Departement Mathematik

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Zurich colloquium in applied and computational mathematics

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Frühjahrssemester 2021

Datum / Zeit Referent:in Titel Ort
24. Februar 2021
16:15-17:15 		Dr. José Luis Romero
University of Vienna
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Zurich Colloquium in Applied and Computational Mathematics

Titel Sampling, density, and equidistribution
Referent:in, Affiliation Dr. José Luis Romero, University of Vienna
Datum, Zeit 24. Februar 2021, 16:15-17:15
Ort Zoom Meeting
Abstract The sampling problem concerns the reconstruction of every function within a given class from their values observed only at certain points (samples). A density theorem gives precise necessary or sufficient conditions for such reconstruction in terms of an adequate notion of density of the set of samples. The most classical density theory, due to Shannon and Beurling, concerns bandlimited functions (that is, functions whose Fourier transforms are supported on the unit interval) and provides a sharp geometric characterization of all configurations of points that lead to stable reconstruction. I will present recent variants of such results and their applications in other fields, including the asymptotic equidistribution of Coulomb gases at low temperatures.
Sampling, density, and equidistributionread_more
Zoom Meeting
10. März 2021
16:15-17:15 		Prof. Dr. Mikko Salo
University of Jyväskylä
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Zurich Colloquium in Applied and Computational Mathematics

Titel Why are inverse problems ill-posed?
Referent:in, Affiliation Prof. Dr. Mikko Salo, University of Jyväskylä
Datum, Zeit 10. März 2021, 16:15-17:15
Ort Zoom Meeting
Abstract Many inverse and imaging problems, such as image deblurring or electrical/optical tomography, are known to be highly sensitive to noise. In these problems small errors in the measurements may lead to large errors in reconstructions. Such problems are called ill-posed or unstable, as opposed to being well-posed (a notion introduced by J. Hadamard in 1902). The inherent reason for instability is easy to understand in linear inverse problems like image deblurring. For more complicated nonlinear imaging problems the instability issue is more delicate. We will discuss a general framework for understanding ill-posedness in inverse problems based on smoothing/compression properties of the forward map together with estimates for entropy and capacity numbers in relevant function spaces. The methods apply to various inverse problems involving general geometries and low regularity coefficients. We will use Electrical Impedance Tomography as a guiding example in the presentation. This talk is based on joint work with Herbert Koch (Bonn) and Angkana Rüland (Heidelberg).
Why are inverse problems ill-posed?read_more
Zoom Meeting
24. März 2021
16:15-17:15 		Prof. Dr. Hans Feichtinger
University of Vienna
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Zurich Colloquium in Applied and Computational Mathematics

Titel Gabor Analysis: Background, Concepts and Computational Issues
Referent:in, Affiliation Prof. Dr. Hans Feichtinger, University of Vienna
Datum, Zeit 24. März 2021, 16:15-17:15
Ort Zoom Meeting
Abstract Gabor Analysis goes back to the fundamental paper of D. Gabor from 1946, who expressed (resp. conjectured) that every function can be expanded into a series of time-frequency shifted version of the standard Gaussian, meaning by building blocks of the form

g_{k,n}(x) = exp(2 \pi i b n} g_0(x - ak),

with $g_0(t) = exp(-\pi t^2)$. By choosing $a=1=b$ he was hoping to expand "every signal in a unique way", thus having a natural interpretation of the energy at position $(k,n)$ in phase-space through the (expected) unique coefficients for such an expansion.
Only since 35 years mathematicians have taken care of this problem, which provides a number of interesting challenges, going far beyond the original problem. From the speaker's point of view there is not only the computational problem arising (since Gabor Analysis can also be realized in the context of finite, Abelian groups), but also a variety of functional analytic concepts, including the idea of Banach frames and Banach Gelfand Triples.
The methods developed in the last 25 years are also relevant for the teaching of classical Fourier Analysis (such a course has been held at ETH last semester by the speaker), but also provides a good platform for the formulation of "Conceptual Harmonic Analysis", where the main question concerns the relationship between the continuous formulation of a problem and the corresponding discrete version and its computational realization.
Gabor Analysis: Background, Concepts and Computational Issuesread_more
Zoom Meeting
31. März 2021
16:15-17:15 		Dr. Anirbit Mukherjee
University of Pennsylvania, USA
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Zurich Colloquium in Applied and Computational Mathematics

Titel Some Recent Progresses in the Mathematics of Neural Training
Referent:in, Affiliation Dr. Anirbit Mukherjee, University of Pennsylvania, USA
Datum, Zeit 31. März 2021, 16:15-17:15
Ort Zoom Meeting
Abstract One of the most intriguing mathematical mysteries of our times is to be able to explain the phenomenon of deep-learning. Neural nets can be made to paint while imitating classical art styles or play chess better than any machine or human ever and they seem to be the closest we have ever come to achieving "artificial intelligence". But trying to reason about these successes quickly lands us into a plethora of extremely challenging mathematical questions - typically about discrete stochastic processes. Some of these questions remain unsolved for even the smallest neural nets! In this talk we will describe two of the most recent themes of our work in this direction. Firstly, we will explain how under mild distributional conditions we can construct iterative algorithms which can train a ReLU gate in the realizable setting in linear time while also keeping track of mini-batching. We will show how this algorithm does approximate training when there is a data-poisoning attack on the training labels. Such convergence proofs remain unknown for S.G.D, but we will show via experiments that our algorithm very closely mimics the behaviour of S.G.D. Lastly, we will review this very new concept of "local elasticity" of a learning process and demonstrate how it appears to reveal certain universal phase changes during neural training. Then we will introduce a mathematical model which reproduces some of these key properties in a semi-analytic way. We will end by delineating various open questions in this theme of macroscopic phenomenology with neural nets. This is joint work with Weijie Su (Wharton, Statistics), Sayar Karmakar (U Florida, Statistics) and Phani Deep (Amazon, India)
Some Recent Progresses in the Mathematics of Neural Trainingread_more
Zoom Meeting
14. April 2021
16:15-17:15 		Dr. Gerhard Unger
TU Graz
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Zurich Colloquium in Applied and Computational Mathematics

