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BEGIN:VEVENT
SUMMARY:Consistency of Bayesian inference with Gaussian process priors for
a parabolic inverse problem
DTSTART;VALUE=DATE-TIME:20210924T082000Z
DTEND;VALUE=DATE-TIME:20210924T085000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-827@conference2.aau.at
DESCRIPTION:Speakers: Hanne Kekkonen (Delft University of Technology)\nWe
consider the statistical nonlinear inverse problem of recovering the absor
ption term f > 0 in the heat equation\, with given boundary and initial va
lue functions\, from N discrete noisy point evaluations of the solution u_
f. We study the statistical performance of Bayesian nonparametric procedur
es based on Gaussian process priors\, that are often used in practice. We
show that\, as the number of measurements increases\, the resulting poster
ior distributions concentrate around the true parameter f* that generated
the data\, and derive a convergence rate for the reconstruction error of t
he associated posterior means. We also consider the optimality of the cont
raction rates and prove a lower bound for the minimax convergence rate for
inferring f from the data\, and show that optimal rates can be achieved w
ith truncated Gaussian priors.\n\nhttps://conference2.aau.at/event/62/cont
ributions/827/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/827/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Convergence Rate of Runge-Kutta-Type Regularization for Nonlinear
Ill-Posed Problems under Logarithmic Source Condition
DTSTART;VALUE=DATE-TIME:20210923T082000Z
DTEND;VALUE=DATE-TIME:20210923T085000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-838@conference2.aau.at
DESCRIPTION:Speakers: Christine Böckmann (University of Potsdam\, Institu
te of Mathematics\, Karl-Liebknecht-Str. 24-25\, 14476 Potsdam\, Gemany)\n
We present two families of regularization method for solving nonlinear ill
-posed problems between Hilbert spaces by applying the family of Runge–K
utta methods to an initial value problem\, in particular\, to the asymptot
ical regularization method.\nIn Hohage [1]\, a systematic study of converg
ence rates for regularization methods under logarithmic source condition i
ncluding the case of operator approximations for a priori and a posteriori
stopping rules is provided. \nWe prove the logarithmic convergence rate o
f the families of usual and modified iterative Runge-Kutta methods under t
he logarithmic source condition\, and numerically verify the obtained resu
lts. The iterative regularization is terminated by the a posteriori discre
pancy principle\, Pornsawad\, et al. [2]. Up to now\, the logarithmic conv
ergence rate under logarithmic source condition has only been investigated
for particular examples\, namely\, the Levenberg–Marquardt method [3] a
nd the modified Landweber method [4]. Here\, we extended the results to th
e whole family of Runge-Kutta-type methods with and without modification.\
n \n[1] Hohage\, T.\, Regularization of exponentially ill-posed problems.
Numer. Funct. Anal. Optimiz. 2000\, 21\, 439–464.\n[2] Pornsawad\, P.\,
Resmerita\, E.\, Böckmann\, C.\, Convergence Rate of Runge-Kutta-Type Reg
ularization for Nonlinear Ill-Posed Problems under Logarithmic Source Cond
ition\, Mathematics 2021\, 9\, 1042.\n[3] Böckmann\, C.\, Kammanee\, A.\,
Braunß\, A.\, Logarithmic convergence rate of Levenberg–Marquardt meth
od with application to an inverse potential problem. J. Inv. Ill-Posed Pro
bl. 2011\, 19\, 345–367. \n[4] Pornsawad\, P.\, Sungcharoen\, P.\, Böck
mann\, C.\, Convergence rate of the modified Landweber method for solving
inverse potential problems. Mathematics 2020\, 8\, 608.\n\nhttps://confere
nce2.aau.at/event/62/contributions/838/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/838/
END:VEVENT
BEGIN:VEVENT
SUMMARY:A discrepancy-type stopping rule for conjugate gradients under whi
te noise
DTSTART;VALUE=DATE-TIME:20210922T124000Z
DTEND;VALUE=DATE-TIME:20210922T131000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-842@conference2.aau.at
DESCRIPTION:Speakers: Markus Reiß (Humboldt-Universität zu Berlin)\nWe c
onsider a linear inverse problem of the form $y=Ax+\\epsilon \\dot W$ wher
e the action of the operator (matrix) $A$ on the unknown $x$ is corrupted
by white noise (a standard Gaussian vector) $\\dot W$ of level $\\epsilon>
0$. We study the candidate solutions $\\hat x_m$ provided by the $m$-th co
njugate gradient CGNE iterates. Refining Nemirovskii's trick\, we are able
to provide explicit error bounds for the best (oracle) iterate along the
iteration path. This yields optimal estimation rates over polynomial sourc
e conditions.\nIn a second step we identify monotonic proxies for bias (ap
proximation error) and variance (stochastic error) of the nonlinear estima
tors $\\hat x_m$ and develop a residual-based stopping rule for a data-dri
ven choice $\\hat m$ of the number of iterations. This yields a stochastic
version of the discrepancy principle. Using tools from concentration of m
easure and extending deterministic ideas by Hanke\, we can provide an orac
le-type inequality for the prediction error $E[\\|A(\\hat x_{\\hat m}-x)\\
|^2]$ (non-trivial under white noise)\, which gives rate-optimality up to
a dimensionality effect. Finally\, we provide partial results also for the
estimation error $E[\\|\\hat x_{\\hat m}-x\\|^2]$\, discussing the challe
nges generated by the statistical noise.\n\nhttps://conference2.aau.at/eve
nt/62/contributions/842/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/842/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Diffractive tensor field tomography as an inverse problem for a tr
ansport equation
DTSTART;VALUE=DATE-TIME:20210922T094000Z
DTEND;VALUE=DATE-TIME:20210922T100000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-835@conference2.aau.at
DESCRIPTION:Speakers: Lukas Vierus (Saarland University)\nWe consider a ho
listic approach to find a closed formula for the generalized ray transform
of a tensor field. This means that we take refraction\, attenuation and t
ime-dependence into account. We model the refraction by an appropriate Rie
mannian metric which leads to an integration along geodesics. The absorpti
on appears as an attenuation coefficient in an exponential factor. The der
ived explicit integral formula solves a transport equation whose boundary
conditions are given by the measured data. Deriving the weak formulation o
f the problem\, we obtain solutions in Sobolev-Bochner spaces. Whereas it
fails to guarantee a unique solution of the implied initial boundary value
problem (IBVP)\, it is possible to prove uniqueness of viscosity solution
s by using the Lax-Milgram-theorem. For this\, however\, certain restricti
ons on the refractive index and the attenuation coefficient must be assume
d. Considering the parameter-to-solution map as the forward operator\, the
inverse problem can be solved by minimizing a Tikhonov functional. Here t
he adjoint operator can also be identified as a solution of an IBVP.\n\nht
tps://conference2.aau.at/event/62/contributions/835/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/835/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Variational Data Assimilation and low-rank solvers
DTSTART;VALUE=DATE-TIME:20210922T070000Z
DTEND;VALUE=DATE-TIME:20210922T075000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-867@conference2.aau.at
DESCRIPTION:Speakers: Melina Freitag ()\nWeak constraint four-dimensional
variational data assimilation is an important method for incorporating obs
ervations into a (usually\nimperfect) model. The resulting minimisation pr
ocess takes place in very high dimensions. In this talk we present two app
roaches for reducing the dimension and thereby the computational cost and
storage of this optimisation problem. The first approach formulates the li
nearised system as a saddle point problem. We present a low-rank approach
which exploits the structure of the saddle point system using techniques a
nd theory from solving large scale matrix equations and low-rank Krylov su
bspace methods. The second approach uses projection methods for reducing t
he system dimension. Numerical experiments with the linear advection-diffu
sion equation\, and the nonlinear Lorenz-95 model demonstrate the effectiv
eness of both approaches.\n\nhttps://conference2.aau.at/event/62/contribut
ions/867/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/867/
END:VEVENT
BEGIN:VEVENT
SUMMARY:The tangential cone condition for EIT
