chemnitz symposium on inverse problems 2014 · chemnitz symposium on inverse problems 2014...
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Department of Mathematics
Analysis – Inverse Problems
Chemnitz Symposium on
Inverse Problems 2014
Conference Guide
September 18 – 19, 2014
Chemnitz, Germany
General informationTimetableAbstracts
List of participants
Imprint
Editor:Prof. Bernd HofmannProfessorship of Analysis – Inverse ProblemsDepartment of MathematicsTechnische Universitat Chemnitz
Editorial staff:Dr. Jens Flemming
Office:Reichenhainer Straße 39/4109126 Chemnitz, GermanyPostal address:09107 Chemnitz, GermanyPhone: +49 (0) 371 / 531-22000Fax: +49 (0) 371 / 531-22009Email: [email protected]
ISSN 2190-7900
General information
Goal
Our symposium will bring together experts from the German andinternational ‘Inverse Problems Community’ and young scientists.The focus will be on ill-posedness phenomena, regularization theoryand practice, and on the analytical, numerical, and stochastic treat-ment of applied inverse problems in natural sciences, engineering,and finance.
Location
Technische Universitat ChemnitzStraße der Nationen 62 (Bottcher-Bau)Conference hall ‘Altes Heizhaus’09111 Chemnitz, Germany
Selection of invited speakers
Andrew M. Stuart (Warwick, Great Britain)Elena Resmerita (Klagenfurt, Austria)Bastian von Harrach (Stuttgart, Germany)Markus Grasmair (Trondheim, Norway)
Scientific board
Bernd Hofmann (Chemnitz, Germany)Peter Mathe (Berlin, Germany)Sergei V. Pereverzyev (Linz, Austria)Masahiro Yamamoto (Tokyo, Japan)
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Organizing committee
Bernd HofmannStephan W. AnzengruberSteven BurgerJens FlemmingKerstin Seidel
Conference contacts
Bernd HofmannJens Flemming
Technische Universitat ChemnitzDepartment of MathematicsReichenhainer Straße 4109126 Chemnitz, Germany
Email: [email protected]: http://www.tu-chemnitz.de/mathematik/ip-symposium
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Timetable
Overview for Thursday, September 18
09.00–09.05 Introduction
09.05–10.40 Session 1A. Stuart, P. Mathe, E, Resmerita
10.40–11.00 Coffee break11.00–12.15 Session 2
B. Harrach, M. Grasmair, V. Michel
12.15–13.15 Lunch break
13.15–14.55 Session 3M. Yamamoto, C. Clason, F. Werner, R. Kowar
14.55–15.15 Coffee break15.15–16.55 Session 4
T. Reginska, M. Schlottbom, C. Gerhards, I. R. Bleyer
17.10–21.30 Excursion
Overview for Friday, September 19
09.00–09.10 Laudation
09.10–10.25 Session 1S. V. Pereverzyev, M. Hegland, S. W. Anzengruber
10.25–10.45 Coffee break10.45–12.25 Session 2
S. Hollborn, T. Helin, J.-F. Pietschmann,
S. Pereverzyev Jr.
12.25–13.20 Lunch break
13.20–14.35 Session 3H. Kekkonen, S. S. Nanthakumar, F. Margotti,
R. Winkler, S. Orzlowski
14.35–14.45 Coffee break
14.45–16.00 Session 4Z. Purisha, P. Tkachenko, D. Saxenhuber, D. Gerth,
S. Burger
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Program for Thursday, September 18
09.00–09.05 Introduction
09.05–09.50 Opening lectureAndrew Stuart (Warwick, Great Britain)Well-posed Bayesian geometric inverse problems
09.50–10.15 Peter Mathe (Berlin, Germany)Merging regularization theory into Bayesianinverse problems
10.15–10.40 Elena Resmerita (Klagenfurt, Austria)An entropic Landweber-type method for linearill-posed problems
10.40–11.00 Coffee break
11.00–11.25 Bastian von Harrach (Stuttgart, Germany)Inverse coefficient problems andshape reconstruction
11.25–11.50 Markus Grasmair (Trondheim, Norway)Bregman distance, source conditions, andvariational inequalities in Tikhonov regularisation
11.50–12.15 Volker Michel (Siegen ,Germany)The regularized (orthogonal) functionalmatching pursuit - a best basis algorithm forinverse problems in geomathematics andmedical imaging
12.15–13.15 Lunch break
13.15–13.40 Masahiro Yamamoto (Tokyo, Japan)Inverse problems of moving sources inwave equation
13.40–14.05 Christian Clason (Essen, Germany)Stochastic inverse problems with impulsive noise
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14.05–14.30 Frank Werner (Gottingen, Germany)Statistical inverse problems in fluorescencemicroscopy
14.30–14.55 Richard Kowar (Innsbruck, Austria)On time reversal in photoacoustic tomography fortissue similar to water
14.55–15.15 Coffee break
15.15–15.40 Teresa Reginska (Warsaw, Poland)Solution-functional and data-functionalregularization strategies for determining the laserbeam quality parameters
15.40–16.05 Matthias Schlottbom (Munster, Germany)Identification of nonlinear heat conduction laws inheat transfer problems
16.05–16.30 Christian Gerhards (Siegen, Germany)Combining downward continuation and localapproximation on different spheres by optimizedkernels
16.30–16.55 Ismael Rodrigo Bleyer (Helsinki, Finland)Digital speech: an application of the dbl-RTLSmethod for solving GIF problem
17.10–21.30 Excursion to Wasserschloss Klaffenbachand conference dinner
Time periods include 5 minutes for discussion.
