16:10
The ill-posed lateral Cauchy problem in elastodynamics
Dr. Leonidas Mindrinos | Radon Institute for Computational and Applied Mathematics | Austria
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Authors:
Roman Chapko | Ivan Franko National University of Lviv | Ukraine
B. Tomas Johansson | Aston University | United Kingdom
Dr. Leonidas Mindrinos | Radon Institute for Computational and Applied Mathematics | Austria
In this work we consider the inverse problem of reconstructing the Cauchy data on the inner boundary of a doubly-connected elastic medium from measurements on the outer boundary. The problem is formulated in two dimensions and the medium is considered as homogeneous. We use the Laguerre transform to write the inverse problem as a sequence of stationary problems and then we apply the integral equation method to obtain a sequence of boundary integral equations. The ill-posed linear system is solved using Tikhonov regularization. Numerical results are also presented.
16:30
All-at-once versus Reduced Landweber-Kaczmarz for parameter identification in time-dependent inverse problems: a potential application in Magnetic Particle Imaging
M. Sc. Tram Nguyen | Alpen-Adria-Universität Klagenfurt | Austria
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Author:
M. Sc. Tram Nguyen | Alpen-Adria-Universität Klagenfurt | Austria
In this study, we consider a general time-space system over a finite time horizon and parameter identification by using Landweber-Kaczmarz regularization. The problem is investigated in two different modeling settings: An All-at-once and a Reduced version, together with two observation scenarios: continuous and discrete observations. Segmenting the time line into several subintervals leads to the idea of applying the Kaczmarz method. We give a perspective on applying this to Magnetic Particle Imaging.
16:50
Beyond the Bakushinskii veto 2: Discretisation, white noise and nonlinear problems
M. Sc. Tim Jahn | Goethe-Universität Frankfurt | Germany
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Authors:
M. Sc. Tim Jahn | Goethe-Universität Frankfurt | Germany
Prof. Dr. Bastian von Harrach | Goethe-Universität Frankfurt
Prof. Dr. Roland Potthast | Deutscher Wetterdienst
In 'Beyond the Bakushinskii veto: Regularising without knowing the noise
distribution' it was introduced the natural approach to solve an
ill-posed problem Kx=y: Given unbiased and indentically distributed
measurements Y1,...,Yn of the right hand side y, use (Y1+...+Yn)/n as
an approximation of y and the estimated error sigma/sqrt(n) (where sigma
is the square root of the sample variance) together with a deterministic
regularisation method.
Since the true data y is usually an element of an inifinite dimensional
Hilbert space, it is more realistic to assume that one measures only a
finite dimensional discretisation Py. In many cases one has some
knowledge of the discretisation error ||y-Py|| (for example if one
discretise a function by finite elements). Thus, given the
discretisation error, one should measure n times such that sigma/sqrt(n)
= ||y-Py|| and then use (Y1+...+Yn)/n togehter with the sum of the
measurement error and the disretisation error, sigma/sqrt(n)+||y-Py||,
as the estimated error. We show how to obtain convergence when the
discretisation error tends to 0 for a wide range of error distributions,
inluding unbounded noise (White noise).
17:10
The monotonicity method for the stationary elastic wave equation
Dr. Sarah Eberle | Goethe-Universität Frankfurt am Main | Germany
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Authors:
Dr. Sarah Eberle | Goethe-Universität Frankfurt am Main | Germany
Prof. Dr. Bastian von Harrach
Dr. Houcine Meftahi
Tahar Rezgui
The main motivation of this problem is the non-destructive testing of elastic structures
for material inclusions. We consider the inverse problem of recovering an isotropic elastic
tensor from the Neumann-to-Dirichlet map. We show that the shape of a region where
the elastic tensor differs from a known background elastic tensor can be detected by a
simple monotonicity relation. In addition, we prove a Lipschitz stability. Our approach relies
on the monotonicity of the Neumann-to-Dirichlet operator with respect to the elastic
tensor and the techniques of localized potentials.