DTSTART;VALUE=DATE-TIME:20210922T082000Z
DTEND;VALUE=DATE-TIME:20210922T085000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-836@conference2.aau.at
DESCRIPTION:Speakers: Stefan Kindermann (Johannes Kepler University Linz)
\nThe tangential cone conditions (TCCs) are sufficient conditions on a non
linear forward operator for proving convergence of various iterative nonli
near regularization schemes such as Landweber iteration. Especially for pa
rameter identification problems with boundary data\, they have not been ve
rified yet\, even though numerical results for nonlinear iterative regular
ization method\nusually show the expected convergence behavior. In this t
alk we analyze the tangential cone conditions for the classical impedance
tomography problem (EIT) and state sufficient conditions when they hold\,
although a general result on the validity of the TCCs remains open. An imp
ortant tool is the use of Loewner monotonicity\, which allows us to prove
the TCC in situations\, e.g.\, when the conductivities are pointwisely ab
ove or below the true conductivity. This talk is a summary of the arXiv ar
ticle [1]\n\n[1] S. Kindermann\, On the tangential cone condition for elec
trical impedance tomography\, \nPreprint on arXiv\, 2021. https://arxiv.or
g/abs/2105.02635\n\nhttps://conference2.aau.at/event/62/contributions/836/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/836/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Monotonicity-Based Regularization for Shape Reconstruction in Line
ar Elasticity
DTSTART;VALUE=DATE-TIME:20210922T092000Z
DTEND;VALUE=DATE-TIME:20210922T094000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-841@conference2.aau.at
DESCRIPTION:Speakers: Sarah Eberle (Goethe University Frankfurt)\nWe deal
with the shape reconstruction of inclusions in elastic bodies and solve th
e inverse problem by means of a monotonicity-based regularization. In more
detail\, we show how the monotonicity methods can be converted into a reg
ularization method for a data-fitting functional without losing the conver
gence properties of the monotonicity methods. In doing so\, we introduce c
onstraints on the minimization problem of the residual based on the monoto
nicity methods and prove the existence and uniqueness of a minimizer as we
ll as the convergence of the method for noisy data. In addition\, we compa
re numerical reconstructions of inclusions based on the monotonicity-based
regularization with a standard approach (one-step linearization with Tikh
onov-like regularization)\, which also shows the robustness of our method
regarding noise in practice.\n\nhttps://conference2.aau.at/event/62/contri
butions/841/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/841/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Uniqueness and global convergence for inverse coefficient problems
with finitely many measurements
DTSTART;VALUE=DATE-TIME:20210922T085000Z
DTEND;VALUE=DATE-TIME:20210922T092000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-833@conference2.aau.at
DESCRIPTION:Speakers: Bastian Harrach (Goethe University Frankfurt)\nSever
al applications in medical imaging and non-destructive material testing le
ad 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 prob
lem\, for which unique reconstructability results\, stability estimates an
d global convergence of numerical methods are very hard to achieve.\n\nIn
this talk we will consider an inverse coefficient problem with finitely ma
ny measurements and a finite desired resolution. We will present a criteri
on based on monotonicity\, convexity and localized potentials arguments th
at allows us to explicitly estimate the number of measurements that is req
uired to achieve the desired resolution. We also obtain an error estimate
for noisy data\, and overcome the problem of local minima by rewriting the
problem as an equivalent uniquely solvable convex non-linear semidefinite
optimization problem.\n\n**References**\n 1. B. Harrach\, Uniqueness\, st
ability and global convergence for a discrete inverse elliptic Robin trans
mission problem\, *Numer. Math.* **147** (2021)\, pp. 29-70\, [https://doi
.org/10.1007/s00211-020-01162-8][1]\n 2. B. Harrach\, Solving an inverse e
lliptic coefficient problem by convex non-linear semidefinite programming\
, arXiv preprint (2021)\, [arXiv:2105.11440][2]\n\n\n [1]: https://doi.or
g/10.1007/s00211-020-01162-8\n [2]: http://arxiv.org/abs/2105.11440\n\nht
tps://conference2.aau.at/event/62/contributions/833/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/833/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bayesian non-linear inversion problems and PDEs: progress and chal
lenges
DTSTART;VALUE=DATE-TIME:20210922T115000Z
DTEND;VALUE=DATE-TIME:20210922T124000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-866@conference2.aau.at
DESCRIPTION:Speakers: Richard Nickl (University of Cambridge)\nWe review t
he Bayesian approach to inverse problems\, and describe recent progress in
our theoretical understanding of its performance in non-linear situations
. Statistical and computational guarantees for such algorithms will be pro
vided in high-dimensional\, non-convex scenarios\, and model examples from
elliptic and transport (X-ray type) PDE problems will be discussed. The c
onnection between MCMC and other existing iterative methods will be touche
d upon\, and several open mathematical problems will be described.\n\nhttp
s://conference2.aau.at/event/62/contributions/866/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/866/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Infinite-dimensional inverse problems with finite measurements
DTSTART;VALUE=DATE-TIME:20210923T070000Z
DTEND;VALUE=DATE-TIME:20210923T075000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-865@conference2.aau.at
DESCRIPTION:Speakers: Giovanni Alberti (University of Genova)\nIn this tal
k I will discuss uniqueness\, stability and reconstruction for infinite-di
mensional nonlinear inverse problems with finite measurements\, under the
a priori assumption that the unknown lies in\, or is well-approximated by\
, a finite-dimensional subspace or submanifold. The methods are based on t
he interplay of applied harmonic analysis\, in particular sampling theory
and compressed sensing\, and the theory of inverse problems for partial di
fferential equations. Several examples\, including the Calderón problem a
nd scattering\, will be discussed.\n\nhttps://conference2.aau.at/event/62/
contributions/865/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/865/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stable determination of a rigid scatterer in elastodynamics
DTSTART;VALUE=DATE-TIME:20210924T070000Z
DTEND;VALUE=DATE-TIME:20210924T075000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-864@conference2.aau.at
DESCRIPTION:Speakers: Eva Sincich (University of Trieste)\nWe deal with an
inverse elastic scattering problem for the shape determination of a rigid
scatterer in the time-harmonic regime. We prove a local stability estimat
e of log log type for the identification of a scatterer by a single far-fi
eld measurement. \nThe needed a priori condition on the closeness of the s
catterers is estimated by the universal constant appearing in the Friedric
hs inequality. \n\nThis is based on a joint work with Luca Rondi and Moura
d Sini.\n\nhttps://conference2.aau.at/event/62/contributions/864/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/864/
END:VEVENT
BEGIN:VEVENT
SUMMARY:A Generative Variational Model for Inverse Problems in Imaging
DTSTART;VALUE=DATE-TIME:20210922T145000Z
DTEND;VALUE=DATE-TIME:20210922T151000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-863@conference2.aau.at
DESCRIPTION:Speakers: Andreas Habring (University of Graz)\nIn recent year
s deep/machine learning methods using convolutional networks have become i
ncreas- ingly popular also in inverse problems mainly due to their practic
al performance [1]. In many cases these methods outperform conventional re
gularization methods\, such as total variation regulariza- tion\, in parti
cular when applied to more complicated data such as images containing text
ure. A major downside of machine learning methods\, however\, is the need
for large sets of training data\, which are often not available in the nec
essary extent. Moreover\, the level of analytic understanding of machine l
earning methods\, in particular in view of an analysis for inverse problem