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Program for Friday, September 19
The first morning session is dedicated to the 60th anniversaryof PD Dr. Peter Mathe.
09.00–09.10 Bernd Hofmann (Chemnitz, Germany)Laudation
09.10–09.35 Sergei V. Pereverzyev (Linz, Austria)Aggregation of regularized approximations
09.35–10.00 Markus Hegland (Canberra, Australia)Weighted function spaces for highdimensionalapproximation and multiparameter regularisation
10.00–10.25 Stephan W. Anzengruber(Chemnitz, Germany)A NURBS-based gradient method for sparseangle tomography
10.25–10.45 Coffee break
10.45–11.10 Stefanie Hollborn (Mainz, Germany)Backscatter data in electric impedancetomography
11.10–11.35 Tapio Helin (Helsinki, Finland)Inverse scattering in half-space with randomboundary conditions
11.35–12.00 Jan-Frederik Pietschmann(Darmstadt, Germany)Identification of chemotaxis models withvolume filling
12.00–12.25 Sergiy Pereverzyev Jr. (Innsbruck, Austria)Multi-penalty regularization for detectingrelevant variables
12.25–13.20 Lunch break
13.20–13.35 Hanne Kekkonen (Helsinki, Finland)White noise paradox in Bayesian inverse problems
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13.35–13.50 Srivilliputtur Subbiah Nanthakumar(Weimar, Germany)Inverse problem of multiple inclusions detectionin piezoelectric structures using XFEM andLevel sets
13.50–14.05 Fabio Margotti (Karlsruhe, Germany)Inexact newton regularization methods inBanach spaces
14.05–14.20 Robert Winkler (Karlsruhe, Germany)Adaptive sensitivity-based regularization forNewton-type inversion in electrical impedancetomography
14.20–14.35 Sarah Orzlowski (Siegen, Germany)Regularized joint inversion of EEG and MEG databy a best basis algorithm
14.35–14.45 Coffee break
14.45–15.00 Zenith Purisha (Helsinki, Finland)Recovering shape of 2D pipe with corrosion andattenuation coefficient with limited data
15.00–15.15 Pavlo Tkachenko (Linz, Austria)Multi-parameter regularization of ill-posedspherical pseudo-differential equations in C-space
15.15–15.30 Daniela Saxenhuber (Linz, Austria)Atmospheric tomography for ELT adaptive optics
15.30–15.45 Daniel Gerth (Linz, Austria)The method of the approximate inverse foratmospheric tomography
15.45–16.00 Steven Burger (Chemnitz, Germany)An autoconvolution problem connected withSD-SPIDER
Time periods include 5 minutes for discussion.
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Abstracts
Speaker Page
Stephan W. Anzengruber 10Ismael Rodrigo Bleyer 11Steven Burger 12Christian Clason 13Christian Gerhards 14Daniel Gerth 15Markus Grasmairk 16Bastian Harrach 17Markus Hegland 18Tapio Helin 19Stefanie Hollborn 20Hanne Kekkonen 21Richard Kowar 22Fabio Margotti 23Peter Mathe 24Volker Michel 25Srivilliputtur Subbiah Nanthakumar 27Sarah Orzlowski 28Sergei V. Pereverzyev 29Sergei Pereverzyev Jr. 30Jan-Frederik Pietschmann 31Zenith Purisha 32Teresa Reginska 33Elena Resmerita 35Daniela Saxenhuber 36Matthias Schlottbom 37Andrew Stuart 38Pavlo Tkachenko 39Frank Werner 40Robert Winkler 41Masahiro Yamamoto 42
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A NURBS-based gradient method forsparse angle tomography
Stephan W. Anzengruber
In sparse angle tomography one attempts to reconstruct an im-age from highly undersampled Radon data. The resulting severelyill-posed problem becomes more stable, if the solution is known tobe piecewise constant. This has been used in Mumford-Shah-typeregularization methods [2, 3], for example, but recently also in [1]where the interfaces in the image are parametrized by NURBS (Non-Uniform Rational B-splines). The latter approach even yields re-constructions in vector-graphics format, but the dependence of theimage on the design parameters of the interface curves is non-linearand not differentiable.
In this talk we will see that Radon data, however, remain differ-entiable almost everywhere and then use gradient-type methods tofind NURBS design parameters for the interface curves from sparseangle tomographic data.
References
[1] Z. Purisha and S. Siltanen. Tomographic Inversion using NURBSand MCMC. Proceedings of the 3rd Annual Int. Conference on Com-putational Mathematics, Geometry and Statistics, 2011.
[2] R. Ramlau and W. Ring. A MumfordShah level-set approach for theinversion and segmentation of X-ray tomography data. Journal ofComputational Physics, 221(2), 2007.
[3] M. Storath, A. Weinmann, J. Frikel and M. Unser. A splitting ap-proach to Potts regularization of inverse imaging problems.arXiv :1405.5850v1, 2014.
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Digital speech: an application of the
dbl-RTLS method for solving GIF problem
Ismael Rodrigo Bleyer
“Digital Speech Processing” refers to the study of a speech sig-nal. Namely, these signals are processed in a digital representation,as for example, synthesis, analysis, enhancement, compression andrecognition may refers to this process.
In this talk we are interested on solving the core problem knownas “Glottal Inverse Filtering” (GIF). Commonly this problem canbe modelled by convolving a pressure function (input signal) withan impulse response function (filter). Our approach is done in adeterministic setup based on the dbl-RTLS (double regularised totalleast squares) approach introduced recently. Additionally we reviewbriefly the methodology and we give the first numerical realisations,based on an alternating minimisation procedure.
