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CIP Events and Seminars

Layout Image

CIP Events and Seminars

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Friday 7th October at 1pm.

Roland Griesmaier (Universität Würzburg)

Location: Maths 500, Gordon Street 25

Title: Uncertainty principles for far field patterns and applications to inverse source problems

 

Classical uncertainty principles in signal processing limit the amount of simultaneous concentration of a signal with respect to time and frequency. In the inverse source problem, the far field radiated by a source f is its restricted (to the unit sphere) Fourier transform, and the operator that maps the restricted Fourier transform of f(x) to the restricted Fourier transform of its translate f(x+c) is called the far field translation operator.

In this talk we discuss an uncertainty principle, where the role of the Fourier transform is replaced by the far field translation operator. Combining this principle with a regularized Picard criterion, which characterizes the non-evanescent far fields radiated by a compactly supported limited power source provides extensions of several results about splitting a far field radiated by well-separated sources into the far fields radiated by each source component.

We also combine the regularized Picard criterion with a more conventional uncertainty principle for the map from  a far field to its Fourier coefficients. This leads to a data completion algorithm which tells us that we can deduce missing data if we know a priori that the source has small support.

All of these results can be combined so that we can simultaneously complete the data and split the far fields into the components radiated by well-separated sources. We discuss both  l^2 (least squares) and l^1 (basis pursuit) algorithms to accomplish this. Perhaps the most significant point is that all of these algorithms come with explicit bounds on their condition numbers which are sharp in their dependence on geometry and wavenumber.

This is joint work with John Sylvester (University of Washington).

 

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Friday 14th October at 1pm.

Kim Knudsen, Danmarks Tekniske Universtitet, Copenhagen, (Denmark)

Location: Maths 500, Gordon Street 25

Title: HD Electrical Impedance Tomography by Acoustic Modulation

 

Abstract:

The inverse problem in Electrical Impedance Tomography (EIT), mathematically known as the Calderon problem, is known to be extremely ill-posed, and hence any reconstruction from noisy data suffers from low resolution and low contrast. Recently new ideas have emerged that appear to resolve the issues by utilizing interior information that in principle can be computed using so-called hybrid data from other imaging devices. One such combined tomographic modality is known as Acousto-Electric Tomography and makes use of both ultrasonic waves and EIT simultaneously. The combination the physical waves gives rise to new and challenging mathematical questions of both theoretical and computational nature.

 

In this talk the basic difficulties in EIT will be discussed and the mathematical problem of Acousto-Electric Tomography (and similar kinds) will be introduced. The fundamental questions will be posed and (partially) answered. In particular we will through numerical examples discus some recent results regarding the stability and instability of the reconstruction problem.

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Friday 21st October at 1pm.

Olga Chervova, UCL

Location: Maths 500, Gordon Street 25

Title: Time reversal method with stabilising boundary conditions for Photoacoustic Tomography

We study an inverse initial source problem that models Photoacoustic Tomography measurements with array detectors, and introduce a method that can be viewed as a modification of the so called back and forth nudging method. We show that the method converges at an exponential rate under a natural visibility condition, with data given only on a part of the boundary of the domain of wave propagation. This is a joint work with Lauri Oksanen.

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Friday 28th October at 1pm.

Alden Waters, UCL

Location: Maths 500, Gordon Street 25

A deterministic optimal design problem for the heat equation

In everyday language, this talk addresses the question about the optimal shape and location of a thermometer of a given volume to reconstruct the temperature distribution in an entire room. For random initial conditions, this problem was considered by Privat, Trelat and Zuazua (ARMA, 2015), and we remove both the randomness and geometric assumptions in their article. Analytically, we obtain quantitative estimates for the wellposedness of an inverse problem, in which one determines the solution in the whole domain from its restriction to a subset of given volume. Using wave packet decompositions from microlocal analysis, we conclude that there exists a unique optimal such subset, that it is semi-analytic and can be approximated by solving a sequence of finite-dimensional optimization problems. This talk will also address future applications to inverse problems.

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Friday 4th November at 1pm.

Tuomo Valkonen, University of Liverpool (UK)

Location: Maths 500, Gordon Street 25 

Title: Block-proximal methods for image processing

Abstract: In recent years, stochastic coordinate descent has become popular in big data optimisation. The success of these methods is based on two features: a) an efficient splitting of very large problems onto huge computing clusters, and b) being able to exploit the local blockwise structure of the problems under consideration to obtain faster convergence compared to standard methods. Early methods in the literature have had restrictive conditions that have stopped them from being applied to interesting image processing problems, but recently there has been significant progress in applicable methods. Here we present a class of such methods. As our example image processing problems are small enough to not benefit from being split on a computing cluster, in our numerical studies we concentrate primarily on deterministic variants of our methods with the property b): efficient use of local structure of the problem. We demonstrate improved performance compared to standard methods.  

 

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Monday 7th November at 1pm.

Matias Courdurier, Pontifcia Universidad Catolica de Chile

Location: Roberts G06, Malet Place Engineering Building, UCL

Title

Reconstructing the Source and Attenuation in
SPECT Using Ballistic and Scattering Measurements.

