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Zurich colloquium in applied and computational mathematics

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Autumn Semester 2018

Date / Time Speaker Title Location
19 September 2018
16:15-17:15 		Dr. Philipp Petersen
Oxford University
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Zurich Colloquium in Applied and Computational Mathematics

Title Deep Neural Networks and Partial Differential Equations: Approximation Theory and Structural Properties
Speaker, Affiliation Dr. Philipp Petersen, Oxford University
Date, Time 19 September 2018, 16:15-17:15
Location HG E 1.2
Abstract Novel machine learning techniques based on deep learning, i.e., the data-driven manipulation of neural networks, have reported remarkable results in many areas such as image classification, game intelligence, or speech recognition. Driven by these successes, many scholars have started using them in areas which do not focus on traditional machine learning tasks. For instance, more and more researchers are employing neural networks to develop tools for the discretisation and solution of partial differential equations. Two reasons can be identified to be the driving forces behind the increased interest in neural networks in the area of the numerical analysis of PDEs. On the one hand, powerful approximation theoretical results have been established which demonstrate that neural networks can represent functions from the most relevant function classes with a minimal number of parameters. On the other hand, highly efficient machine learning techniques for the training of these networks are now available and can be used as a black box.
In this talk, we will give an overview of some approaches towards the numerical treatment of PDEs with neural networks and study the two aspects above. We will recall some classical and some novel approximation theoretical results and tie these results to PDE discretisation. Afterwards, providing a counterpoint, we analyse the structure of network spaces and deduce considerable problems for the black box solver. In particular, we will identify a number of structural properties of the set of neural networks that render optimisation over this set especially challenging and sometimes impossible.
Deep Neural Networks and Partial Differential Equations: Approximation Theory and Structural Propertiesread_more
HG E 1.2
10 October 2018
16:15-17:15 		Prof. Dr. Steffen Börm
Institut für Informatik, Universität Kiel
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Zurich Colloquium in Applied and Computational Mathematics

Title Hybrid compression of boundary element matrices for high-frequency Helmholtz problems
Speaker, Affiliation Prof. Dr. Steffen Börm, Institut für Informatik, Universität Kiel
Date, Time 10 October 2018, 16:15-17:15
Location HG E 1.2
Abstract Fast summation methods are well-established for non-local operators arising in the context for electrostatics, molecular dynamics, or linear elastostatics, where they can reduce the complexity from O(n²) to O(n)or O(n log n). In the case of high-frequency Helmholtz equations, the situation is significantly more challenging: although the kernel function is still analytic, standard approximations, e.g., by polynomials, converge only slowly. This problem can be solved by splitting the kernel function into a smooth factor and a plane wave and approximating only the smooth factor, e.g., by interpolation. We obtain a fast summation scheme, but the storage requirements are fairly high: on one hand, multiple directions have to be handled simultaneously to reach a suitable accuracy. On the other hand, the computational domain has to be split into fairly small subdomains. Fortunately, we can combine the interpolation-based approximation with algebraic techniques to reduce the storage requirements, and this allows us to handle large problems efficiently. This talk gives an introduction to the directional interpolation approach, illustrates its properties in numerical examples, describes the algebraic re-compression, and demonstrates that it can significantly improve the overall performance of the method.
Hybrid compression of boundary element matrices for high-frequency Helmholtz problemsread_more
HG E 1.2
17 October 2018
16:15-17:15 		Prof. Dr. Lourenco Beirao da Veiga
Università di Milano-Bicocca
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Zurich Colloquium in Applied and Computational Mathematics