s in function space\, is still far from the one of conventional variationa
l methods.\nIn this talk\, we propose a novel regularization method for so
lving inverse problems in imaging\, which is inspired by the architecture
of convolutional neural networks as seen in many in deep learning approach
es. In the model\, the unknown is generated from a variable in latent spac
e via multi-layer convolutions and non-linear penalties. In contrast to co
nventional deep learning methods\, however\, the convolution kernels are l
earned directly from the given (possibly noisy) data\, such that no traini
ng is required.\nIn the talk\, we will motivate the model and provide theo
retical results about existence/stability of solutions and convergence for
vanishing noise in function space. Afterwards\, in a discretized setting\
, we will show practical results of our method in comparison to a state of
the art deep learning method [1].\n\n[1] V. Lempitsky\, A. Vedaldi\, and
D. Ulyanov\, Deep image prior\, in 2018 IEEE/CVF Conference on Computer Vi
sion and Pattern Recognition.\n\nhttps://conference2.aau.at/event/62/contr
ibutions/863/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/863/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Regularization as an approximation problem
DTSTART;VALUE=DATE-TIME:20210924T092000Z
DTEND;VALUE=DATE-TIME:20210924T095000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-862@conference2.aau.at
DESCRIPTION:Speakers: Daniel Gerth ()\nClassically\, regularization method
s are often divided into three frameworks: variational regularization\, it
erative regularization\, and regularization by projection. In this talk we
consider regularization as an approximation problem in the classical Hilb
ert space setting. This enables us to treat all three categories in the sa
me framework which we demonstrate on Tikhonov regularization and Landweber
iteration. Our approach provides new insight on the way regularization wo
rks\, helps understanding parameter choice rules\, naturally includes disc
rete (finite dimensional) problems and\, maybe most importantly\, yields a
numerically observable and computable quantity\, namely a source element
for the regularized solutions\, that contains information about the smooth
ness of the unknown solution and the noise.\n\nhttps://conference2.aau.at/
event/62/contributions/862/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/862/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stochastic EM methods with Variance Reduction for Penalised PET Re
constructions
DTSTART;VALUE=DATE-TIME:20210924T132000Z
DTEND;VALUE=DATE-TIME:20210924T134000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-861@conference2.aau.at
DESCRIPTION:Speakers: Zeljko Kereta (UCL)\nExpectation-maximization (EM) i
s a popular and well-established method for image reconstruction in positr
on emission tomography (PET) due to its simple form and desirable properti
es. But\, it often suffers from slow convergence\, and full batch computat
ions are often infeasible due to large data sizes in modern scanners. Orde
red subsets EM (OSEM) is an effective mitigation scheme that provides sign
ificant acceleration during initial iterations\, but it has been observed
to enter a limit cycle. Another difficulty for EM methods is the incorpor
ation of a regularising penalty\, which poses additional difficulties for
the maximisation step.\nIn this work\, we investigate two classes of algor
ithms for accelerating OSEM based on variance reduction for penalised PET
reconstructions. The first is a stochastic variance reduced EM algorithm\,
termed as SVREM\, which is an extension of the classical EM to the stocha
stic context that combines classical OSEM with insights from variance redu
ction techniques for gradient descent and facilitates the computation of t
he M-step through parabolic surrogates for the penalty. The second views O
SEM as a preconditioned stochastic gradient ascent\, and applies variance
reduction techniques\, i.e.\, SAGA and SVRG\, to estimate the update direc
tion. We present several numerical experiments to illustrate the efficienc
y and accuracy of the two methodologies. The numerical results show that t
hese approaches significantly outperform existing OSEM type methods for pe
nalised PET reconstructions\, and hold great potential.\n\nhttps://confere
nce2.aau.at/event/62/contributions/861/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/861/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Generalized conditional gradient methods for variational inverse p
roblems with convex regularizers
DTSTART;VALUE=DATE-TIME:20210922T151000Z
DTEND;VALUE=DATE-TIME:20210922T153000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-860@conference2.aau.at
DESCRIPTION:Speakers: Marcello Carioni (University of Cambridge)\nIn this
talk\, we propose and analyze a generalized conditional gradient method fo
r infinite dimensional variational inverse problems written as the sum of
a smooth\, convex loss function and a\, possibly non-smooth\, convex regul
arizer.\nOur method relies on the mutual update of a sequence of extremal
points of the unit ball of the regularizer and a sparse iterate given as a
suitable linear combination of such extreme points. \nWe show that under
standard hypotheses on the minimization problem\, our algorithm converges
sublinearly to a solution of the inverse problem. Moreover\, we demonstrat
e that by imposing additional assumptions on the structure of the minimize
rs\, the associated dual variables and the nondegeneracy of the problem\,
we can improve such convergence result to a linear rate.\nThen we apply ou
r generalized conditional gradient method to solve dynamic inverse problem
s regularized with the Benamou-Brenier energy. Relying on recent results a
bout the characterization of the extremal points for the ball of the Benam
ou-Brenier energy\, we show that our algorithm can be applied to this spec
ific example to reconstruct the motion of heavily undersampled dynamic dat
a together with the presence of noise.\n\nhttps://conference2.aau.at/event
/62/contributions/860/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/860/
END:VEVENT
BEGIN:VEVENT
SUMMARY:From displacement field to parameter estimation: theory and applic
ation
DTSTART;VALUE=DATE-TIME:20210923T144000Z
DTEND;VALUE=DATE-TIME:20210923T150000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-859@conference2.aau.at
DESCRIPTION:Speakers: Ekaterina Sherina (University of Vienna)\nDiseases l
ike cancer or arteriosclerosis often cause changes of tissue stiffness on
the micrometer scale. Elastography is a common technique for medical diagn
ostics developed to detect these changes. We consider a complex problem of
estimating both the internal displacement field and the material paramete
rs of an object which is being subjected to a deformation. In particular\
, we present our recently developed elastographic optical flow method (EOF
M) for motion detection from optical coherence tomography images. This met
hod takes into account experimental constraints\, such as appropriate boun
dary conditions\, the use of speckle information\, as well as the inclusio
n of structural information derived from knowledge of the background mater
ial. Furthermore\, we present numerical results based on both simulated an
d experimental data from an elastography experiment and discuss the materi
al parameter estimation from these data.\n\nhttps://conference2.aau.at/eve
nt/62/contributions/859/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/859/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Recent analytical progress on some nonlinear tomography problems
DTSTART;VALUE=DATE-TIME:20210923T150000Z
DTEND;VALUE=DATE-TIME:20210923T152000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-858@conference2.aau.at
DESCRIPTION:Speakers: Jan Bohr (University of Cambridge)\nWe consider a cl
ass of nonlinear inverse problems\, encompassing e.g. Polarimetric Neutron
Tomography (PNT)\, where one seeks to recover a magnetic field by probing
it with Neutron beams and measuring the resulting spin change. In recent
years there has been great progress on fundamental theoretical questions r
egarding injectivity and stability properties for PNT and we survey some o
f the latest results\, including a novel range characterisation for the fo
rward map. One of the drivers behind these results is the desire to give r
igorous guarantees for the statistical performance of Bayesian algorithms.