M. Courdurier, joint work with J.C. Quintana, CIB-UC, F. Monard,
A. Osses & F. Remero.


Abstract

In the imaging technique of Single-Photon Emission Computed Tomography (SPECT)
a source of gamma-ray radiation is distributed inside the patient. The emitted
gamma-ray photons are attenuated and scattered as they traverse the body of the
patient and the radiation leaving the patient is measured.

The transport of photons inside the patient can be modeled using the radiative
transfer equation (RTE) and the standard image reconstruction problem in SPECT
can be described as the reconstruction of the source map from the knowledge of
the RTE solution outside the patient (see e.g. [1]).

The most common approach to this reconstruction problem only considers
the ballistic photons, and when the total attenuation map in known, the
reconstruction of the source map is obtained by inversion of the attenuated
Radon transform (see e.g. [3]).

In this talk I will present an approach in which we try to reconstruct both
the source and the attenuation maps. To tackle this problem we assume an
enlarged set of measurements, namely the ballistic and first order scattering
photons, and based on a standard decomposition in orders of scattering we
propose a model for these measurements and we study the inversion of the
linearized equation and the original non-linear problem (see [2]).

References

1.        G. Bal. Inverse transport theory and applications, Inv. Prob. 25, 053001(2009).
2.        M. Courdurier, F. Monard, A. Osses, F. Romero. Simultaneous source and
attenuation reconstruction in SPECT using ballistic and single scattering data.
Inverse Problems, 31, (2015).
3.        R. Novikov. An inversion formula for the attenuated x-ray transformation.
Arkiv för Matematik,  40, 145–67 (2002).
4.        P. Stefanov, The Identification Problem for the attenuated X-ray transform,
Amer. J. Math., 136(5): 1215-1247 (2014).

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Friday 18th November at 1pm.

Bernadette Hahn, University of Wurzburg (Germany)

Location: Maths 500, Gordon Street 25 

Title: Inverse Problems in Dynamic Imaging

Abstract : The acquisition of tomographic data takes a considerable amount of time. For example, in computerized tomography, the X-ray source has to be rotated around the investigated object. Temporal changes of the object during this time period lead to inconsistent data. Hence, the application of standard methods for solving inverse problems causes motion artifacts in the images which can severely impede a reliable diagnostics.

To reduce the artifacts, the reconstruction method has to take into account the dynamic behavior of the specimen. Thus, the development of motion estimation and compensation algorithms is an important challenge in dynamic imaging. In addition, it is essential to understand how the object’s deformation affect the overall information content within the data. For example, certain singularities of the specimen might not the gathered by the measured data, although they would be visible if the object was stationary during the scanning. The presented talk addresses these challenges with a special focus on computerized tomography.

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Friday 25th November, Double seminar 1-3pm

Location: Maths 505, Gordon Street 25

Mikko Salo, University of Jyväskylä, Finland

Title: The Calderón problem for the fractional Laplacian

Abstract: We show global uniqueness in an inverse problem for the fractional Schrödinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial data problem where the measurements are taken in an arbitrary open subset of the exterior. The results apply in any dimension $\geq 2$ and are based on a strong approximation property of the fractional equation that extends earlier work. This special feature of the nonlocal equation renders the analysis of related inverse problems radically different from the traditional Calderón problem.
This is a joint work with T. Ghosh (HKUST) and G. Uhlmann (Washington)


Samuli Siltanen, University of Helsinki, Finland

Title: Electrical impedance tomography imaging via the Radon transform

Abstract: In Electrical Impedance Tomography (EIT) one attempts to recover the electric conductivity inside a domain from electric boundary measurements. This is a nonlinear and ill-posed inverse problem. The so-called Complex Geometric Optics (CGO) solutions have proven to be a useful tool for both analysis and practical reconstruction tasks in EIT. A new property of CGO solutions is presented, showing that a one-dimensional Fourier transform in the spectral variable provides a connection to parallel-beam tomography of the conductivity. One of the consequences of this “nonlinear Fourier slice theorem” is a novel capability to recover inclusions within inclusions in EIT.

 

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Friday 3rd February at 1pm.

Location: Gordon Street (25) Maths 500

Thierry Daude, Université de Cergy-Pontoise

Title: On the hidden mechanism behind non-uniqueness for the anisotropic Calderon problem with data on disjoint sets

Abstract: In this talk, we shall show that there is generically non-uniqueness for the anisotropic Calderon problem at fixed frequency when the Dirichlet and Neumann data are measured on disjoint sets of the boundary of a given domain. More precisely, we first show that given a smooth compact connected Riemannian manifold with boundary (M,g) of dimension higher than 3, there exist in the conformal class of g an infinite number of Riemannian metrics such that their corresponding DN maps at a fixed frequency coincide when the Dirichlet and Neumann data are measured on disjoint sets. The conformal factors that lead to these non-uniqueness results for the anisotropic Calderon problem satisfy a nonlinear elliptic PDE of Yamabe type on the original manifold (M,g) and are associated to a natural but subtle gauge invariance of the anisotropic Calderon problem with data on disjoint sets. We then construct a large class of counterexamples to uniqueness in dimension higher than 3 to the anisotropic Calderon problem at fixed frequency with data on disjoint sets "modulo this gauge invariance". This class consists in cylindrical Riemannian manifolds with boundary having two ends (meaning that the boundary has two connected components), equipped with a suitably chosen warped product metric.