Keywords
ill-posed problems, noisy operator, noisy right hand side, regularised to-
tal least squares, double regularisation, alternating minimisation, glottal
inverse filtering, digital speech
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An autoconvolution problemconnected with SD-SPIDER
Steven Burger
In non-linear optics SD-SPIDER is a proposed method to measureultra-short laser pulses. For the reconstruction of the electric field ofthe pulses from the measured data one needs to solve an autoconvo-lution problem. For this problem we present theoretical results andnumerical examples with artificial data.
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Stochastic inverse problemswith impulsive noise
Christian Clason
Impulsive noise models such as salt & pepper noise are frequentlyused in mathematical image restoration, but are usually restricted tothe discrete setting. We introduce an infinite-dimensional stochas-tic impulsive noise model based on marked Poisson point processes,discuss its properties and show regularization properties of inverseproblems subject to such noise. We also treat the conforming dis-cretization of such noise models and compare this to the classicaldiscrete impulsive noise. This is joint work with Laurent Demaret(Helmholtz-Zentrum Mnchen).
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Combining Downward Continuation and
Local Approximation on Different Spheres
by Optimized Kernels
Christian Gerhards
Various problems in gravitation and geomagnetism (that are basedon harmonic potentials) require the combination of satellite dataand ground data at (or near) the Earth’s surface in order to obtainhigh resolution models. While satellite data is available globallyand requires ill-posed downward continuation to the Earth’s surface,ground data does not suffer from this ill-posedness but it is only avail-able locally/regionally. Therefore, satellite data is suitable for thereconstruction of large-scale features, whereas local/regional grounddata can be used for the detection of spatially finer structures.
In this talk, we present a multiscale-based method that pays trib-ute to the different properties of the two types of data. More pre-cisely, the approximation is based on a two-step algorithm: First, aconvolution with a scaling kernel ΦN deals with the downward con-tinuation from the satellite orbit ΩR to the spherical Earth’s surfaceΩr, r < R, while in a second step, the result is locally refined by aconvolution in a subregion Γr ⊂ Ωr with a wavelet kernel ΨN . Thekernels ΦN and ΨN are simultaneously optimized in such a way thatthe former behaves well for the downward continuation while thelatter shows a good localization in Γr. The approach is indicatedfor a scalar and vectorial setting. Furthermore, numerical tests andcomparisons to existing methods are presented.
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The method of the approximate inversefor atmospheric tomography
Daniel Gerth, Bernadette Hahn, Ronny Ramlau
The method of the approximate inverse is a regularization schemeto determine a stable solution of an ill-posed operator equation. In-stead of the original one, an auxiliary problem has to be solved tocompute the so called reconstruction kernels, which are then used toreconstruct a mollified version of the desired function at certain gridpoints with the help of the measured data. Using invariances, all in-dividual reconstruction kernels can easily be calculated from the so-lution of a single auxiliary problem that may be precomputed, hencesaving computational effort. We apply this idea to the problem of at-mospheric tomography, used for large ground-based telescopes whichrely on adaptive optics systems to remove the effects of atmosphericdistortions in order to achieve a good image quality. Adaptive opticssystems physically correct atmospheric turbulences via deformablemirrors in real-time. The optimal shape of the deformable mirrors isdetermined from wavefront measurements of natural guide stars aswell as laser guide stars. Numerical results will be demonstrated forthe European Extremely Large Telescope (E-ELT).
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Bregman distance, source conditions,
and variational inequalities
in Tikhonov regularisation
Markus Grasmair
In classical Tikhonov regularisation for the stable solution of alinear operator equation Fu = vδ on a Hilbert space, it is possibleto derive a range of convergence rates if the true solution u† satisfiesa source condition of the form u† = (F ∗F )νω with 0 < ν ≤ 1.The main step in the derivation of these rates is the formulationof an interpolation inequality involving the similarity term and theregularisation term; for the higher order rates obtained with 1/2 <ν ≤ 1, this may be done using a dual formulation of the Tikhonovfunctional. In the more general setting of regularisation on Hilbertspaces with a non-quadratic regularisation term, this approach canbe used for obtaining convergence rates with respect to the Bregmandistance, which measures the deviance of the regularisation termfrom an affine approximation. The rates obtained in that manner,however, are for ν < 1/2 typically of lower order than the rates withcomparable conditions for quadratic regularisation. Higher orderrates can be obtained though, if the norm can be estimated fromabove by the Bregman distance. Another possibility for derivingconvergence rates is the usage of variational inequalities involvingthe similarity term and the Bregman distance. In the quadraticsetting, these conditions almost provide a characterisation of therange of (F ∗F )ν . In this talk we will discuss the differences andlimitations of these two approaches. In particular, we will focuson the setting of total variation regularisation and provide somegeometrical interpretation of the results in this situation.
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Inverse coefficient problemsand shape reconstruction
Bastian Harrach
Novel imaging methods commonly lead to the inverse problem ofdetermining one or several coefficient function(s) in a partial differ-ential equation from (partial) knowledge of its solutions. Arguably,the most famous such problem is the Calderon problem, where thediffusion coefficient in an elliptic PDE is to be reconstructed fromthe Neumann- and Dirichlet-boundary data of its solutions.
The mathematical challenges behind inverse coefficient problemsreach from theoretical uniqueness questions to the construction ofconvergent numerical algorithms and stability issues. In this talk wewill present recent work on these subjects that is based on combiningmonotony relations between the coefficients and the Neumann/Di-richlet-data with so-called localized potentials. We will show howto derive simple uniqueness results, avoid non-linearity effects in thecontext of shape-reconstruction, and deal with imprecisely knownbackground media and measurement noise.