The talk is based on joint work with Gabriel Paternain and Richard Nickl.
\n\nhttps://conference2.aau.at/event/62/contributions/858/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/858/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Variational analysis of a dynamic PET reconstruction model with op
timal transport regularization
DTSTART;VALUE=DATE-TIME:20210923T101000Z
DTEND;VALUE=DATE-TIME:20210923T103000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-857@conference2.aau.at
DESCRIPTION:Speakers: Marco Mauritz (University of Münster\, Institute fo
r Analysis and Numerics)\nWe consider the dynamic Positron Emission Tomogr
aphy (PET) reconstruction method proposed by Schmitzer et al. $[1]$ that p
articularly aims to reconstruct the temporal evolution of single or small
numbers of cells by leveraging optimal transport. Using a MAP estimate the
cells' evolution is reconstructed by minimizing a functional $\\mathcal{E
}_n$ - composed of a Kulback-Leibler-type data fidelity term and the Benam
our-Brenier functional - over the space of positive Radon measures. This c
hoice of the regularization ensures temporal consistency between different
time points. \n\nThe PET measurements in our forward model are described
by Poisson point processes with a given intensity $q_n$. In the talk we sh
ow $\\Gamma$-convergence of the stochastic functionals $\\mathcal{E}_n$ to
a deterministic limit functional for $q_n\\to \\infty$. This helps unders
tanding the properties of the considered reconstruction method for an incr
easing SNR. To compute the $\\Gamma$-limit we show convergence of Poisson
point processes for intensities growing to infinity as well as convergence
of the optimal transport regularization. The latter requires the approxim
ation of arbitrary Radon measures by ones satisfying the continuity equati
on while controlling the Benamou-Brenier energy.\n\nReference:\n$[1]$ B. S
chmitzer\, K. P. Schäfers\, and B. Wirth. Dynamic Cell Imaging in PET wit
h\nOptimal Transport Regularization. IEEE Transactions on Medical Imaging\
,\n2019.\n\nhttps://conference2.aau.at/event/62/contributions/857/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/857/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Direct regularized reconstruction for the three-dimensional Calder
ón problem
DTSTART;VALUE=DATE-TIME:20210924T130000Z
DTEND;VALUE=DATE-TIME:20210924T132000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-856@conference2.aau.at
DESCRIPTION:Speakers: Aksel Rasmussen (Technical University of Denmark)\nE
lectrical Impedance Tomography gives rise to the severely ill-posed Calder
ón problem of determining the electrical conductivity distribution in a b
ounded domain from knowledge of the associated Dirichlet-to-Neumann map fo
r the governing equation. The electrical conductivity of an object is of i
nterest in many fields\, notably medical imaging\, where applications may
vary from stroke detection to early detection of breast cancer.\nThe uniqu
eness and stability questions for the three-dimensional problem were large
ly answered in the affirmative in the 1980's using complex geometrical opt
ics solutions\, and this led further to a direct reconstruction method rel
ying on a non-physical scattering transform.\n\nIn this talk we look at a
direct reconstruction algorithm for the three-dimensional Calderón proble
m in the scope of regularization. Indeed\, a suitable and explicit truncat
ion of the scattering transform gives a stable and direct reconstruction m
ethod that is robust to small perturbations of the data. Numerical tests o
n simulated noisy data illustrate the feasibility and regularizing effect
of the method\, and suggest that the numerical implementation performs bet
ter than predicted by theory.\n\nhttps://conference2.aau.at/event/62/contr
ibutions/856/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/856/
END:VEVENT
BEGIN:VEVENT
SUMMARY:A model reduction approach for inverse problems with operator valu
ed data
DTSTART;VALUE=DATE-TIME:20210924T085000Z
DTEND;VALUE=DATE-TIME:20210924T092000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-855@conference2.aau.at
DESCRIPTION:Speakers: Matthias Schlottbom (University of Twente)\nWe study
the efficient numerical solution of linear inverse problems with operator
valued data which arise\, e.g.\, in seismic exploration\, inverse scatter
ing\, or tomographic imaging. The high-dimensionality of the data space im
plies extremely high computational cost already for the evaluation of the
forward operator\, which makes a numerical solution of the inverse problem
\, e.g.\, by iterative regularization methods\, practically infeasible. To
overcome this obstacle\, we develop a novel model reduction approach that
takes advantage of the underlying tensor product structure of the problem
and which allows to obtain low-dimensional certified reduced order models
of quasi-optimal rank. The theoretical results are illustrated by applica
tion to a typical model problem in fluorescence optical tomography.\n\nhtt
ps://conference2.aau.at/event/62/contributions/855/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/855/
END:VEVENT
BEGIN:VEVENT
SUMMARY:The ensemble Kalman filter applied to inverse problems: a neural n
etwork based one-shot formulation
DTSTART;VALUE=DATE-TIME:20210924T114000Z
DTEND;VALUE=DATE-TIME:20210924T120000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-854@conference2.aau.at
DESCRIPTION:Speakers: Simon Weissmann (Heidelberg University)\nThe ensembl
e Kalman filter (EnKF) is a widely used metheodology for data assimilation
problems and has been recently generalized to inverse problems\, known as
ensemble Kalman inversion (EKI). We view the method as a derivative free
optimization method for a least-squares misfit functional and we present v
arious variants of the scheme such as regularized EKI methods. This opens
up the perspective to use the method in various areas of applications such
as imaging\, groundwater flow problems\, biological problems as well as i
n the context of the training of neural networks. In particular\, we will
present application of the EKI to recent machine learning approaches\, whe
re we consider the incorporation of neural networks into inverse problems.