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Friday 10th February at 1pm.

Location: Gordon Street (25) Maths 500

Ghislain Haine, Institut Supérieur de l'Aéronautique et de l'Espace

Title: Back and forth observers: application to TAT Abstract: In this presentation, we will introduce the back and forth nudging algorithm in the case of linear systems. First, we detailed each step in the finite dimensional setting, involving only matrices. After, we point out the major drawbacks appearing in the infinite dimensional case, and give a brief explanation of "why it still work". Finally, we apply the algorithm to thermo-acoustic tomography, by showing that it fits a particular case for which the algorithm allows to deal with ill-posed inverse problem. 3D simulations will be provided to illustrate the efficiency of the reconstruction.

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Friday 3rd March at 1pm.

Location: Gordon Street (25) Maths 500

Paul Ledger, Swansea University

Title: From coins to landmines: how polarizability tensors can assist in metal detection

Abstract: Discriminating between between different buried hidden conducting objects has a range of important applications ranging from searching for buried treasure to identifying landmines. Still further applications exist in the early detection of concealed terrorist threats. Traditional approaches to the metal detection problem involve determining the conductivity and permeability distributions in the eddy current approximation of Maxwell’s equations and lead to an ill-posed inverse problem. On the other hand, practical engineering solutions in hand held metal detectors use simple thresholding and are not able to discriminate between small objects close to the surface and larger objects buried deeper underground.

In this talk, I will discuss an alternative approach in which prior information about the form of the conducting object has been introduced.  This allows us to describe the magnetic field perturbations due to the presence of a conducting (permeable) object in the form of an asymptotic expansion as the object size tends to zero. The asymptotic expansion separates the object's position from its shape and material description offering considerable advantages in case of isolated objects. In particular, the leading order term of the asymptotic expansion contains a rank 2 magnetic polarizability tensor whose coefficients can be computed by solving a vector valued transmission problem and which describes the object’s shape and its material parameters. The talk will include a discussion of how the transmission problem can be solved numerically using a hp-finite element discretisation and hence how the magnetic polarizability tensor can be computed.

The talk will also explore the interesting properties exhibited by the magnetic polarizability tensor, which characterises conducting objects. These properties offer possibilities to discriminate between objects with different topologies (eg between rings and coins) and also between objects with different conductivities and permeabilities.

 

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Previous seminars in 2015-2016

See also:

    2013-2014 series

    2014-2015 series

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Friday 18th September. 1.15p.m. Franz-Erich Wolter, Leibnitz University of Hannover (Germany)

location : Malet Place Engineering Building, Room 120

Title: "Shape and Image Cognition, Construction and Compression via Tools from Differential Geometry"

Professor Franz-Erich Wolter, Welfenlab, Institute of Man-Machine-Communication,
Faculty of Computer Science, Leibniz University of Hannover; Germany


Abstract : We shall describe how concepts from differential geometry have been providing powerful tools creating major advances in geometric modeling, geometry processing and image analysis dealing with the topics presented in the title of this address. This talk includes a retrospective compiling contributions of the author's works showing how concepts from local and global differential geometry have introduced new methods into geometric modeling and shape interrogation and classification finally ending with modern state of the art research on geometry processing and image processing. A major part of this seminar starting with works in the late nineties at the author's laboratory is dedicated to discussing how "efficient finger prints" useful for indexing and clustering digital data collections can be derived from spectra of Laplace operators being naturally associated with geometric objects such as surfaces and solids as well as (colored) images including medical 2d- and 3d-images. Recently the latter works obtained particular attention in the area of medical imaging. Geometric aspects of the Laplacian operator lead to generalizations of the Laplacian operating on line bundles allowing to compute eigen-functions whose iso-surfaces yield smooth non-orientable Seifert surfaces spanning knot complements in three-dimensional manifolds.

Next we focus on cut loci, the medial axis and its inverse in Euclidean and Riemannian worlds.  This work starts with basic medial axis results presented by the the author in the early nineties. Those results state: The Medial Axis Transform can be used to reconstruct, modify and design a given shape ("Shape Reconstruction Theorem"). Under some weak assumptions the medial axis contains the essence of the topological shape of the geometric object as it is a deformation retract of the given shape ("Topological Shape Theorem"). Therefore the medial axis contains the homotopy type of the given shape. We present recent results showing how geodesic Voronoi diagrams, geodesic medial axis and its inverse can be computed in 3d- or higher-dimensional Riemannian spaces. The "medial axis inverse" allows to construct a medial modeler providing efficient features for shape optimization with respect to shape dependent mechanical properties.

