinsic difficulty with these minimization problems comes from the numerical intractability of the considered norms, from the tuning of the numerous model parameters, and, overall, from the
complexity of extracting noise from a textured image, given the strong similarity
between these two components.
In this talk, I will present a two-stage variational model for the additive
decomposition of images into piecewise constant, smooth, textured and
white noise components. Then, I will discuss how the challenging separation of noise from textured images can be successfully overcome by integrating a whiteness constraint in the model, and how the selection of the regularization parameters
can be performed based on a novel multi-parameter cross-correlation principle. Finally, I will present numerical results that show
the potentiality of the proposed model for the decomposition of textured
images corrupted by several kinds of additive white noises.
Heuristic parameter choice from local minimum points of the quasioptimality function for the class of regularization methods University of Tartu, Estonia
We consider an operator equation
\begin{equation*}
Au=f, \quad f\in R(A),\tag{1}
\end{equation*}
where $A\in L(H, F)$ is the linear continuous operator between real Hilbert spaces $H$ and $F$. In general this problem is ill-posed: the range $R(A)$ may be non-closed, the kernel $N(A)$ may be non-trivial.
Instead of an exact right-hand side $f_*$ we have only an approximation $f \in F$. For the regularization of problem (1) we consider the following class of regularization methods:
\begin{equation*}
u_r = (I- A^* A g_r (A^* A )) u_0 + g_r ( A^* A) A^* f.
\end{equation*}
Here $u_0$ is the initial approximation, $r$ is the regularization parameter, $I$ is the identity operator and the generating function $g_r(\lambda)$ satisfies the conditions
\begin{equation*}
\sup_{0\leq \lambda \leq \|A^*A\|} \left| g_r (\lambda)\right| \leq \gamma r, \quad r\geq 0, \quad \gamma>0.
\end{equation*}
\begin{equation*}
\sup_{0\leq \lambda \leq \|A^*A\|} \lambda^p \left| 1-\lambda g_r (\lambda)\right| \leq \gamma_p r^{-p}, \quad r\geq 0, \quad 0\leq p \leq p_0, \quad \gamma_p>0.
\end{equation*}
Examples of methods of this class are (iterated) Tikhonov method, Landweber iteration method, implicite iteration method, method of asymptotical regularization, the truncated singular value decomposition methods etc.
If the noise level of data is unknown, for the choice of the regularization parameter $r$ heuristic rule is needed. We propose to choose $r$ from the set
$L_{min}$ of the local minimum points of the quasioptimality criterion function
\begin{equation*}
\psi_{Q}(r)=r \|A^*(I-AA^*g_r(AA^*))^{\frac{2}{p_0}}(Au_r-f)\|
\end{equation*}
on the set of parameters $\Omega=\left\{r_{j}: \,r_{j}=q r_{j-1},\, j=1,2,...,M, \, q>1 \right\} $.
Then
the following error estimates hold:
a) \begin{equation*}
\min_{r \in L_{min}}\left\|u_r-u_{*}\right\| \leq C \min_{r_0 \leq r \leq r_M} \left\{ \left\| u_r^{+}-u_{*}\right\|+\left\| u_r-u_r^{+}\right\| \right\}.
\end{equation*}
Here $u_{*}$ and
$u_r^{+}$ are the exact and regularized solutions of equation $Au=f_*$
and the constant $C \leq c_q \ln(r_M / r_0) $ can
be computed for each individual problem $Au=f$.
b) Let $u_{*}=(A^{*}A)^{p/2}v, \, \left\|v\right\| \leq \rho $. If $r_0=1, \, r_M=c \left\|f-f_{*}\right\|^{-2}, \, c=(2 \left\|u_{*}\right\|)^{2}$, then
\begin{equation*}
\min_{r \in L_{min}}\left\|u_r-u_{*}\right\| \leq c_{p,q} \rho^{1/(p+1)} \left| \ln \left\| f-f_{*} \right\| \right| \left\| f-f_{*} \right\|^{p/(p+1)}, 0<p \leq 2 p_0 .