Title An introduction to virtual elements in 3D
Speaker, Affiliation Prof. Dr. Lourenco Beirao da Veiga, Università di Milano-Bicocca
Date, Time 17 October 2018, 16:15-17:15
Location HG E 1.2
Abstract The Virtual Element Method (VEM), is a very recent technology introduced in [Beirao da Veiga, Brezzi, Cangiani, Manzini, Marini, Russo, 2013, M3AS] for the discretization of partial differential equations, that has shared a good success in recent years. The VEM can be interpreted as a generalization of the Finite Element Method that allows to use general polygonal and polyhedral meshes, still keeping the same coding complexity and allowing for arbitrary degree of accuracy. The Virtual Element Method makes use of local functions that are not necessarily polynomials and are defined in an implicit way. Nevertheless, by a wise choice of the degrees of freedom and introducing a novel construction of the associated stiffness matrixes, the VEM avoids the explicit integration of such shape functions. In addition to the possibility to handle general polytopal meshes, the flexibility of the above construction yields other interesting properties with respect to more standard Galerkin methods. For instance, the VEM easily allows to build discrete spaces of arbitrary C^k regularity, or to satisfy exactly the divergence-free constraint for incompressible fluids. The present talk is an introduction to the VEM, aiming at showing the main ideas of the method. We consider for simplicity a simple elliptic model problem (that is pure diffusion with variable coefficients) but set ourselves in the more involved 3D setting. In the first part we introduce the adopted Virtual Element space and the associated degrees of freedom, first by addressing the faces of the polyhedrons (i.e. polygons) and then building the space in the full volumes. We then describe the construction of the discrete bilinear form and the ensuing discretization of the problem. Furthermore, we show a set of theoretical and numerical results. In the very final part, we will give a glance at more involved problems, such as magnetostatics (mixed problem with more complex spaces interacting) and large deformation elasticity (nonlinear problem).
An introduction to virtual elements in 3Dread_more
HG E 1.2
24 October 2018
16:15-17:15 		Prof. Dr. Claudia Schillings
Universitaet Mannheim
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Zurich Colloquium in Applied and Computational Mathematics

Title On the Convergence of Laplace's Approximation and Its Implications for Bayesian Computation
Speaker, Affiliation Prof. Dr. Claudia Schillings, Universitaet Mannheim
Date, Time 24 October 2018, 16:15-17:15
Location HG E 1.2
Abstract Inverse problems arise in various fields of sciences and engineering. Methods to efficiently incorporate data into models are needed to reduce the overall uncertainty and to ensure the reliability of the simulations under real world conditions. The Bayesian approach to inverse problems provides a rigorous framework for the incorporation and quantification of uncertainties in measurements, parameters and models. The concentration of the posterior is a highly desirable situation in practice, since it relates to informative or large data. However, sampling methods for Bayesian inference show numerical instabilities in the case of concentrated posterior distributions. In this talk, we will discuss convergence results of Laplace’s approximation and analyze the use of the approximation within sampling methods. This is joint work with Bjoern Sprungk (U Goettingen) and Philipp Wacker (FAU Erlangen).
On the Convergence of Laplace's Approximation and Its Implications for Bayesian Computationread_more
HG E 1.2
7 November 2018
16:15-17:15 		Dr. Richard Küng
Caltech, Pasadena, USA
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Zurich Colloquium in Applied and Computational Mathematics

Title Low rank matrix recovery from group orbits
Speaker, Affiliation Dr. Richard Küng, Caltech, Pasadena, USA
Date, Time 7 November 2018, 16:15-17:15
Location HG E 1.2
Abstract We prove that low-rank matrices can be recovered efficiently from a small number of measurements that are sampled from orbits of a certain matrix group. As a special case, our theory makes statements about the phase retrieval problem. Here, the task is to recover a vector given only the amplitudes of its inner product with a small number of vectors from an orbit. Variants of the group in question have appeared under different names in many areas of mathematics. In coding theory and quantum information, it is the complex Clifford group; in time-frequency analysis the oscillator group; and in mathematical physics the metaplectic group. It affords one particularly small and highly structured orbit that includes and generalizes the discrete Fourier basis: While the Fourier vectors have coefficients of constant modulus and phases that depend linearly on their index, the vectors in said orbit have phases with a quadratic dependence. Our proof methods could be adapted to cover orbits of other groups.
Low rank matrix recovery from group orbitsread_more
HG E 1.2
5 December 2018
16:15-17:15 		Prof. Dr. Elena Cordero
Universita degli Studi di Torino
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Zurich Colloquium in Applied and Computational Mathematics