Time permitting the seminar will  also touch on some recent  results on computations of singularities of the  geodesic exponential  map solving the geodesic initial value problem by assigning to a given initial direction v of length IvI the corresponding geodesic path.

Short Biography: Dr. F.-E. Wolter has been a chaired full professor of computer science at Leibniz Universität Hannover (LUH) since the winter term of academic year 1994-1995, where he heads the Institute of Man-Machine-Communication and directs the Division of Computer Graphics and Geometric Modeling called Welfenlab. Before coming to Hannover, Dr. Wolter held faculty positions at the University of Hamburg (in 1994), MIT (1989-1993) and Purdue University (1987-1989). Prior to this he developed industrial expertise as a software and development engineer with AEG in Germany (1986-1987). Dr. Wolter obtained his Ph.D. in 1985 from the department of mathematics at the Technical University of Berlin in the area of Riemannian manifolds. In 1980 he graduated in mathematics and theoretical physics from the Free University of Berlin. At MIT Dr. Wolter co -developed the geometric modeling system Praxiteles for the US Navy. Since then he has been publishing various papers that broke new ground applying concepts from differential geometry and topology on problems and design of new methods used in geometric modeling and CAD systems as well as shape and image analysis. These works include pioneering contributions on medial axis theory, the computation of medial axes and Voronoi diagrams and geodesics in Riemannian space as well as pioneering works on Laplace spectra as finger prints for multi dimensional geometric objects and images with applications in biomedical imaging. The works on Laplacian operators recently lead to computational studies on generalizations of Laplacians operating on line bundles allowing to compute eigen-functions whose iso-surfaces yield smooth orientable and non-orientable Seifert surfaces in three-dimensional manifolds. Other ongoing works  include  pioneering applications of computational differential geometry on  computing, analysing and visualising singular sets of solutions to systems of algebro differential equations located in higher dimensional Euclidean spaces as well as precise numerical computations and lucid presentations of singularities of  Riemannian exponential maps.  During the last decade research on Virtual Reality systems with an emphasis on haptic and tactile perception has been subject of Dr. Wolter's research in Hannover. More recently his research includes the development of medical imaging systems and bio-mechanical simulation systems including hearing mechanics of the cochlear.

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Friday 25th September. 1.15p.m. Jean-Pierre Puel, Laboratoire de Mathematiques de Versailles, (France) 

location : Malet Place Engineering Building, Room 120

Title: "Relations between some inverse problems for evolution equations and controllability"

Abstract : We intend to show some relations between several types of inverse problems and controllability methods and results.  Actually we will present some results concerning data assimilation with a quite novel point of view for parabolic problems first and then for hyperbolic problems. In a second step, we will apply this to the question of source recovery for hyperbolic equations from boundary measurements and we will also show the limits of this question with some counterexamples. In a third part we will present some results concerning the recovery of a potential in a hyperbolic equation (or a Schrödinger equation) from boundary measurements and we will present some important open problems.

 

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Friday 9th October. 2.00p.m. Giovanni Alberti, ETH 

location : Roberts Room 110

Title: Disjoint sparsity for signal separation and applications to quantitative photoacoustic tomography

This is joint work with H Ammari.

Abstract : The main focus of this talk is the reconstruction of the signals $f$ and $g_{i}$, $i=1,\dots,N$, from the knowledge of their sums $h_{i}=f+g_{i}$, under the assumption that $f$ and the $g_{i}$s can be sparsely represented with respect to two different dictionaries $A_{f}$ and $A_{g}$. This generalises the well-known "morphological component analysis'' to a multi-measurement setting. The main result states that $f$ and the $g_{i}$s can be uniquely and stably reconstructed by finding sparse representations of $h_{i}$ for every $i$ with respect to the concatenated dictionary $[A_{f},A_{g}]$, provided that enough incoherent measurements $g_{i}$s are available. The incoherence is measured in terms of their mutual disjoint sparsity.

This method finds applications in the reconstruction procedures of several hybrid imaging inverse problems, where internal data are measured. These measurements usually consist of the main unknown multiplied by other unknown quantities, and so the disjoint sparsity approach can be directly applied. As an example, I  will show how to apply the method to the reconstruction in quantitative photoacoustic tomography, also in the case when the Grüneisen parameter, the optical absorption and the diffusion coefficient are all unknown.

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Friday 16th October. 2.00p.m. Yavar Kian, Aix-Marseille Universite (France) 

location : Cruciform B404 LT 2

Title: Determination of a time-dependent coefficient for the wave equation from partial data

Abstract :  Let $\Omega$ be a $\mathcal C^2$ bounded domain of $\R^n$, $n\geq2$, and fix $Q=(0,T)\times\Omega$ with $T>0$.

We consider the inverse problem of determining a time-dependent coefficient of order zero $q$, appearing in a Dirichlet initial-boundary value problem for a wave equation $\partial_t^2u-\Delta_x u+q(t,x)u=0$ in $Q$, from partial observations on $\partial Q$. We obtain both results of uniqueness and stability.