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Weighted function spaces
for highdimensional approximation
and multiparameter regularisation
Markus Hegland
In a recent paper [J.Complexity,29(1),p.76-91,2013] with Greg Wa-silkowski, spaces of multivariate functions were studied in which ap-proximation does not suffer under the curse of dimensionality. Orig-inally, these function spaces were used in data mining to show con-vergence of ANOVA decompositions. The arguments used relate tothe concentration of measure.
It was recently suggested to the author that the these functionspaces – which are reproducing kernel Hilbert spaces – may also beused for inverse problems and in particular provide a framework formultiparameter regularisation.
In this talk I will review the original results on approximationand then discuss opportunities for multivariate and high-dimensionalregularisation.
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Inverse scattering in half-spacewith random boundary conditions
Tapio Helin
An inverse acoustic scattering problem in half-space is studied. Weassume an impedance boundary value condition, where the Robin co-efficient is a Gaussian random function λ. The covariance operator ofλ is assumed to be a classical pseudodifferential operator. Our data isthe amplitude of the backscattered field averaged over the frequencyband. We assume that such data is generated by a single realizationof the random Robin coefficient. It is shown that in such a case theprincipal symbol of the covariance operator of λ is uniquely deter-mined. We have an anisotropic stochastic model for the principalsymbol, which leads to the analysis of a novel anisotropic sphericalRadon transform and its invertibility. This is joint work with MattiLassas and Lassi Paivarinta.
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Backscatter Datain Electric Impedance Tomography
Stefanie Hollborn
This talk presents the concept of backscatter data in electric im-pedance tomography. These sparse data are the analogue to so-calledbackscattering data in inverse scattering. They arise in practice if thesame single pair of electrodes is used to drive currents and measurevoltage differences subsequently at various locations on the boundaryof the object to be imaged. A single insulating cavity within an oth-erwise homogeneous object is uniquely determined by its backscatterdata. However, this is in general not true for a perfectly conductinginclusion. The talk presents the uniqueness proof for an insulatingcavity, and it outlines a reconstruction algorithm based thereon. Theresults presented here arise from joint research with Martin Hankeand Nuutti Hyvnen.
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White noise paradoxin Bayesian inverse problems
Hanne Kekkonen, Matti Lassas, Samuli Siltanen
Our aim is to provide new analytic insight to the relationshipbetween the continuous and practical inversion models corrupted bywhite Gaussian noise.
Let us consider an indirect noisy measurement m of a physicalquantity u
m(x) = Au(x) + δε(x),
where δ > 0 is the noise magnitude. If ε was an L2-function,Tikhonov regularization gives an estimate
Tα(m) = argminu∈Hr
‖Au−m‖2L2 + α‖u‖2Hr
for u where α = α(δ) is the regularization parameter. Here pe-nalization of the Sobolev norm ‖u‖Hr covers the cases of standardTikhonov regularization (r = 0) and first derivative penalty (r = 1).
Realizations of white Gaussian noise are almost never in L2, butdo belong to Hs with probability one if s < 0 is small enough. Thatis why we present a modification of Tikhonov regularization theorycovering the case of white Gaussian measurement noise. We will alsoconsider the question in which space does the estimate convergenceto a correct solution when the noise variance goes to zero and whatis the speed of the convergence.
The results are important for removing the apparent paradox aris-ing from the use of Tikhonov regularization in discrete case andthe infinite L2-norm of the natural limit of white Gaussian noise inRn as n → ∞. Rigorous connection between discrete and infinite-dimensional limit model is often useful. Examples include erroranalysis for numerical inversion and computational speed-ups basedon consistent switching between different discretizations related tomultigrid methods.
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On time reversal
in photoacoustic tomography
for tissue similar to water
Richard Kowar
This paper is concerned with time reversal in photoacoustic tomog-raphy (PAT) of dissipative media that are similar to water. Underan appropriate condition, it is shown that the time reversal methodin [2, 1] based on the non-causal thermo-viscous wave equation canbe used if the non-causal data is replaced by a time shifted set ofcausal data.
We investigate a similar imaging functional for time reversal andan operator equation with the time reversal image as right handside. If required, an enhanced image can be obtained by solvingthis operator equation. Although time reversal (for noise-free data)does not lead to the exact initial pressure function, the theoreticaland numerical results of this paper show that regularized time rever-sal in dissipative media similar to water is a valuable method. Wenote that the presented time reversal method can be considered asan alternative to the causal approach in [?] and a similar operatorequation may hold for their approach.
Keywords
time reversal, photoacoustic tomography, inverse and ill-posed problems
AMS subject classifications
35R30, 35L05, 47A52, 45A05, 65J20
References
[1] H. Ammari and E. Bretin and J. Garnier and A. Wahab,Time reversal in attenuating acoustic media, Mathematical and sta-tistical methods for imaging, 151-163, Contemp. Math., 548, Amer.Math. Soc., Providence, RI, 2011.
[2] A. Wahab, Modeling and imaging of attenuation in biological me-dia, PhD Dissertation, Centre de Mathemathiques Appliquee, EcolePolytechnique Palaiseau, Paris, 2011.
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Inexact Newton regularization methodsin Banach spaces
Fabio Margotti, Andreas Rieder
Inexact Newton methods have proven to be a powerful class ofiterative methods to solve nonlinear inverse and ill-posed problemsin Hilbert spaces. To realize such a method one must linearize theequation around the current iteration and then apply a regularizationtechnique to solve the resulting linear system. We propose here theadaptation of some classical regularization methods of gradient- andTikhonov-type to solve the linearized system in the Banach spacessetting.