\end{equation*}
We consider some algorithms for parameter choice from the set $L_{min}$.
Choice of the regularization parameter in case of over- or underestimated noise level of data University of Tartu, Estonia
We consider an operator equation
\begin{equation*}
Au=f_{*}, \quad f_{*}\in R(A),
\end{equation*}
where $A\in L(H,F)$ is the linear continuous operator between real Hilbert spaces $H$ and $F$.
We assume that instead of the exact right-hand side $f_{*}$ we have only an
approximation $f\in F$ with supposed noise level $\delta$.
To get the regularized solution we consider Tikhonov method $u_\alpha=(\alpha I+A^{*}A)^{-1}A^{*}f,$
where $\alpha>0$ is the regularization parameter.
In article [1] is shown that at least one local minimum point $m_k $ of the quasioptimality criterion function
\begin{equation*}
\psi_{Q}(\alpha)=\alpha \left\|du_{\alpha}/d\alpha\right\|=\alpha^{-1}\left\|A^{*}(Au_{2,\alpha}-f)\right\|, \quad u_{2,\alpha}=(\alpha I+A^{*}A)^{-1}(\alpha u_{\alpha}+A^{*}f),
\end{equation*}
is always a good regularization parameter. We will use this fact to choose proper regularization parameter in case of a possible over- or underestimation of the noise level.
If the actual noise level $ \left\| f - f_*\right\|$ can be less than $\delta$, then we propose the following rule.
Rule 1. Let $c>1$ and the parameter $\alpha(\delta)$ is choosen according to the modified discrepancy principle or monotone error rule (see [2]). For the regularization parameter choose smallest local minimum point $m_k \leq \alpha(\delta) $ of the quasioptimality criterion function for which holds
\begin{equation*}
\max_{\alpha,\alpha', m_k \leq \alpha' < \alpha \leq \alpha(\delta)} \frac{\psi_{Q}(\alpha')}{\psi_{Q}(\alpha)} \leq c. \qquad \qquad \qquad(1)
\end{equation*}
If such local minimum point does not exist, then choose $\alpha(\delta)$.
If the actual noise level can be both larger or smaller than $\delta$ then we propose the following rule.
Rule 2. Let $c>1$ and the parameter $\alpha(\delta)$ is chosen according to the balancing principle (see [2]).
If there exists local minimum point $m_{k_0} > \alpha(\delta)$ for which $\psi_{Q}(\alpha(\delta)) > c \psi_{Q}(m_{k_0})$, then choose $m_{k_0}$ for the regularization parameter. Otherwise, choose smallest local minimum point $m_k \leq \alpha(\delta) $ for which holds inequality (1). If such local minimum point does not exist, then choose $\alpha(\delta)$.
[1] T. Raus, U. Hämarik. Heuristic parameter choice in Tikhonov method from minimizers of the quasi-optimality function. In: Hofmann, Bernd, Leitao, Antonio, Zubelli, Jorge P. (Ed.). New Trends in Parameter Identification for Mathematical Models (1−18). Birkhäuser De Gruyter:227 - 244, 2018.
[2] T. Raus, U. Hämarik. About the Balancing Principle for Choice of the Regularization Parameter. Numerical Functional Analysis and Optimization, 30:9-10,
951 - 970, 2008.
Multi-Penalty TV Regularisation for Image Denoising: A Study University of Vienna, Austria
A common method for image denoising would be through TV regularisation, i.e.,
$$
\frac{1}{2}\|k\ast x-y^\delta\|^2+\alpha\operatorname{TV}(x)\to\min_x,
$$
with $k=\operatorname{id}$ and $\alpha>0$ as the parameter determining the trade-off between the accuracy and computational stability of your solution. The noise level $\|y-y^\delta\|\le\delta$ is usually unknown, and therefore in this talk, I have opted to present a numerical study of the performance of several heuristic (i.e., noise-level free) parameter choice rules for total variation regularisation, both isotropic and anisotropic, with a focus on image denoising.