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Friday 6th November. 2.00p.m. Muriel Boulakia, Universite Pierre et Marie Curie, (France) 

location : Chadwick G07

Title: Numerical simulations and inverse problems in cardiac electrophysiology


Abstract: In this talk, we are interested by the electrical activity of the heart. The evolution of the cardiac potentials is represented by the so-called bidomain model coupled with an ionic model. I will first focus on the direct problem and show numerical simulations able to reproduce realistic electrocardiograms, the main tool used by medical doctors to measure the electrical activity in the heart. Then, I will present theoretical and numerical results related to inverse problems. In particular, I will state theoretical stability estimates for the identification of some parameters in the cardiac model and I will present results on the numerical identification of parameters from electrocardiograms.

The works I will present are joint works with Miguel Fernandez, Jean-Frederic Gerbeau, Elisa Schenone and Nejib Zemzemi.

 

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Friday 20th November. 2.00p.m. Hanming Zhou, Dept. Pure Maths, and Mathematical Statistics, Cambridge, (UK) 

location : Gordon Street (25) Maths 706

Title: Invariant distributions and the geodesic ray transform

Abstract: We establish an equivalence principle on simple manifolds between the solenoidal injectivity of the geodesic ray transform acting on symmetric $m$-tensors and the existence of invariant distributions with prescribed projection over the set of solenoidal $m$-tensors. We show the equivalence in both $L^2$ and smooth cases.

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Friday 27th November. 2.15p.m. Teemu Saksala, University of Helsinki, (Finland) 

location : Gordon Street (25) Maths 706

Title: Determination of a  Riemannian manifold from the distance difference functions

Abstract :  We consider the problem on an $n$ dimensional manifold $N$ with a Riemannian metric $g$ that corresponds to the travel time of a wave between two points. The Riemannian distance of points $x,y\in N$ is denoted by $d(x,y)$. For simplicity we assume that the manifold $N$ is compact and has no boundary. Instead of considering measurements on boundary, we assume that the manifold contains an unknown open part $M\subset N$ and the metric is known outside this set.

When a spontaneously point produces a wave at some unknown point $x\in M$ at some unknown time $t\in \mathbb{R}$, the produced wave is observed at the point $z\in N\setminus M$ at time $T_{x,t}(z)=d(z,x)+t$. These observation times at two points $z_1,z_2\in N\setminus M$ determine the {\it distance difference function}

$$

D_x(z_1,z_2)=T_{x,t}(z_1)-T_{x,t}(z_2)=d(z_1,x)-d(z_2,x).

$$

Physically, this function corresponds to the difference of times at $z_1$ and $z_2$ of the waves produced by the point source at $(x,t)$. An assumption there are a large number point sources and that we do measurements over a long time can be modeled by the assumption that we are given the family of functions

$$

\{D_x\ ;\ x\in M\}\subset C((N\setminus M)\times (N\setminus M)),             (*)

$$

 

We will formulate and prove an uniqueness result related to data (*) 

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Friday 11th December 2p.m. André Massing, Umeå University (Sweden)

location : Gordon Street (25) Maths 706

Title: CutFEM: Discretizing Geometry and Partial Differential Equations in Multidimensional Multiphysics Problems

Abstract: We consider a cut finite element framework for the numerical solution of partial differential equations (PDEs) posed on and and coupled through domains of different topological dimensionality. A prominent use case are flow and transport problems in porous media when large-scale networks of fractures and channels are modelled as 2D or 1D geometries embedded into a 3D bulk domain. Another important example is the modeling of cell motility where reaction-diffusion systems on the cell membrane and inner cell are coupled to describe the active reorganization of the cytoskeleton. But with complex lower-dimensional and possibly evolving geometries, traditional PDE discretization technologies are severely limited by their strong requirements on the domain discretization.

In this talk, we focus on the cut finite element framework as one possible and general approach to discretize coupled PDE systems on complex domains. To allow for a flexible discretization and easy coupling between PDEs in the bulk and on lower dimensional manifold-type domains, the lower-dimensional geometries are embedded in an unfitted manner into a three dimensional background mesh consisting of tetrahedra. Since the embedded geometry is not aligned with the background mesh, we use the trace of finite element functions defined on the tetrahedra as trial and test functions in the discrete variational formulations.  As the resulting linear system may be severely ill-conditioned due to possibly small intersections between the embedded manifold and the background mesh, we discuss several possibilities for adding (weakly) consistent stabilizations terms to the original bilinear form.  The proposed discretization schemes have optimal convergence properties and give raise to discrete linear systems which are well-conditioned independent of the intersection configuration. Along with presentation of the framework we will give a number of numerical examples which illustrate the theoretical findings and the applicability of the framework to complex modeling problems.

This work has been done in collaboration with Erik Burman (UCL), Peter Hansbo (Jönköping University, Sweden), and Mats G. Larson (Umeå University,Sweden).

