All seminars take place via Zoom. See below for additional
information (e.g. title, abstract and zoom link). Further info
about past and future meetings are available at the webpage:
https://num-gssi.github.io/seminar/
Please feel free to distribute this announcement as you see fit.
Hope to see you all on Wednesday and Friday!
Francesco Tudisco and Nicola Guglielmi
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November 18, 2020 (Wednesday) at 17h00 (Italian time)
Christian Lubich
Dynamical low-rank approximation
This talk reviews differential equations and their numerical
solution on manifolds of low-rank matrices or of tensors with a
rank structure such as tensor trains or general tree tensor
networks. These low-rank differential equations serve to
approximate, in a data-compressed format, large time-dependent
matrices and tensors or multivariate functions that are either
given explicitly via their increments or are unknown solutions to
high-dimensional evolutionary differential equations, with
multi-particle time-dependent Schrödinger equations and kinetic
equations such as Vlasov equations as noteworthy examples of
applications.
Recently developed numerical time integrators are based on
splitting the projection onto the tangent space of the low-rank
manifold at the current approximation. In contrast to all standard
integrators, these projector-splitting methods are robust to the
unavoidable presence of small singular values in the low-rank
approximation. This robustness relies on exploiting geometric
properties of the manifold of low-rank matrices or tensors: in
each substep of the projector-splitting algorithm, the
approximation moves along a flat subspace of the low-rank
manifold. In this way, high curvature due to small singular values
does no harm.
This talk is based on work done intermittently over the last
decade with Othmar Koch, Bart Vandereycken, Ivan Oseledets, Emil
Kieri, Hanna Walach and Gianluca Ceruti.
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November 20, 2020 (Friday) at 17h30 (Italian time)
Patricia Diaz De Alba
Numerical treatment for inverse electromagnetic problems
Electromagnetic induction surveys are among the most popular
techniques for non-destructive investigation of soil properties,
in order to detect the presence of both ground inhomogeneities and
particular substances. Frequency-domain electromagnetic
instruments allow the collection of data in different
configurations, that is, varying the intercoil spacing, the
frequency, and the height above the ground.
Based on a non-linear forward model used to describe the
interaction between an electromagnetic field and the soil, the aim
is to reconstruct the distribution of either the electrical
conductivity or the magnetic permeability with respect to depth.
To this end, the inversion of both the real (in-phase) and the
imaginary (quadrature) components of the signal are studied by a
regularized damped Gauss-Newton method. The regularization part of
the algorithm is based on a low-rank approximation of the Jacobian
of the non-linear model. Furthermore, in many situations, a
regularization scheme retrieving smooth solutions is blindly
applied, without taking into account the prior available
knowledge. An algorithm for a regularization method that promotes
the sparsity of the reconstructed electrical conductivity or
magnetic permeability distribution is available. This
regularization strategy incorporates a minimum gradient support
stabilizer into a truncated generalized singular value
decomposition scheme. The whole inversion algorithm has been
enclosed in a MATLAB package, called FDEMtools, allowing the user
to experiment with synthetic and experimental data sets, and
different regularization strategies, in order to compare them and
draw conclusions.
The numerical effectiveness of the inversion procedure is
demonstrated on synthetic and real datasets by using FDEMtools
package.
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—
Francesco Tudisco
Assistant Professor
School of Mathematics
GSSI Gran Sasso Science Institute
Web:
https://ftudisco.gitlab.io