We replace the complex forward model by a neural network acting as a phys
ics-informed surrogate model\, which will be trained in a one-shot fashion
. This means we train the unknown parameter and the neural network at once
\, i.e. the neural network is only trained for the underlying unknown para
meter. We connect the neural network based one-shot formulation to the Bay
esian approach for inverse problems and apply the ensemble Kalman inversio
n in order to solve the optimization problem. Furthermore\, we provide nu
merical experiments to highlight the promising direction of neural network
based one-shot formulation together with the application of the ensemble
Kalman inversion.\n\nhttps://conference2.aau.at/event/62/contributions/854
/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/854/
END:VEVENT
BEGIN:VEVENT
SUMMARY:An inverse source problem for vector field
DTSTART;VALUE=DATE-TIME:20210923T125000Z
DTEND;VALUE=DATE-TIME:20210923T131000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-853@conference2.aau.at
DESCRIPTION:Speakers: David Omogbhe (Johann Radon Instute for Computationa
l and Applied Mathematics(RICAM))\nWe consider an inverse source problem i
n the stationary radiating transport through a two dimensional absorbing a
nd scattering medium. The attenuation and scattering properties of the med
ium are assumed known and the unknown vector field source is isotropic. Fo
r scattering kernels of finite Fourier content in the angular variable\, w
e show how to recover the isotropic vector field sources from boundary mea
surements. The approach is based on the Cauchy problem for a Beltrami-lik
e equation associated with $A$-analytic maps in the sense of Bukhgeim. Thi
s is a joint work with Kamran Sadiq (RICAM).\n\nhttps://conference2.aau.at
/event/62/contributions/853/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/853/
END:VEVENT
BEGIN:VEVENT
SUMMARY:On the geometric structures of Laplacian eigenfunctions and applic
ations to inverse scattering problems
DTSTART;VALUE=DATE-TIME:20210924T112000Z
DTEND;VALUE=DATE-TIME:20210924T114000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-852@conference2.aau.at
DESCRIPTION:Speakers: XINLIN CAO ()\nIn this talk\, we present some novel
findings on the geometric structures of Laplacian eigenfunctions and their
deep relationship to the quantitative behaviours of the eigenfunctions. T
he studies reveal that the intersecting angle between two lines (nodal lin
es\, singular lines and generalized singular lines) is closely related to
the vanishing order of the eigenfunction at the intersecting point in R^2.
And in R^3\, the analytic behaviors of a Laplacian eigenfunction depends
on the geometric quantities at the corresponding corner point (edge corner
and vertex corner). The theoretical findings can be applied directly to s
ome physical problems including the inverse obstacle scattering problem. T
aking two-dimensional case for example\, it is shown in a certain polygona
l setup that one can recover the support of the unknown scatterer as well
as the surface impedance parameter by finitely many far-field patterns. In
deed\, at most two far-field patterns are sufficient for some important ap
plications.\n\nhttps://conference2.aau.at/event/62/contributions/852/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/852/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ill-posedness effects for well-posed problems
DTSTART;VALUE=DATE-TIME:20210923T120000Z
DTEND;VALUE=DATE-TIME:20210923T123000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-851@conference2.aau.at
DESCRIPTION:Speakers: Arnd Rösch (Universität Duisburg-Essen)\nIn this t
alk we study the discretization of a well-posed nonlinear problem.\nIt may
happen that discretized solutions do not converge. However\, this effect
disappears for a suitable chosen optimal control problem.\n\nhttps://confe
rence2.aau.at/event/62/contributions/851/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/851/
END:VEVENT
BEGIN:VEVENT
SUMMARY:An Inverse Magnetization Problem on the Sphere with Localization C
onstraints
DTSTART;VALUE=DATE-TIME:20210922T100000Z
DTEND;VALUE=DATE-TIME:20210922T102000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-850@conference2.aau.at
DESCRIPTION:Speakers: Xinpeng Huang (TU Bergakademie Freiberg\, Institute
of Geophysics and Geoinformatics)\nWe study an inverse magnetization probl
em arising in geo- and planetary magnetism. This problem is non-unique and
the null space can be characterized by the Hardy-Hodge decomposition. The
additional assumption that the underlying magnetization is spatially loca
lized in a subdomain of the sphere (which can be justified when interested
\, e.g.\, in regional magnetic anomalies) ameliorates the non-uniqueness i
ssue so that only the tangential divergence-free contribution remains unde
termined. In a previous reconstruction approach\, we addressed the localiz
ation by including an additional penalty term in the minimizing functional
. This\, however\, requires the coestimation of the undetermined divergenc
e-free contribution. Here\, we present a first attempt at more directly in
cluding the localization constraint without requiring such a coestimation.
In addition\, we show that the localization constraint is closely connect
ed to the problem of extrapolation in Hardy spaces.\n\nhttps://conference2
.aau.at/event/62/contributions/850/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/850/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Radon-based image reconstruction in magnetic particle imaging usin
g an FFL-scanner
DTSTART;VALUE=DATE-TIME:20210923T142000Z
DTEND;VALUE=DATE-TIME:20210923T144000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-849@conference2.aau.at
DESCRIPTION:Speakers: Stephanie Blanke (Universität Hamburg)\nReliable an
d fast medical imaging techniques are indispensable for diagnostics in cli
nical everyday life. A promising example of those is given by magnetic par
ticle imaging (MPI) invented by Gleich and Weizenecker [1]. MPI is a trace
r-based imaging method allowing for the reconstruction of the spatial dist
ribution of magnetic nanoparticles via exploiting their non-linear magneti
zation response to changing magnetic fields. We dedicate ourselves towards
MPI using a field-free line (FFL) for spatial encoding [2]. For data acqu
isition the FFL is moved through the field of view resulting in a scanning
geometry resembling the one in computerized tomography. Indeed\, in the i
deal setting\, corresponding MPI data can be traced back to the Radon tran
sform of the particle concentration [3]. We jointly reconstruct Radon data
and particle concentration by means of total variation regularization and
have a look at some numerical examples. We conclude with problems that ar
ise when leaving the ideal setting. For example\, in practice\, we are con
fronted with imperfections of the applied magnetic fields leading to defor
med low-field volumes and\, when ignored\, image artifacts.\n\n*References
:*\n[1] Gleich B and Weizenecker J 2005 Tomographic imaging using the nonl
inear response of magnetic particles *Nature* 435 1214-1217 \n(https://doi
.org/10.1038/nature03808)\n[2] Weizenecker J\, Gleich B\, and Borgert J 20
08 Magnetic particle imaging using a field free line *J. Phys. D: Appl. Ph
ys.* 41 105009 \n(https://doi.org/10.1088/0022-3727/41/10/105009)\n[3] Kno
pp T\, Erbe M\, Sattel T F\, Biederer S\, and Buzug T M 2011 A Fourier sli
ce theorem for magnetic particle imaging using a field-free line *Inverse
Problems* 27 095004 \n(https://doi.org/10.1088/0266-5611/27/9/095004)\n\nh
ttps://conference2.aau.at/event/62/contributions/849/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/849/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Holmgren-John unique continuation for viscoelastic equation
DTSTART;VALUE=DATE-TIME:20210923T135000Z
DTEND;VALUE=DATE-TIME:20210923T142000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-848@conference2.aau.at
DESCRIPTION:Speakers: Gen Nakamura ()\nWe concern on the Holmgren-John un
ique continuation theorem for a visco-elastic equation with a memory term
when the coefficients of the equation are analytic. This is a special cas
e of the general unique continuation property (UCP) for the equation if it
s coefficients are analytic. This equation describes visco-elastic behavio
r of a medium. In this talk we will present the UCP for the viscoelastic e