 

 

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Monday 18th January. 4.00p.m. Jürgen Frikel, DTU, Copenhagen, 

location :  Darwin B05

Title:  Incomplete data reconstructions in tomography 

Abstract: In this talk, we consider the limited view reconstruction problem in x-ray and photoacoustic tomography from an integral geometric perspective. We outline that these problems are severely ill-posed and explain that only specific features can be reconstructed reliably and that artifacts can be generated. We also discuss the strength of artifacts and present an artifact reduction methodology. Finally, we present some numerical examples.

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Friday 22nd January. 2.00p.m.  Sara Garbarino, Dipartmento di Matematica, Universita di Genova (Italy)

location :  Roberts 110

Title:  An inverse problem approach to compartmental analysis in Positron Emission Tomography

Abstract: Fluoro-Deoxy-Glucose Positron Emission Tomography (FDG-PET) experiments involve two kinds of problems: the first one is the reconstruction of the concentration of tracer, that leads to a quite classical imaging inverse problem, connected to the inversion of the Radon transform. On the other hand, the second problem is the quantification of descriptors of FDG kinetics, leading to a compartmental inverse problem. 

In this talk we discuss the uniqueness problem related to the compartmental inverse problem and present a regularized Newton-based method for its reduction. We describe different applications utilizing FDG-PET data measured by a PET device for small animal models..

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Friday 29th January. 1.15p.m.  Bastian von Harrach, University of Frankfurt (Germany)

location :  Roberts 110

Title: Inverse problems and medical imaging 

Abstract: Medical diagnosis has been revolutionized by noninvasive imaging methods such as computerized tomography (CT) and magnetic resonance imaging (MRI). These great technologies are based on mathematics. If the patient's interior was known then we could numerically simulate the outcome of physical measurements performed on the patient. Medical imaging requires solving the corresponding inverse problem of determining the patient's interior from the performed measurements. 

In this talk, we will give an introduction to inverse problems in medical imaging, and discuss the challenges in newly emerging techniques such as electrical impedance tomography (EIT), where electrical currents are driven through a patient to image its interior. EIT leads to the inverse problem of determining the coefficient in a partial differential equation from (partial) knowledge of its solutions. We will describe recent advances on this problem that are based on monotonicity relations with respect to matrix (resp. operator) definiteness and the concept of localized potentials.

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Friday 12th February. 1.15p.m.  Wei Chen, University of Cambridge (UK)

location :  Roberts 110

Title: Sparse Bayesian Learning for Multitask Compressive Sensing

Abstract: Compressive sensing (CS) has triggered an enormous research activity since its first appearance. CS exploits the signal’s sparsity or compressibility in a particular domain and integrates data compression and acquisition, thus allowing exact reconstruction through relatively few non-adaptive linear measurements. Recently, beyond CS that considers the element-sparse model and mainly reconstruction, increasing attention has been attracted by multi-task CS models, CS classification, and decentralized CS processing. In this talk, I will present our research outcomes that address these interesting problems via Bayesian methods. The algorithms developed are applied to imaging reconstruction, motion segmentation and sensor networks, and shown to outperform existing approaches.

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Friday 19th Feb, 1.15 Marko Vauhkonen, Dept. of Applied Physics, University of Eastern Finland 

P.O.Box 1627, FI-70211 Kuopio, Finland, marko.vauhkonen@uef.fi

location :  Roberts 110

Title: Electromagnetic Flow Tomography

Abstract: Electromagnetic flow meters (EMFMs) are a gold standard in measuring flow velocity in process industry. The EMFM can measure the mean flow velocity of conductive liquids and slurries. A drawback of this approach is that the velocity field cannot be determined. Asymmetric axial flows, often encountered in multiphase flows, near valves and in pipe elbows and T-junctions, are problematic and can lead to serious systematic errors with EMFMs.

Compared to standard EMFM the electromagnetic flow tomography (EMFT) system consists of a pair or pairs of coils and a set of electrodes attached on the surface of the pipe. The aim in EMFT is to reconstruct two- or three-dimensional (2D,3D) flow velocity field inside the pipe based on the known magnetic field(s) and the set of voltage measurements around the pipe.

In this talk, the proposed EMFT approach is explained and the first results of the 2D velocity fieldreconstructions from simulations and experiments are shown. In the flow reconstructions, it is assumed that vx = vy = 0 and only the vz-component along the pipe is estimated. In the reconstructions, 16 electrodes and two pairs of coils are used resulting in total of 32 voltage measurements. Different asymmetric velocity fields encountered near pipe elbows, solids-in-water flows in inclined pipes and in stratified or multiphase flows are tested. The results suggest that the proposed reconstruction method  could be used to estimate velocity fields in complicated pipe flows in which the conventional electromagnetic flow meters have limited accuracy.

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Note Date : Thursday 25th February. 2.00p.m.  Jakob Sauer Jorgensen (Denmark Technical University)

location :  Malet Place Engineering Building 120

Title: How few projections are enough? Predicting performance of sparsity regularization in X-ray tomography

Abstract:  Sparsity regularization in X-ray tomography has shown large promise for accurate reconstruction from reduced data, potentially leading to substantially reduced patient X-ray exposure in medical imaging and shorter scan times in materials science. Compressed sensing (CS) has motivated these developments by offering guarantees of accurate reconstruction of sparse images from reduced data under suitable assumptions on the sampling matrix. However, existing CS guarantees, e.g., based on the restricted isometry property (RIP), do not apply to structured and sparse sampling matrices in X-ray tomography. The empirical success of sparsity regularization in X-ray tomography therefore remains theoretically unexplained.