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Merging regularization theoryinto Bayesian inverse problems
Peter Mathe
We discuss Bayesian linear inverse problems
yδ = Kx+ δη
in Hilbert space with Gaussian noise η and the noise level δ. Undera Gaussian prior for the unknown solution x, say with mean m0 andcovariance Cδα, depending on the noise level δ, and an additionaltuning parameter α, the posterior distribution, given the data yδ
will also be Gaussian. The focus is on a fast concentration of theposterior probability around the unknown true solution as expressedin the concept of posterior contraction rates (as δ → 0).
As previous analysis on posterior contraction rates revealed, thiscan be achieved by a proper choice of α, depending on the solu-tion smoothness (a priori choice) or adaptive (oracle-type, a poste-riori choice). We refer to B. T. Knapik, A. W. van der Vaart, andJ. H. van Zanten, Bayesian Inverse Problems with Gaussian Prior,Ann. Statist. (39), 2011, Theorem 4.1, for details and a discussion.In particular these authors discuss the saturation of the contrac-tion rate. Since then this phenomenon was also observed in otherstudies. Heuristically, this saturation can be explained by relatingthe Bayesian approach to Tikhonov regularization. The question iswhether and how to overcome this resistance of the posterior con-traction rates to higher smoothness.
It will be shown that this actually can be achieved by centeringthe prior distribution appropriately, i.e., with a choice for the priormean m0 := m0(α, y
δ). The idea and the analysis use results fromregularization theory. This is joint work with S. Agapiou, Universityof Warwick.
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The Regularized (Orthogonal) Functional
Matching Pursuit — a Best Basis Algorithm
for Inverse Problems in Geomathematics
and Medical Imaging
Volker Michel
We present a novel algorithm, the RFMP, and its enhancement,the ROFMP, for the regularization of ill-posed inverse problems ofthe following kind: Given y ∈ Rl and linear and continuous function-als F1, . . . ,F l : H(D) → R, find f ∈ H(D) such that F jf = yj forall j = 1, . . . , l. Here, H(D) is a certain Hilbert space of functionson D, e.g. an L2-space, a Sobolev space, or a reproducing kernelHilbert space. Examples of such problems occur in geomathemat-ics and medical imaging, e.g. for the evaluation of satellite data ofthe Earth, the exploration of the Earth’s interior, the observation ofclimate-based water mass transports, and the localization of partic-ular brain activities.
The ROFMP and the RFMP are improvements of methods byMallat & Zhang, by Vincent & Bengio, and by Pati et al. which weredeveloped for data interpolation without a regularization. The basicidea is as follows: We initially define a so-called dictionaryD ⊂ H(D)which consists of trial functions of different kinds which might beuseful for the problem to be solved. Then, we iteratively construct asequence of approximations in the sense that Fn+1 := Fn+αn+1dn+1,F0 := 0, and αn+1 ∈ R as well as dn+1 ∈ D are chosen such thatthe regularized data misfit is minimized. The ROFMP improves thisselection process and yields a better sparsity of the solution.
We mention theoretical properties (like a convergence theorem)and demonstrate some numerical results for practical problems.
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References
[1] D. Fischer, V. Michel: Sparse regularization of inverse gravimetry –case study: spatial and temporal mass variations in South America,Inverse Problems, 28 (2012), 065012 (34pp).
[2] V. Michel: RFMP — an iterative best basis algorithm for inverseproblems in the geosciences, in: Handbook of Geomathematics (W.Freeden, M.Z. Nashed, and T. Sonar, eds.), 2nd edition, accepted,2013.
[3] V. Michel, R. Telschow: A Non-linear approximation method on thesphere, preprint, Siegen Preprints on Geomathematics, 10, 2014.
[4] R. Telschow: An Orthogonal Matching Pursuit for the Regularizationof Spherical Inverse Problems, PhD Thesis, submitted, University ofSiegen, 2014.
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Inverse problem of multiple inclusions
detection in Piezoelectric Structures
using XFEM and Level sets
Srivilliputtur Subbiah Nanthakumar,
Tom Lahmer, Timon Rabczuk
An algorithm to solve the inverse problem of detecting inclusion in-terfaces in a piezoelectric structure is proposed. The inverse problemis solved iteratively in which the Extended Finite Element Method(XFEM) is used for the analysis of the structure in each iteration.The combination of shape derivatives and level set method used instructural optimization is employed to solve the inverse problem.The formulation is presented for two and three dimensional struc-tures and inclusions made of different materials are detected by us-ing multiple level sets. The results obtained prove that the iterativeprocedure proposed can determine the location and the approximateshape of material subdomains in the piezoelectric structure by us-ing boundary displacement and electric potential measurements. Weshow numerically that the solutions of the inverse problem obtainedare robust with respect to errors in the data.
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Regularized joint inversion of EEG andMEG data by a best basis algorithm
Sarah Orzlowski
Brain activities and conversations in the cerebrum can be de-scribed via electric signals, i. e. neuronal currents, which induce anelectric potential on the scalp measured by electroencephalography(EEG) and a magnetic field outside the head measured via magne-toencephalography (MEG). The reconstruction of the neuronal sig-nals, the neuronal currents, from these sets of data is an ill-posedinverse problem, since the radial component of the current cannotbe reconstructed. Hence, we decompose the current into suitablefunctions and analyze the null spaces of the associated operators.