quation when the relaxation tensor is analytic and allowed to be fully ani
sotropic. We will describe the UCP in terms of a distance defined by \nthe
travel time of the slowest wave associated to the elastic part of this eq
uation.\n\nThe collaborators of this study are Maarten de Hoop (Rice Unive
rsity)\, Matthias Eller (Georgetown University) and Ching-Lung Lin (Nation
al Cheng-Kung University).\n\nhttps://conference2.aau.at/event/62/contribu
tions/848/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/848/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adaptive Spectral Decomposition for Inverse Scattering Problems
DTSTART;VALUE=DATE-TIME:20210923T123000Z
DTEND;VALUE=DATE-TIME:20210923T125000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-847@conference2.aau.at
DESCRIPTION:Speakers: Yannik G. Gleichmann ()\nA nonlinear optimization me
thod is proposed for inverse scattering problems\, when the unknown medium
is characterized by one or several spatially varying parameters. The inve
rse medium problem is formulated as a PDE-constrained optimization problem
and solved by an inexact truncated Newton-type method. Instead of a grid-
based discrete representation\, each parameter is projected to a separate
fnite-dimensional subspace\, which is iteratively adapted during the optim
ization. Each subspace is spanned by the first few eigenfunctions of a lin
earized regularization penalty functional chosen a priori. The (small and
slowly increasing) finite number of eigenfunctions effectively introduces
regularization into the inversion and thus avoids the need for standard Ti
khonov-type regularization and\, in practice\, appears more robust to miss
ing data or added noise. Numerical results illustrate the accuracy and eff
iciency of the resulting adaptive spectral regularization for inverse scat
tering problems for the wave equation in time domain.\n\nhttps://conferenc
e2.aau.at/event/62/contributions/847/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/847/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frame Decompositions and Inverse Problems
DTSTART;VALUE=DATE-TIME:20210923T085000Z
DTEND;VALUE=DATE-TIME:20210923T091000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-846@conference2.aau.at
DESCRIPTION:Speakers: Simon Hubmer (Johann Radon Institute Linz)\nThe sing
ular-value decomposition (SVD) is an important tool for the analysis and s
olution of linear ill-posed problems in Hilbert spaces. However\, it is of
ten difficult to derive the SVD of a given operator explicitly\, which lim
its its practical usefulness. An alternative in these situations are frame
decompositions (FDs)\, which are a generalization of the SVD based on sui
tably connected families of functions forming frames. Similar to the SVD\,
these FDs encode information on the structure and ill-posedness of the pr
oblem and can be used as the basis for the design and implementation of ef
ficient numerical solution methods. Crucially though\, FDs can be derived
explicitly for a wide class of operators\, in particular for those satisfy
ing a certain stability condition. In this talk\, we consider various theo
retical aspects of FDs such as recipes for their construction and some pro
perties of the reconstruction formulae induced by them. Furthermore\, we p
resent convergence and convergence rates results for continuous regulariza
tion methods based on FDs under both a-priori and a-posteriori parameter c
hoice rules. Finally\, we consider the practical utility of FDs for solvin
g inverse problems by considering two numerical examples from computerized
and atmospheric tomography.\n\nhttps://conference2.aau.at/event/62/contri
butions/846/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/846/
END:VEVENT
BEGIN:VEVENT
SUMMARY:iPALM-based unsupervised energy disaggregation
DTSTART;VALUE=DATE-TIME:20210922T133000Z
DTEND;VALUE=DATE-TIME:20210922T135000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-845@conference2.aau.at
DESCRIPTION:Speakers: Christian Aarset (University of Graz)\nWith smart en
ergy meters increasingly available to private households\, new application
s arise\, such as identifying main power consuming devices and predicting
human activity. One major obstacle is that smart energy meters typically p
rovide *aggregated* data\, where each source of energy consumption is summ
ed. Further\, obtaining training data can be intrusive. To counteract this
\, we propose an unsupervised minimization approach based on the Inertial
Proximal Alternating Linearized Minimization (iPALM) algorithm\, utilising
convolutional sparse coding to represent individual device energy signatu
res as atoms convolved with sparse coefficient vectors.\n\nhttps://confere
nce2.aau.at/event/62/contributions/845/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/845/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Parameter identification for PDEs: From neural-network-based learn
ing to discretized inverse problems
DTSTART;VALUE=DATE-TIME:20210922T141000Z
DTEND;VALUE=DATE-TIME:20210922T143000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-844@conference2.aau.at
DESCRIPTION:Speakers: Tram Nguyen ()\nWe investigate the problem of learni
ng an unknown nonlinearity in parameter-dependent PDEs. The nonlineartiy i
s represented via a neural network of an unknown state. The learning-infor
med PDE model has three unknowns: physical parameter\, state and nonlinear
ity. We propose an all-at-once approach to the minimization problem. (Join
t work: Martin Holler\, Christian Aarset)\nMore generally\, the representa
tion via neural networks can be realized as a discretization scheme. We st
udy convergence of Tikhonov and Landweber methods for the discretized inve
rse problems\, and prove convergence when the discretization error approac
hes zero. (Joint work: Barbara Kaltenbacher)\n\nhttps://conference2.aau.at
/event/62/contributions/844/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/844/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Regularisation of certain non-linear problems in $L^{\\infty}$
DTSTART;VALUE=DATE-TIME:20210923T095000Z
DTEND;VALUE=DATE-TIME:20210923T101000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-843@conference2.aau.at
DESCRIPTION:Speakers: Lukas Pieronek (KIT)\nIn many cases the parameters o
f interest in inverse problems arise as coefficients of PDE models for whi
ch $L^{\\infty}$ is one of the most natural spaces. Despite its formal con
nection to the regular and regularisation-approved $L^p$-spaces\, $L^{\\in
fty}$ itself is non-smooth\, non-reflexive and non-separable. Hence\, stan
dard Banach space methods generally fail and the need of discretisation in
practice makes it even hopeless to aim for good reconstructions in the st
rong topology. In this talk we present a novel regularisation method which
generates uniformly bounded iterates as approximate solutions to locally
ill-posed equations and for which the regularisation property then holds w
ith respect to weak-$\\ast$ convergence. Numerical examples will complete
our analysis.\n\nhttps://conference2.aau.at/event/62/contributions/843/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/843/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Algorithmic improvements via a dictionary learning add-on
DTSTART;VALUE=DATE-TIME:20210924T122000Z
DTEND;VALUE=DATE-TIME:20210924T124000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-840@conference2.aau.at
DESCRIPTION:Speakers: Naomi Schneider (University of Siegen\, Geomathemati
cs Group)\nIn the last 10 years\, the Inverse Problem Matching Pursuits (I
PMPs) were proposed as alternative solvers for linear inverse problems on
the sphere and the ball\, e.g. from the geosciences. They were constantly
further developed and tested on diverse applications\, e.g. on the downwar
d continuation of the gravitational potential. This task remains a priorit
y in geodesy due to significant contemporary challenges like the climate c
hange.\nIt is well-known that\, for linear inverse problems on the sphere\
, there exist a variety of global as well as local basis systems\, e.g. sp
herical harmonics\, Slepian functions as well as radial basis functions an
d wavelets. All of these system have their specific pros and cons. Nonethe
less\, approximations are often represented in only one of the systems. \n
On the contrary\, as matching pursuits\, the IPMPs realize the following l
ine of thought: an approximation is built in a so-called best basis\, i.e.