This is a prelude, however, to the more ominous multi-parameter choice problem [2] through the example of semiblind deconvolution [1], in which one only has an approximation $k_\eta$ of a blurring kernel $k$, with $\|k-k^\eta\|\le\eta$ (and $\eta$ is known, thus the expression "semi"-blind). The functional we would like to minimise would then be
$$
\frac{1}{2}\|k\ast x-y^\delta\|^2+\alpha\operatorname{TV}(x)+\beta\|k-k^\eta\|^2\to\min_{x,k}.
$$
To quote a famous science-fiction film: "now there are two of them" ($\alpha$ and $\beta$, that is).
[1] A. Buccini, M. Donatelli, R. Ramlau, A Semiblind Regularization Algorithm for Inverse Problems with Application to Image Deblurring, SIAM Journal on Scientific Computing, 2018. https://epubs.siam.org/doi/10.1137/16M1101830.
[2] M. Fornasier, V. Naumova, S. V. Pereverzyev, Parameter Choice Strategies for Multipenalty Regularization, SIAM Journal on Numerical Analysis, 2014. https://epubs.siam.org/doi/10.1137/130930248.
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| 4:00pm - 6:00pm | CT08: Contributed talks Location: VG2.105 Session Chair: Stephan F Huckemann |
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On Adaptive confidence Ellipsoids for sparse high dimensional linear models Cambridge University, United Kingdom
In high-dimensional linear models the problem of constructing adaptive confidence sets for the full parameter is known to be generally impossible. We propose re-weighted loss functions under which constructing fully adaptive confidence sets for the parameter is shown to be possible. We give necessary and sufficient conditions on the loss functions for which adaptive confidence sets exist, and exhibit a concrete rate-optimal procedure for construction of such confidence sets.
Sparsity-promoting hierarchical Bayesian inverse problems and uncertainty quantification Massachusetts Institute of Technology, United States of America
Recovering sparse generative models from limited and noisy measurements presents a significant and complex challenge. Given that the available data is frequently inadequate and affected by noise, it is crucial to assess the resulting uncertainty in the relevant parameters. Notably, this uncertainty in the parameters directly impacts the reliability of predictions and decision-making processes.
In this talk, we explore the Bayesian framework, which facilitates the quantification of uncertainty in parameter estimates by treating involved quantities as random variables and leveraging the posterior distribution. Within the Bayesian framework, sparsity promotion and computational efficiency can be attained with hierarchical models with conditionally Gaussian priors and gamma hyper-priors. However, most of the existing literature focuses on the numerical approximation of maximum a posteriori (MAP) estimates, and less attention has been given to sampling methods or other means for uncertainty quantification. To address this gap, our talk will delve into recent advancements and developments in uncertainty quantification and sampling techniques for sparsity-promoting hierarchical Bayesian inverse problems.
Parts of this talk are joint work with Anne Gelb (Dartmouth), Youssef Marzouk (MIT), and Jonathan Lindbloom (Dartmouth).
Recursive Update of Linearization Model Error for Conductivity Reconstruction from ICDI 1Technical University of Denmark, Denmark; 2The Chinese University of Hong Kong, China
Conductivity Reconstruction serves as one of the most critical tasks of medical imaging, while approaches concerning interior current density information (ICDI) have drawn a lot of attention recently. However, they face challenges due to the nonlinearity between the conductivity and the interior current density and the high contrast of the conductivity. In this work, we propose a novel Bayesian framework to tackle these difficulties. We incorporate and iteratively update the model error introduced by linearization in the framework, and we also reform the linearization operator recursively to obtain better approximation. Numerical implementation shows that our method outperforms other approaches in terms of both relative errors of estimates and Kullback-Leibler divergence between distributions.
Fractional graph Laplacian for image reconstruction 1University of Study of Insubria, Italy; 2University of Cagliari, Italy
Image reconstruction problems, like image deblurring and computer tomography, are usually ill-posed and require regularization.
A popular approach to regularization is to substitute the original problem with an optimization problem that minimizes the sum of two terms, an $\ell^2$ term and an $\ell^q$ term with $0<q\leq 1$. The first penalizes the distance between the measured data and the reconstructed one, the latter imposes sparsity on some features of the computed solution.