It is now our aim to reconstruct and localize the electric currentsin the brain by means of the decomposition and the novel regular-ized functional matching pursuit algorithm (RFMP), which has beenused so far for tomographic inverse problems in geophysics. Thisalgorithm needs an appropriate set of trial functions, the so-calleddictionary, which we construct for the particular problem, a regular-ization term, and a regularization parameter, which has to be chosenwisely in order to obtain a suitable solution. With this algorithm,separate inversions of the EEG and MEG data are conducted primar-ily, followed by a joint inversion of both data sets. Some numericalresults for a test case are demonstrated.
28
Aggregation of Regularized Approximations
Sergei V. Pereverzyev
Recent satellite missions monitoring the Earth’s gravity or mag-netic fields supply a large amount of data with a fairly good globalcoverage. In order to obtain high-resolution gravitational or geo-magnetic models it become necessary to combine these data. Forexample, different types of observations such as Satellite-to-SatelliteTracking (SST) or Gradiometer Measurements (SGG) enter into thedetermination of gravity field, leading to several observation equa-tions, which have the same solution, namely the gravitational poten-tial at the Earth’s surface. A naive way would be to estimate thispotential with the use of only one of the observation equations. Atthe same time, the goal is to benefit from the diversity of observa-tion models. In the talk we are going to discuss the possibility toachieve this goal by aggregating regularized solutions of all obser-vation equations. The idea of aggregation was originally proposedin the context of statistical regression analysis that corresponds tothe case of observation equations with identity operators. Here wedevelop this idea in the context of inverse problems.
The presentation is based on the results of joint research with ChenJieyang and Pavlo Tkachenko, both from RICAM. The research ispartially supported by the Austrian Science Foundation FWF.
29
Multi-penalty regularization fordetecting relevant variables
Sergiy Pereverzyev Jr.,
Katerina Hlavackova-Schindler, Valeriya Naumova
In this talk we present a new method for detecting relevant vari-ables from a priori given high-dimensional data under the assumptionthat input-output dependence is described by a nonlinear functiondepending on a few variables. The method is based on the inspec-tion of the behavior of discrepancies of a multi-penalty regularizationwith a component-wise penalization for small and large values of reg-ularization parameters. We provide the justification of the proposedmethod under a certain condition on sampling operators. The effec-tiveness of the method is demonstrated in the example with syntheticdata and in the reconstruction of gene regulatory networks. In thelatter example, the obtained results provide a clear evidence of thecompetitiveness of the proposed method.
30
Identification of Chemotaxis Modelswith Volume Filling
Jan-Frederik Pietschmann,
Herbert Egger, Matthias Schlottbom
Chemotaxis refers to the directed movement of cells in responseto a chemical signal called chemoattractant. A crucial point in themathematical modelling of chemotatic processes is the correct de-scription of the chemotactic sensitivity f and of the production rateg of the chemoattractant. In this talk, we investigate the identifi-cation of these non-linear parameters in a chemotaxis model withvolume filling given by
∂tρ = ∇ · (∇ρ− f(ρ)∇c) in Ω× (0, T ),
0 = ∆c− c+ g(ρ) in Ω for a.e. t ∈ (0, T ),
with Ω ⊂ R2 and complemented by appropriate initial and boundaryconditions. We give an affirmative answer to the question of uniqueidentifiability of either f or g given distributed measurements ofthe cell density ρ = ρ(x, t). Furthermore, we discuss the numericalrealization of Tikhonov regularization for its stable solution. Ourtheoretical findings are supported by numerical tests and concludeby presenting some open problems.
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Recovering shape of 2D pipe with corrosionand attenuation coefficient with limited data
Zenith Purisha, Samuli Siltanen
The inverse problem of reconstructing 2D pipe with corrosion us-ing Bayesian Inversion and Non-Uniform Rational B-Splines (NURBS)is considered. This method is implemented to assess the shape of cor-rosion and the attenuation coefficient of the inside material out oflimited measurement. Instead of points of curve, control points inNURBS become our parameters of interest. Accordingly, MarkovChain Monte Carlo is recommended to manage the emersion of non-linearity. The outside boundary of the pipeline and the attenuationcoefficient of steel are as a priori information. The potential draw-back of the proposed method is heavy computation.
Keywords
tomographic; corrosion; NURBS; Bayesian inversion; MCMC
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Solution-functional and data-functional
regularization strategies for determining
the laser beam quality parameters
Teresa Reginska
The laser beam is an electromagnetic field so complicated that itsanalytical description is practically impossible. Especially, semicon-ductor lasers have the large divergence of the beam, the multimodestructure of the field and often the astigmatism. An experimentaldetermination of the whole radiation field is also impossible. Practi-cally, we are able to measure the electromagnetic field only on somesubsets of physical space (e.g. on some surfaces).
Therefore, numerical solving the ill-posed problem of reconstruc-tion of the radiation field from experimental data given on a partof domain boundary is very important. The presentation will con-cern the simplified mathematical model for collimated laser beamthat is described by Cauchy problem for the Helmholtz equation onbounded domain, especially on cuboid with measurements accessi-ble on the one its side only. The case of infinite strip was alreadyconsidered in [1, 2] where numerical analysis of spectral type regu-larization for the problem was presented. The case of rectangulardomain was considered in [3] where using a certain series represen-tation of solution, we formulate a finite dimensional spectral typeregularization method. The obtained error bound depends on theregularization parameter, measurement error and, additionally, onan a priori bound for the solution traces on the part of boundary,where no measurements are performed.
Now, we are interested in determining certain laser beam qualityparameters that can be described by corresponding functionals. Onestrategy consists in direct applying the regularized radiation field inthe functionals (solution-functional strategy). Weaknesses of thisapproach will be explained. Another strategy (data-functional one)
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based on additional measurements and a priori assumptions willbe presented. In this approach quality parameters are estimateddirectly from measurement data.