a mixture of diverse trial functions. Such a basis is chosen iteratively
from an intentionally overcomplete dictionary which contains several types
of the mentioned global and local functions. The choice of the next best
basis element aims to reduce the Tikhonov functional. \nIn practice\, an a
-priori\, finite set of trial functions was usually used which was highly
inefficient. We developed a learning add-on which enables us to work with
an infinite dictionary instead while simultaneously reducing the computati
onal cost. Moreover\, it automatized the dictionary choice as well. The ad
d-on is implemented as constrained non-linear optimization problems with r
espect to the characteristic parameters of the different basis systems. In
this talk\, we explain the learning add-on and show recent numerical resu
lts with respect to the downward continuation of the gravitational potenti
al.\n\nhttps://conference2.aau.at/event/62/contributions/840/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/840/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stability estimates for a special class of anisotropic conductivit
ies with an ad-hoc functional
DTSTART;VALUE=DATE-TIME:20210924T124000Z
DTEND;VALUE=DATE-TIME:20210924T130000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-839@conference2.aau.at
DESCRIPTION:Speakers: Sonia Foschiatti (Università degli Studi di Trieste
)\nThe Calderon problem\, known also as the inverse conductivity problem\,
regards the determination of the conductivity inside a domain by the know
ledge of the boundary data. For the isotropic case\, the stability issue i
s almost solved. However\, for the anisotropic case things get more compli
cated\, since Tartar observation that any diffeomorphism of the domain whi
ch keeps the boundary points fixed has the property of leaving the Dirichl
et-to-Neumann map unchanged\, whereas the conductivity tensor is modified.
In this talk we will introduce a special class of anisotropic conductivit
ies for which we can prove a stability estimate. The novelty of this resul
t lies in the fact that the stability is proved using an ad-hoc functional
. As a corollary\, we derive a Lipschitz stability estimate in terms of th
e classical Dirichet-to-Neumann map. This talk is based on a joint work wi
th Eva Sincich and Romina Gaburro.\n\nhttps://conference2.aau.at/event/62/
contributions/839/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/839/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Beating the Saturation Phenomenon of Stochastic Gradient Descent
DTSTART;VALUE=DATE-TIME:20210924T110000Z
DTEND;VALUE=DATE-TIME:20210924T112000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-837@conference2.aau.at
DESCRIPTION:Speakers: Zehui Zhou (Department of Mathematics\, The Chinese
University of Hong Kong)\nStochastic gradient descent (SGD) is a promising
method for solving large-scale inverse problems\, due to its excellent sc
alability with respect to data size. The current mathematical theory in th
e lens of regularization theory predicts that SGD with a polynomially deca
ying stepsize schedule may suffer from an undesirable saturation phenomeno
n\, i.e.\, the convergence rate does not further improve with the solution
regularity index when it is beyond a certain range. In this talk\, I will
present our recent results on beating this saturation phenomenon:\n(i) (B
y using small initial stepsize.) We derive a refined convergence rate anal
ysis of SGD\, which shows that saturation actually does not occur if the i
nitial stepsize of the schedule is sufficiently small.\n(ii) (By using Sto
chastic variance reduced gradient (SVRG)\, a popular variance reduction te
chnique for SGD.) We prove that\, for a suitable constant step size schedu
le\, SVRG can achieve an optimal convergence rate in terms of the noise le
vel (under suitable regularity condition)\, which means the saturation doe
s not occur.\n\nhttps://conference2.aau.at/event/62/contributions/837/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/837/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Convergence rates for oversmoothing Banach space regularization
DTSTART;VALUE=DATE-TIME:20210923T093000Z
DTEND;VALUE=DATE-TIME:20210923T095000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-834@conference2.aau.at
DESCRIPTION:Speakers: Philip Miller (Institute for Numerical and Applied M
athematics\, University of Göttingen\, Germany)\nWe show convergence rate
s results for Banach space regularization in the case of oversmoothing\, i
.e. if the penalty term fails to be finite at the unknown solution. We pre
sent a flexible approach based on K-interpolation theory which provides mo
re general and complete results than classical variational regularization
theory based on various types of source conditions for true solutions cont
ained in the penalty's domain. In particular\, we prove order optimal conv
ergence rates for bounded variation regularization. Moreover\, we show a r
esult for sparsity promoting wavelet regularization and demonstrate in num
erical simulations for a parameter identification problem in a differentia
l equation that our theoretical results correctly predict rates of converg
ence for piecewise smooth unknown coefficients.\n\nhttps://conference2.aau
.at/event/62/contributions/834/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/834/
END:VEVENT
BEGIN:VEVENT
SUMMARY:On a regularization of unsupervised domain adaptation in RKHS
DTSTART;VALUE=DATE-TIME:20210922T143000Z
DTEND;VALUE=DATE-TIME:20210922T145000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-832@conference2.aau.at
DESCRIPTION:Speakers: Duc Hoan Nguyen (Johann Radon Institute)\nWe analyze
the use of the so-called general regularization scheme in the scenario of
unsupervised domain adaptation under the covariate shift assumption. Lear
ning algorithms arising from the above scheme are generalizations of impor
tance weighted regularized least squares method\, which up to now is among
the most used approaches in the covariate shift setting. We explore a lin
k between the considered domain adaptation scenario and estimation of Rado
n-Nikodym derivatives in reproducing kernel Hilbert spaces\, where the gen
eral regularization scheme can also be employed and is a generalization of
the kernelized unconstrained least-squares importance fitting. We estimat
e the convergence rates of the corresponding regularized learning algorith
ms and discuss how to resolve the issue with the tuning of their regulariz
ation parameters. The theoretical results are illustrated by numerical exa
mples\, one of which is based on real data collected for automatic stenosi
s detection in cervical arteries.\n\nhttps://conference2.aau.at/event/62/c
ontributions/832/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/832/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vector Spline Approximation on the $3d$-Ball for Ill-Posed Functio
nal Inverse Problems in Medical Imaging
DTSTART;VALUE=DATE-TIME:20210924T120000Z
DTEND;VALUE=DATE-TIME:20210924T122000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-831@conference2.aau.at
DESCRIPTION:Speakers: Sarah Leweke (University of Siegen)\nHuman brain act
ivity is based on electrochemical processes\, which can only be measured i
nvasively. For this reason\, induced quantities such as magnetic flux dens
ity (via MEG) or electric potential differences (via EEG) are measured non
-invasively in medicine and research. The reconstruction of the neuronal c
urrent from the measurements is a severely ill-posed problem though the vi
sualization of the cerebral activity is one of the main tools in brain sci
ence and diagnosis.\n\nUsing an isotropic multiple-shell model for the geo
metry of the human head \nand a quasi-static approach for modelling the el
ectro-magnetic processes\, a singular-value decomposition of the continuou
s forward operator between infinite-dimensional Hilbert spaces is derived.