In this work, we propose to use the fractional Laplacian of a properly constructed graph in the $\ell^q$ term to compute extremely accurate reconstructions of the desired images. A simple model with a fully plug-and-play method is used to construct the graph and enhanced diffusion on the graph is achieved with the use of a fractional exponent in the Laplacian operator. Since this is a global operator, we propose to replace it with an approximation in an appropriate Krylov subspace.
We show that the algorithm is a regularization method under some reasonable assumptions. Some selected numerical examples in image deblurring and computer tomography show the performances of our proposal.
[1] D. Bianchi, A. Buccini, M. Donatelli, E. Randazzo. Graph Laplacian for image deblurring. Electronic Transactions on Numerical Analysis, 55:169-186, 2021.
[2] A. Buccini, M. Donatelli. Graph Laplacian in $\ell^2-\ell^q$ regularization for image reconstruction. Proceedings - 2021 21st International Conference on Computational Science and Its Applications, ICCSA 2021:29-38, 2021.
[3] S. Aleotti, A. Buccini, M. Donatelli. Fractional Graph Laplacian for image reconstruction.In progress, Applied Numerical Mathematics, 2023
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| Date: Friday, 08/Sept/2023 | |
| 4:00pm - 6:00pm | CT13: Contributed talks Location: VG2.105 Session Chair: Martin Halla |
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Joint Born Inversion of Acoustic and Electromagnetic Wave fields 1Centrum Wiskunde & Informatica, Netherlands, The; 2Delft University of Technology, Delft, The Netherlands
Imaging by inversion of acoustic or electromagnetic wave fields have applications in a wide variety of areas, such as non-destructive testing, biomedical applications, and geophysical exploration. However, each modality suffers from its own application specific limitations with respect to resolution and sensitivity. To exploit the advantages of both imaging modalities, methods to combine them include image fusion, usage of spatial priors and application of joint or multi-physics inversion methods. In this work, a joint inversion algorithm based on structural similarity is presented. In particular, a joint Born inversion (BI) algorithm has been developed and tested successfully. With standard BI, an error functional based on the L2-norm of the mismatch between the measured and modeled wave field is minimized iteratively. To accomplish joint BI, we extend the standard error functional with an additional penalty term based on the L2-norm of the difference between the gradients of the acoustic and electromagnetic contrasts.
Imaging of Gravity Dam-Foundation contact by a shape optimization method using non-destructive seismic waves 1INRIA, France; 2EDF R&D, France
The knowledge of concrete-rock foundation interface is a key factor to evaluate the stability of gravity dams as well as understanding their mechanical behavior under water pressure. Being an inaccessible part of the structure, the exploration of this region is a complex procedure. Coring techniques can be used, but they only give limited information about a specific location and can be damaging in some situations. Hence the usefulness of non-destructive seismic waves.
We model several non-destructive seismic waves and we propose an inversion scheme for ob- taining the shape of the interface. Our approach consists in solving an inverse problem using “full-wave inversion” type techniques from wave measurements simulated by the finite element method. The inverse problem is modeled as an optimization for a least square cost functional with perimeter regularization associated with sparse data collected on the dam wall. We model different type of measurements such as elastic waves when the source is on the dam wall or acoustic waves when the source is in the water. Moreover, in order to numerically model the radiation conditions in the rock and in the water we employ PML techniques.
We present some validating results on realistic experiments. We demonstrate in particular how our proposed methodology is capable of accurately reconstructing the interface wile classical reverse time migration techniques fail. We then discuss sensitivity with respect to the position and the number of sensors, the wave number as well as to the propagation medium (for example shape of the dam) and the properties of the materials.