References
[1] T. Reginska and K. Reginski, Approximate solution of a Cauchyproblem for the Helmholtz equation, Inverse Problems, 22 (2006),975–989.
[2] T. Reginska and U. Tautenhahn, Conditional stability estimates andregularization with applications to Cauchy problems for the Helm-holtz equation, Numerical Functional Analysis and Optimization, 30(2009), 1965–1097.
[3] T. Reginska, Regularization methods for a mathematical model oflaser beams, EJMCA, 1, Issue 2, (2014), 39–49. 2014.
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An entropic Landweber-type methodfor linear ill-posed problems
Elena Resmerita
Iterative methods which approximate non-negative solutions of lin-ear ill-posed problems are of interest in numerous applications. Thistalk presents such a procedure which has a relatively simple closedformula.
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Atmospheric tomographyfor ELT Adaptive Optics
Daniela Saxenhuber, Ronny Ramlau
The problem of atmospheric tomography arises in ground-basedtelescope imaging with Adaptive Optics (AO), where one aims tophysically correct atmospheric turbulences via deformable mirrors inreal-time, i.e. at around 500 Hertz. The calculation of the optimalmirror deformations from wavefront sensor measurements is an ill-posed inverse problem. Furthermore, there is a strong increase in thecomputational load for atmospheric reconstruction for the upcominggeneration of Extremely Large Telescopes (ELT). Thus, in order toachieve a reconstruction within the required time frame, numericallycheap iterative solvers that yield a superior reconstruction quality areneeded. In this talk, we propose to use a three step approach witha Gradient-based method for the atmospheric tomography problem.We will present numerical results of our algorithm in the context ofthe European Extremely Large Telescope (E-ELT) that is currentlyunder construction.
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Identification of nonlinear heat conductionlaws in heat transfer problems
Matthias Schlottbom,
Herbert Egger, Jan-Frederik Pietschmann
We consider the identification of nonlinear heat conduction laws instationary heat transfer problems. Only a single additional measure-ment of the temperature on a curve on the boundary is required todetermine the unknown parameter function on the range of observedtemperatures. We first present a new proof of Cannon’s uniquenessresult. For a stable solution we then utilize a reformulation of the in-verse problem as a linear ill-posed operator equation with perturbeddata and operators. We can then prove convergence and convergencerates of the regularized reconstructions under mild assumptions onthe exact parameter. These are, in fact, already needed for the anal-ysis of the forward problem and no additional source conditions arerequired. Numerical tests are presented to illustrate the theoreticalstatements. We will indicate some generalizations of our results toparabolic problems, and to problems involving lower order terms.
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Well-Posed BayesianGeometric Inverse Problems
Andrew Stuart,
Marco Iglesias, Kui Lin, Yulong Lu
There are numerous inverse problems where geometric character-istics, such as interfaces, are key unknown features of the overall in-version. Applications include the determination of layers and faultswithin subsurface formations, and the detection of unhealthy tis-sue in medical imaging. We discuss a theoretical and computationalBayesian framework relevant to the characterization of such inverseproblems.
We start with models in which the geometry is defined via a finitenumber of parameters, utilizing descriptions which are sufficientlygeneral to include layered media, faults and channels, and to allowfor spatial variability within the different components of the field [1].We then proceed to demonstrate how to use level set formulations ofthe Bayesian inverse problem, allowing for more complex geometricinterfaces which are described by an infinite set of parameters [2].
For a groundwater flow inverse problem, in which hydraulic headmeasurements are used to condition the prior information on thepermeability, we show that the Bayesian formulation gives rise toa well-posed posterior distribution, suitable for numerical interro-gation. We then describe numerical experiments which explore theposterior distribution, using state-of-the art MCMC methods.
References
[1] M.A. Iglesias, K. Lin, A.M. Stuart, “Well-Posed Bayesian GeometricInverse Problems Arising in Subsurface Flow”, Inverse Problems, ToAppear. http://arxiv.org/abs/1401.5571
[2] M.A. Iglesias, Y. Lu, A.M. Stuart, “A level-set approach to Bayesiangeometric inverse problems”, In Preparation, 2014.
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Multi-parameter regularization of ill-posed
spherical pseudo-differential equations
in C-space
Pavlo Tkachenko
In this talk, a two-step regularization method is considered tosolve an ill-posed spherical pseudo-dierential equation. For the firststep of regularization we use the approximation of continuous func-tions by means of spherical polynomials minimizing a functional witha penalty term in reproducing kernel Hilbert space. The secondstep is the regularized collocation method. The error bound forthe constructed multi-parameter regularization is established in C-norm. We discuss an a posteriori parameters choice rule and presentsome numerical experiments to confirm the compensatory propertyof our method.The present research was performed in joint coopera-tion with Dr. Hui Cao (China), Prof. Dr. Ian H. Sloan (Australia),and Prof. Dr. Sergei Pereverzyev (Austria)
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Statistical Inverse Problemsin fluorescence microscopy
Frank Werner
In this talk we discuss several inverse problems arrising in the ap-plication of fluorescence microscopy. In the most clasical setup, onetries to reconstruct a unknown fluorophore marker density u fromphoton counts of the density g1 = k ∗ u. Here k is a convolutionkernel (called point-spread function) and g denotes exact data. Un-fortunately, one is not only interested in approximations of u, butone wants to determine a so-called molecular map. This is knowingthe number n of (active) molecules or markers on each grid point x.If all markers at x share the same brightness p (x), then u = p · n.Obviously, p and n cannot be identified given only g1. This can beovercome by exploiting the effect of antibunching, which yields addi-tional photon counts of a density g2 which is depends quadraticallyon p and n. We will discuss problems in determining n from g1 andg2, including non-integer reconstuctions. Additionally we will focuson statistical tests for the hypothesis n (x) = 0 against a meaningfulalternative.