In this seminar I will describe our attempts of establishing a connection between image sparsity and the amount of undersampling admitted by sparsity-regularized reconstruction methods. In particular, I will describe simulation results suggesting how one may predict the sufficient number of projections to acquire based on the sparsity level of the scanned object. I will then describe preliminary results of validation with real data using the SophiaBeads data set recently released from the University of Manchester.

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Friday 26th Feb, 1.15p.m. Sylvain Ervedoza, Université Paul Sabatier (Toulouse III, France)

location : Gordon Street (25) Maths 706

Title: Local exact controllability for compressible Navier-Stokes equations around constant trajectories

Abstract: In this talk, I will present a recent result obtained on the local exact controllability of the 3d compressible Navier-Stokes equation around a constant trajectory with non-zero velocity, when the control is exerted on the whole boundary of the domain. The proof of this result is based on an observability inequality for the adjoint of the linearized system. The main ingredient to obtain this inequality is the tricky combination of the equations to reduce the problem to a closed subsystem easier to deal with, and obtained by the introduction of a new quantity corresponding to the effective viscous flux for the adjoint equations. This subsystem will allow to understand independently the hyperbolic and parabolic effects of the system. We will then introduce Carleman estimates with weight functions following the characteristics allowing to handle the non-linearity of the model using a fixed point argument. This result has been obtained in collaboration with Olivier Glass (Univ. Paris Dauphine) and Sergio Guerrero (Univ. Pierre et Marie Curie).

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Monday 29th February. 4.00p.m.,  Richard Kowar, Innsbruck University (Austria)

location :  Darwin B.05

Title: Investigation of time reversal in photoacoustic imaging for dissipative media with time relaxations

Abstract:  This talk is concerned with investigations of time reversal in photoacoustic and thermoacoustic tomography (PAT and TAT) of dissipative media that are similar to water and which undergoes one or several time relaxation processes. We compare the non-causal thermoviscous wave equation, the wave equation of Nachman, Smith and Waag [1] and the wave equation of Kowar, Scherzer and Bonnefond (cf. [2]) and discuss their "time reversal properties". Moreover, we discuss shortly the numerical implementation of the wave equation of Nachman, Smith and Waag and present one simple numerical example.

[1] A. I. Nachman and J. F. III Smith and and R. C. Waag: An equation for acoustic propagation in inhomogeneous media with relaxation losses. J. Acoust. Soc. Am. 88 (3), Sept. 1990.

[2] R. Kowar and O. Scherzer and X. Bonnefond: Causality analysis of frequency-dependent wave attenuation. Math. Appl. Sci. 2011., (17pp), arXiv:0906.4678, DOI: 10.1002/mma.1344.

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Friday 4th March. 1.15p.m.  Matthias Schlottbom, University of Muenster (Germany)

location :  Roberts 110

Title: On Galerkin schemes for time-dependent radiative transfer

Abstract:  The numerical solution of time dependent radiative transfer problems is challenging, both, due to the high dimension and the anisotropic structure of the underlying integro-partial di fferential equation. Starting from an appropriate variational formulation, we propose a general strategy for designing numerical methods based on a Galerkin discretization in space and angle combined with appropriate time stepping schemes. This allows us to systematically incorporate boundary conditions and to inherit basic properties like exponential stability from the continuous level. We also present the basic approximation error estimates. The starting point for our considerations is to rewrite the radiative transfer problem as a system of evolution equations which has a similar structure as more standard rst order hyperbolic systems in acoustics or electrodynamics. This allows us to generalize the main arguments of the numerical analysis of such applications to the radiative transfer problems under investigation. We also discuss a particular discretization scheme based on a truncated spherical harmonic expansion in angle and a finite element discretization in space.

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Note time : Friday 11th March, 3.00p.m. Thomas Blumensath, University of Southampton (UK)

Note location : Gordon Street (25) Maths 107

Title:  The projected Landweber algorithm with non-convex constraints

Abstract: The projected Landweber algorithm is a classical method used to solve ill-posed inverse problems. Over the years it has been extended to many non-linear settings as well as to problems with convex constraints. More recently, it has been shown that under certain conditions, it can also be used with many non-convex constraints. Non-convex constraints such as sparsity and low-rank structures have become extremely popular in recent years and the projected Landweber algorithm offers an attractive approach to tackle many of these.

In this talk I will provide an overview of the projected Landweber algorithm with non-convex constraints, highlighting several theoretical and empirical results that shine a light on its performance in different settings and discuss different extensions and applications.  