Due to a full characterization of the operator null space\, it is reveale
d that only the harmonic and solenoidal component of the neuronal current
affects the measurements. Uniqueness of the problem can be achieved by a m
inimum-norm condition. The instability of the inverse problem caused by ex
ponentially decreasing singular values requires a stable and robust regula
rization method.\n\nThe few available measurements per time step ($\\appro
x 100$) are irregularly distributed with larger gaps in the facial area. O
n these grounds\, a vector spline method for regularized functional invers
e problems based on reproducing kernel Hilbert spaces is derived for deali
ng with these difficulties. Combined with several parameter choice methods
\, numerical results are shown for synthetic test cases with and without a
dditional Gaussian white noise. The relative normalized root mean square e
rror of the approximation as well as the relative residual do not exceed t
he noise level. Finally\, also results for real data are demonstrated. The
y can be computed with only a short delay time and are reasonable with res
pect to physiological expectations.\n\nhttps://conference2.aau.at/event/62
/contributions/831/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/831/
END:VEVENT
BEGIN:VEVENT
SUMMARY:A Bregman Learning Framework for Sparse Neural Networks
DTSTART;VALUE=DATE-TIME:20210922T131000Z
DTEND;VALUE=DATE-TIME:20210922T133000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-830@conference2.aau.at
DESCRIPTION:Speakers: Tim Roith (Friedrich-Alexander-Universität Erlangen
-Nürnberg)\nI will present a novel learning framework based on stochastic
Bregman iterations. It allows to train sparse neural networks with an inv
erse scale space approach\, starting from a very sparse network and gradua
lly adding significant parameters. Apart from a baseline algorithm called
LinBreg\, I will also speak about an accelerated version using momentum\,
and AdaBreg\, which is a Bregmanized generalization of the Adam algorithm.
I will present a statistically profound sparse parameter initialization s
trategy\, stochastic convergence analysis of the loss decay\, and addition
al convergence proofs in the convex regime. The Bregman learning framework
can also be applied to Neural Architecture Search and can\, for instance\
, unveil autoencoder architectures for denoising or deblurring tasks.\n\nh
ttps://conference2.aau.at/event/62/contributions/830/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/830/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Deterministic Dynamics of Ensemble Kalman Inversion
DTSTART;VALUE=DATE-TIME:20210924T134000Z
DTEND;VALUE=DATE-TIME:20210924T140000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-829@conference2.aau.at
DESCRIPTION:Speakers: Leon Bungert (University of Bonn)\nThe Ensemble Kalm
an inversion (EKI) is a powerful tool for the solution of Bayesian inverse
problems of type $y=Au^\\dagger+\\varepsilon$\, with $u^\\dagger$ being a
n unknown parameter and $y$ a given datum subject to measurement noise $\\
varepsilon$. It evolves an ensemble of particles\, sampled from a prior me
asure\, towards an approximate solution of the inverse problem. In this ta
lk I will provide a complete description of the dynamics of EKI\, utilizin
g a spectral decomposition of the particle covariance. In particular\, I w
ill demonstrate that\, despite the common folklore that EKI creates sample
s from the posterior measure\, this is only true for its mean field limit
and will suggest modifications of EKI that overcome this drawback.\n\nhttp
s://conference2.aau.at/event/62/contributions/829/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/829/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Constrained consensus-based optimization via penalization
DTSTART;VALUE=DATE-TIME:20210923T131000Z
DTEND;VALUE=DATE-TIME:20210923T133000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-828@conference2.aau.at
DESCRIPTION:Speakers: Giacomo Borghi (RWTH Aachen University)\nConstrained
optimization problems represent a challenge when the objective function i
s non-differentiable\, multimodal and the feasible region lacks regularity
. In our talk\, we will introduce a swarm-based optimization algorithm whi
ch is capable of handling generic non-convex constraints by means of a pen
alization technique. The method extends the class of consensus-based optim
ization (CBO) methods to the constrained settings\, a class where a swarm
of interactive particles explores the objective function landscape followi
ng a consensus dynamics.\nIn our algorithm\, we perform a time discretizat
ion of the system evolution and tune the parameters to effectively avoid n
on-admissible regions of the domain. While the particle dynamics may appea
r simple\, recovering convergence guarantees represents the real difficult
y when dealing with swarm-based methods. In the talk\, we will present the
essential mean-field tools that allowed us to theoretically analyze the a
lgorithm and obtain convergence results of its mean-field counterpart unde
r mild assumptions. To conclude\, we will discuss both the algorithm perfo
rmance on benchmark problems and numerical experiments of the mean-field d
ynamics.\n\nhttps://conference2.aau.at/event/62/contributions/828/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/828/
END:VEVENT
BEGIN:VEVENT
SUMMARY:A modified discrepancy principle to attain optimal rates for polyn
omially and exponentially ill-posed problems under white noise
DTSTART;VALUE=DATE-TIME:20210923T091000Z
DTEND;VALUE=DATE-TIME:20210923T093000Z
DTSTAMP;VALUE=DATE-TIME:20231202T152845Z
UID:indico-contribution-62-826@conference2.aau.at
DESCRIPTION:Speakers: Tim Jahn ()\nWe consider a linear ill-posed equation
in the Hilbert space setting under white noise. Known convergence results
for the discrepancy principle are either restricted to Hilbert-Schmidt op
erators (and they require a self-similarity condition for the unknown solu
tion additional to a classical source condition) or to polynomially ill-po
sed operators (excluding exponentially ill-posed problems). In this work w
e show optimal convergence for a modified discrepancy principle for both p
olynomially and exponentially ill-posed operators (without further restric
tions) solely under either Hölder-type or logarithmic source conditions.
In particular\, the method includes only a single simple hyper parameter\,
which does not need to be adapted to the type of ill-posedness.\n\nhttps:
//conference2.aau.at/event/62/contributions/826/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/826/
END:VEVENT
END:VCALENDAR