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Adaptive sensitivity-based regularization
for Newton-type inversion
in Electrical Impedance Tomography
Robert Winkler, Andreas Rieder
Electrical impedance tomography is the inverse problem of de-termining the electrical conductivity distribution of an object frommeasurements at electrodes on its boundary. Due to the ill-posednessof this problem, small measurement and modelling errors can have ahuge impact on the recovered image. Iterative inversion algorithmsthus incorporate regularization techniques, e.g. smoothness priors,to recover the underlying conductivity.
Using information of the Frechet derivative of the forward op-erator, which is available at no additional computational cost inNewton-type algorithms, we accurately model the impact of conduc-tivity changes to boundary measurements. This sensitivity informa-tion can be incorporated into an adaptive discretization scheme forthe inverse problem. Each conductivity coefficient in the resultingdiscretization is equally sensitive to measured boundary data, thatis, to conductivity information and noise contained therein. Thisallows for more accurate regularization and increases the robustnessof the inversion scheme.
Using an inexact Newton-type method with a discrepancy princi-ple, a robust choice of regularization parameters is reduced to theestimation of the noise level of the measured data. We demonstratethe performance of this scheme with simulated and measured data.
E Somersalo, M Cheney and D Isaacson Existence and Uniqueness forElectrode Models for Electric Current Computed Tomography, 1992, SIAMJ. Appl. Math., 52(4), 10231040.
A Lechleiter and A Rieder Newton regularizations for impedance tomog-raphy: convergence by local injectivity, 2008, Inverse Problems 24.
R Winkler and A Rieder Resolution-controlled conductivity discretization
in Electrical Impedance Tomography, 2014, preprint,
http://www.math.kit.edu/ianm3/~winkler/media/res eit.pdf.
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Inverse problems of moving sourcesin wave equation
Masahiro Yamamoto
We consider a wave equation with source term moving in time.We discuss the uniqueness and stability in determining the sourceintensity and velocity by over-determining boundary data. Also wediscuss the case of zero velocity and propose a numerical method.
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List of participants
Stephan W. AnzengruberTU ChemnitzChemnitz, [email protected]
Ismael Rodrigo BleyerUniversity of HelsinkiHelsinki, [email protected]
Steven BurgerTU ChemnitzChemnitz, [email protected]
Christian ClasonUniversitat Duisburg-EssenEssen, [email protected]
Jens FlemmingTU ChemnitzChemnitz, [email protected]
Christian GerhardsUniversity of SiegenSiegen, [email protected]
Daniel GerthJohannes Kepler Universitat LinzLinz, [email protected]
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Markus GrasmairNorwegian University of Science and TechnologyTrondheim, [email protected]
Sybille Handrock-MeyerTU ChemnitzChemnitz, [email protected]
Bastian HarrachUniversity of StuttgartStuttgart, [email protected]
Markus HeglandAustralian National UniversityCanberra, [email protected]
Tapio HelinUniversity of HelsinkiHelsinki, [email protected]
Roland HerzogTU ChemnitzChemnitz, [email protected]
Bernd HofmannTU ChemnitzChemnitz, [email protected]
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Stefanie HollbornJohannes Gutenberg-Universitat MainzMainz, [email protected]
Hanne KekkonenUniversity of HelsinkiHelsinki, [email protected]
Andrej KlassenUniversity of Duisburg-EssenEssen, [email protected]
Richard KowarUniversity of InnsbruckInnsbruck, [email protected]
Tom LahmerBauhaus University WeimarWeimar, [email protected]
Fabio MargottiKarlsruhe Institute of TechnologyKarlsruhe, [email protected]
Peter MatheWeierstrass Institute for Applied Analysis and StochasticsBerlin, [email protected]
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Volker MichelUniversity of SiegenSiegen, [email protected]
Srivilliputtur Subbiah NanthakumarBauhaus Universitat WeimarWeimar, [email protected]
Sarah OrzlowskiUniversity of SiegenSiegen, [email protected]
Sergei V. PereverzyevJohann Radon Institute for Computational and AppliedMathematics (RICAM)Linz, [email protected]
Sergiy Pereverzyev Jr.University of InnsbruckInnsbruck, [email protected]
Jan-Frederik PietschmannTU DarmstadtDarmstadt, [email protected]
Zenith PurishaUniversity of HelsinkiHelsinki, [email protected]
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Teresa ReginskaInstitute of Mathematics of the Polish Academy of SciencesWarsaw, [email protected]
Elena ResmeritaAlpen-Adria UniversityKlagenfurt, [email protected]
Arnd RoschUniversity of Duisburg–EssenEssen, [email protected]
Daniela SaxenhuberJohannes Kepler UniversityLinz, [email protected]
Matthias SchlottbomWWU MunsterMunster, [email protected]
Hans-Jorg StarkloffWestsachsische Hochschule ZwickauZwickau, [email protected]
Andrew StuartWarwick UniversityCoventry, Great [email protected]
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Pavlo TkachenkoJohann Radon Institute for Computational and AppliedMathematics (RICAM)Linz, [email protected]
Dana UhligTU ChemnitzChemnitz, [email protected]
Frank WernerMax Planck Institute for Biophysical ChemistryGottingen, [email protected]
Robert WinklerKarlsruhe Institute of TechnologyKarlsruhe, [email protected]
Masahiro YamamotoUniversity of TokyoTokyo, [email protected]
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ISSN 2190-7900