Titel Contour integral method for scattering resonance problems
Referent:in, Affiliation Dr. Gerhard Unger, TU Graz
Datum, Zeit 14. April 2021, 16:15-17:15
Ort Zoom Meeting
Abstract The use of boundary integral equations enables a reduction of scattering resonance problems in acoustics and electromagnetics to the boundary of the scatterer. Boundary integral formulations of resonance problems always lead to nonlinear eigenvalue problems with respect to the frequency parameter even if the original resonance problem is a linear one, as e.g. in the case of non-dispersive scatterers. The reason for that is that the frequency parameter occurs nonlinearly in the fundamental solution of the involved partial differential operator. Boundary integral formulations of resonance problems can be considered as eigenvalue problems for holomorphic Fredholm operator-valued functions. For such kind of eigenvalue problems a comprehensive spectral theory exists. Moreover, abstract results on the convergence for the approximation of such kind of eigenvalue problems are available and applicable to standard Galerkin approximations of boundary integral formulations of scattering resonance problems. The resulting Galerkin approximations are eigenvalue problems for holomorphic matrix-valued functions. The contour integral method is a reliable method for the approximation of such kind of algebraic eigenvalue problems. The method is based on the contour integration of the inverse of the occurring matrix-valued function of the eigenvalue problem and utilizes that the eigenvalues are poles of it. By contour integration a reduction of the eigenvalue problem for a holomorphic matrix-valued function to an equivalent linear matrix eigenvalue problem is possible such that the eigenvalues of the linear eigenvalue problem coincide with the eigenvalues of the nonlinear eigenvalue problem inside the contour. For the practical application of this method an efficient approximation of the contour integral over the inverse of the underlying matrix-valued function of the eigenvalue problem is necessary. This can be achieved for example by the composite trapezoidal rule, which requires the solution of several linear systems involving boundary element matrices related to the eigenvalue problem.
Contour integral method for scattering resonance problemsread_more
Zoom Meeting
21. April 2021
16:15-17:15 		Dr. Benjamin Scellier
Google Zurich
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Zurich Colloquium in Applied and Computational Mathematics

Titel A deep learning theory for neural networks grounded in physics
Referent:in, Affiliation Dr. Benjamin Scellier, Google Zurich
Datum, Zeit 21. April 2021, 16:15-17:15
Ort Zoom Meeting
Abstract We present a mathematical framework for machine learning, which allows us to train "physical systems with adjustable parameters" by gradient descent. Our framework applies to a very broad class of systems, namely those whose state or dynamics are described by variational equations. This includes physical systems whose equilibrium state is the minimum of an energy function, and physical systems whose trajectory minimizes an action functional (principle of least action). We present a simple procedure to compute the loss gradients in such systems. This procedure, called equilibrium propagation (EqProp), requires solely locally available information for each trainable parameter. In particular, our framework offers the possibility to build and train neural networks in substrates that directly exploit the laws of physics. As an example, we show how to use our framework to train a class of electrical circuits called nonlinear resistive networks. We also sketch a path to apply our framework to spiking neural networks (specifically spiking electrical circuits), by showing that nonlinear RLC circuits satisfy a principle of least action.
A deep learning theory for neural networks grounded in physicsread_more
Zoom Meeting
28. April 2021
16:15-17:15 		Prof. Dr. Richardo Nochetto
University of Maryland, College Park, USA
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Zurich Colloquium in Applied and Computational Mathematics

Titel What is fractional diffusion?
Referent:in, Affiliation Prof. Dr. Richardo Nochetto, University of Maryland, College Park, USA
Datum, Zeit 28. April 2021, 16:15-17:15
Ort Zoom Meeting
Abstract This is a survey talk about fractional diffusion. It describes its formulation via the integral Laplacian, the regularity of solutions on bounded domains and the approximation by finite element methods. It will emphasize recent research about Besov regularity on Lipschitz domains, BPX preconditioning and local energy error estimates.
What is fractional diffusion?read_more
Zoom Meeting
2. Juni 2021
16:15-17:15 		Prof. Dr. Carsten Carstensen
Institut für Mathematik, Humboldt-Universität
Details

Zurich Colloquium in Applied and Computational Mathematics

Titel Eigenvalue Computation for Symmetric PDEs
Referent:in, Affiliation Prof. Dr. Carsten Carstensen, Institut für Mathematik, Humboldt-Universität
Datum, Zeit 2. Juni 2021, 16:15-17:15
Ort Zoom Meeting
Unterlagen abstractfile_download
Eigenvalue Computation for Symmetric PDEsread_more
Zoom Meeting

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