Title Generalized Born-Jordan Distributions and Applications to the Reduction of Interferences
Speaker, Affiliation Prof. Dr. Elena Cordero, Universita degli Studi di Torino
Date, Time 5 December 2018, 16:15-17:15
Location HG E 1.2
Abstract One of the most popular time-frequency representations is certainly the Wigner distribution. Its quadratic nature, however, causes the appearance of unwanted interferences or artefacts. The desire to suppress these artefacts is the reason why engineers, mathematicians and physicists have been looking for related time-frequency distributions, many of them are members of the Cohen class. Among them, the Born-Jordan distribution has recently attracted the attention of many authors, since the so-called "ghost frequencies" are damped quite well, and the noise is, in general, reduced. The very insight relies on the kernel of such a distribution, which contains the \emph{sinus cardinalis} $\mathrm{sinc}$, the Fourier transform\, of the first B-Spline $B_{1}$. Replacing the function $B_{1}$ with the spline or order $n$, denoted by $B_{n}$, yields the function $(\mathrm{sinc})^{n}$ on the Fourier side, whose decay at infinity increases with $n$. The related Cohen's Kernel is given by $\Theta^{n}(z_{1},z_{2})=\mathrm{sinc}^{n}(z_{1}\cdot _{2})$, $n\in\bN$. We study properties of the time-frequency distribution, called \emph{generalized Born-Jordan distribution of order $n$}, arising from these new kernels. Such representations display a great capacity of damping interferences and the reduction increases with $n$. This talk will show the different facets of this phenomenon, from visual comparisons to rigorous mathematical explanations. This is a joint work with Monika Dörfler, Maurice de Gosson (University of Vienna) and Fabio Nicola (Politecnico di Torino).
Generalized Born-Jordan Distributions and Applications to the Reduction of Interferencesread_more
HG E 1.2
12 December 2018
16:15-17:15 		Prof. Dr. Nilima Nigam
Simon Fraser University
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Zurich Colloquium in Applied and Computational Mathematics

Title A modification of Schiffer's Conjecture, and a proof via Finite Elements
Speaker, Affiliation Prof. Dr. Nilima Nigam, Simon Fraser University
Date, Time 12 December 2018, 16:15-17:15
Location HG E 1.2
Abstract Approximations via conforming and non-conforming finite elements can be used to construct validated and computable bounds on eigenvalues for the Dirichlet Laplacian in certain domains. If these are to be used as part of a proof, care must be taken to ensure each step of the computation is validated and verifiable. In this talk we present a long-standing conjecture in spectral geometry, and its resolution using validated finite element computations. Schiffer's conjecture states that if a bounded domain omega in Rn has any nontrivial Neumann eigenfunction which is a constant on the boundary, then omega must be a ball. This conjecture is open. A modification of Schiffer's conjecture is: for regular polygons of at least 5 sides, we can demonstrate the existence of a Neumann eigenfunction which is does not change sign on the boundary. In this talk, we provide a recent proof using finite element calculations forthe regular pentagon. The strategy involves iteratively bounding eigenvalues for a sequence of polygonal subdomains of the triangle with mixed Dirichlet and Neumann boundary conditions. We use a learning algorithm to find and optimize this sequence of subdomains, and use non-conforming linear FEM to compute validated lower bounds for the lowest eigenvalue in each of these domains. The linear algebra is performed within interval arithmetic. This is joint work with Bartlomiej Siudeja and Ben Green, at U. Oregon.
A modification of Schiffer's Conjecture, and a proof via Finite Elementsread_more
HG E 1.2

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