Structure inversions for sound speed differences in solar-like stars 1Heidelberg Institute for Theoretical Studies, Heidelberg, Germany; 2University of Heidelberg, Heidelberg, Germany; 3Max Planck Institute for Astrophysics, Garching, Germany; 4Yale University, New Haven, CT, USA
Data from the Kepler Space telescope have allowed stellar astrophysicists to measure the frequencies of oscillation modes in many stars. These frequencies carry information about the internal structure of the stars, providing ways to test stellar theory. One method, called structure inversions, seeks to infer differences in internal sound speed between a star and its model using the differences in oscillation frequencies. While this method was used extensively to study the structure of the Sun, the number of other stars studied with structure inversions remains low. In the case of main-sequence stars without a convective core, sound speed inversion results are currently only available for two stars other than the Sun. I will present the results of structure inversions for about 10 solar-like stars and discuss what these results imply about our current understanding of stellar structure.
Detection of geophysical structures using optical flow methodologies for potential data University of Guadalajara, Mexico
The subsurface exploration, as part of the development of the human being's environment, focuses on the location of water and mineral deposits, oil, gas, geological structures, among others. Geophysical methods provide information about natural resources, besides information of structures generated by human beings, namely archaeological structures, from the analysis of their physical properties, density of a source body for instance.
The Euler’s homogeneity equation for geophysical potential data is given by:
\begin{equation*}
(x-x_0)\frac{\partial T}{\partial x} + (y-y_0) \frac{\partial T}{\partial y} +
(z - z_0)\frac{\partial T}{\partial z} = n(B - T),
\end{equation*}
where $(x_0,y_0,z_0 )$ refers to the top of a source object, $(x,y,z)$ refers to the position of the observed potential field $T$, $n$ is the structural index, which depends on the source geometry, and $B$ is the regional value of the total field [1].
The inverse problem we are interested in, consists in locating a set of points $(x_0,y_0,z_0 )$ on the top of the source, from observed potential field data. In the classical Euler deconvolution strategy, this is achieved by solving Euler's homogeneity equation shown above. However, this strategy has an opportunity area in estimating the vertical component $z_0$ of the points composing the top of the source. This limitation is amplified when multiple source objects are present.
In the area of image processing, in particular in optical flow, the movement of pixels between two frames is analyzed. The spatial and temporal displacements are assumed to follow the Lambertian assumption: the pixels intensity remains after displacement. This assumption results in a differential equation very similar to the Euler's homogeneity equation: the Optical Flow equation:
$$\nabla T = 0, $$
where $\nabla$ indicates spatial-temporal derivatives. There exist methodological similarities between standard Euler deconvolution method, and standard Optical Flow methods such as Lucas-Kanade. Thus, our hypothesis is that the improving methods applicable to Optical Flow, such as Horn and Schunck method [2], will benefit analogously to the Euler deconvolution method. By reformulating the Euler deconvolution strategy to be similar to the Horn and Schunck method, the position of the top of the source is estimated by minimizing the energy functional:
\begin{equation*}
\begin{aligned}
E_{HSED}(u,v,w)=\int \int \int_{}^{}&(T_x u +T_y v +T_z w - n(B - T))^2 +
\Big(\lambda_u(u_x^2+u_y^2+u_z^2)\\
& + \lambda_v(v_x^2+v_y^2+v_z^2) + \lambda_w(w_x^2+w_y^2+w_z^2)\Big) d_x d_y d_z ,
\end{aligned}
\end{equation*}
where $u$,$v$ and $w$ are the unknowns, $\lambda_u$, $\lambda_v$ and $\lambda_w$ are regularization parameters and the sub-indices indicate partial derivation. It is noticed that the first term in the integral corresponds to Euler’s equation, meanwhile regularization terms impose smoothness on the source position reconstruction.
In this work it will be shown the results obtained after applying the methodology to synthetic 3D subsurface models. We present evidence that the horizontal location of the sources provided by Horn and Schunck based formulation is comparable to results obtained by standard Euler deconvolution strategy, with the advantage that the depth of the top of the subsurface's source is properly estimated.
[1] D.T . Thompsom., A new technique for making computer-assisted depth estimates from magnetic data.", Vol 47. 1982.
[2] Berthold K.P. Horn, Brian G. Schunck . "Determining Optical Flow", Artificial
Intelligence Laboratory, Massachusetts Institute of Technology, Cam bridge, MA 02139, 1981.
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