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Double seminar : Friday 18th March,

location : Gordon Street (25) Maths 706

1.00p.m. Hoai Minh Nguyen, EPFL (Switzerland)

Title: Some applications of negative index materials

Abstract: Negative index materials are artificial structures whose refractive index are negative over some frequency range. These materials were investigated theoretically by Veselago in 1964, innovated by Pendry in 2000, and their existence was confirmed experimentally by Shelby et al. in  2001. In this talk, I will present some applications of these materials such as  cloaking and superlensing using complementary media and cloaking a source and an object via anomalous localized resonance. Various conditions for the stability of these materials are also mentioned.

2.00p.m. Daniel Sanz Alonso, Warwick

Title: Importance Sampling and Inverse Problems

Abstract: Importance sampling is a simple building block of many sampling algorithms, often claimed to fail in high dimension. I will discuss the performance of importance sampling for posterior approximation in the context of Bayesian inverse problems. Here it is not clear what should be understood by 'dimension'. A notion of dimension was suggested by Bickel and coauthors, and I will relate it to the concept of 'effective number of parameters' from statistics and machine learning. I will then explore how these notions of dimension affect the performance of importance sampling for posterior approximation. This work highlights the necessity of absolute continuity in infinite dimensional settings for robust discretizations. 

In collaboration with Sergios Agapiou, Omiros Papaspiliopoulos, and Andrew Stuart.

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Wednesday 11th May. 1.00p.m.  Mohamed Souiai, Technical University of Munich (Germany)

Joint Seminar with  CMIC

location :  Roberts 106

Title:  Continuous Multi-Label Optimization and Diffusion MRI 

 

Abstract: In this talk I will present my recent works which arrise from applications such as in semantic image segmentation and joint motion segmentation of RGB-D images. These labelling problems can be cast as variational problems in which we can incorporate so called co-occurrence, hierarchical, proximity and label transition priors in order to improve the resulting labelling.

Additionally, I will introduce a novel approach to improving the integrality of the solution to relaxed multi-labelling problems. To this end, we incorporate the entropy of the objective variable as a measure of the relaxation tightness.

Finally, I will also present novel methods for diffusion MRI processing which allow 12-fold shorter scans and model-free disease detection.

(light refreshments provided)

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Friday 20th May. 1.00p.m. Allan Greenleaf, University of Rochester

Location: 66GS room 405

Title: Propagation and recovery of singularities in Calderon's inverse conductivity problem


Abstract:  The well-known ill-posedness of the Calderon inverse problem is responsible for the poor spatial resolution of Electrical Impedance Tomography (EIT) and has been an impetus for the development of hybrid imaging techniques, which compensate for this lack of resolution by coupling with a second type of physical wave, typically modeled by a hyperbolic PDE. I will describe how the inverse conductivity problem already contains within itself a mechanism for efficient  propagation of interior singularities of the conductivity to the boundary, based on propagation of singularities for complex principal type operators. Preliminary numerical simulations indicate that this approach is effective for detecting and resolving complex inclusions in the interior using Dirichlet-to-Neumann data. This is joint work with  Matti Lassas, Mateo Santacesaria,  Samu Siltanen and Gunther Uhlmann.

 

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Friday 19th May. 1.00p.m. Jurriaan Gillissen, Singapore MIT Alliance for Research and Technology (SMART) 

Location : Gordon Street (25) Maths 500

Title: A space-time integral minimization method for reconstructing velocity fields from scalar fields

Abstract:

Scalar imaging velocimetry (SIV) is the technique to reconstruct velocity fields from measured scalar fields. This is achieved by enforcing the scalar transport equation, which, in the conventional approach, requires evaluating the space and time derivatives of the measured scalar field. These operations limit the applicability of SIV to large sampling rates and small noise levels.

To alleviate these restrictions, we propose a novel SIV scheme, that does not involve differentiating measured scalar data. The method solves a minimization problem, where the objective function measures the difference between the reconstructed scalar field and the measured scalar field, while being constrained by the Navier Stokes equation and the scalar transport equation. Merits of the modified SIV scheme are demonstrated for the case of a synthetic two dimensional turbulent flow field.

 

Bio:
Jurriaan received his Ph.D. in Fluid Mechanics and was an assistant professor at the Delft University of Technology in The Netherlands. Jurriaan worked as a senior research fellow in Nanyang University of Technology before joining in 2016 the Singapore MIT Alliance for Research and Technology (SMART) as a research scientist in prof. Yue’s team. At SMART Jurriaan’s research focuses on incorporating measurement data into flow simulations.

Related Seminar Series

 

Applied Mathematics Seminars
Tuesday, 3 pm - 4 pm, Room 505, 25 Gordon Street
Applied Mathematics Seminar Programme

Centre for Medical Image Computing Seminars
Wednesday, 12 noon - 1 pm, Room 1.19, Malet Place Engineering
CMIC Seminar Programme

London Analysis and Probability Seminar
Thursday, 3 pm - 5:30 pm, Room 500, 25 Gordon Street
London Analysis and Probability Seminar Programme

Statistical Science Seminars
Thursday, 4 pm - 5 pm, Room 102, 1-19 Torrington Place
Statistical Science Seminar Programme

 

 

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