Zurich colloquium in applied and computational mathematics

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Spring Semester 2019

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27 February 2019
16:15-17:15

Prof. Dr. Benjamin Stamm
RWTH Aachen University

Event Details

Zurich Colloquium in Applied and Computational Mathematics

Title	Efficient numerical methods for polarization effects in molecular systems
Speaker, Affiliation	Prof. Dr. Benjamin Stamm, RWTH Aachen University
Date, Time	27 February 2019, 16:15-17:15
Location	Y27 H25
Abstract	In this talk we provide two examples of models and numerical methods involving N-body polarization effects. One characteristic feature of simulations involving molecular systems is that the scaling in the number N of atoms or particles is important and traditional computational methods, like domain decomposition methods for example, may behave differently than problems with a fixed computational domain. We will first see an example in the context of the Poisson-Boltzmann continuum solvation model and present a numerical method that relies on an integral equation coupled with a domain decomposition strategy. Numerical examples illustrate the behaviour of the proposed method. In a second case, we consider a N-body problem of interacting dielectric charged spheres whose solution satisfies an integral equation of the second kind. We present results from an a priori analysis with error bounds that are independent of the number particles N allowing for, in combination with the Fast Multipole Method (FMM), a linear scaling method. Towards the end, we finish the talk with applications to dynamic processes and enhanced stabilization of binary superlattices through polarization effects.

Efficient numerical methods for polarization effects in molecular systemsread_more

Y27 H25

6 March 2019
16:15-17:15

Dr. Felix Voigtlaender
Catholic University of Eichstätt-Ingolstadt

Event Details

Zurich Colloquium in Applied and Computational Mathematics

Title	Understanding sparsity properties of frames using decomposition spaces
Speaker, Affiliation	Dr. Felix Voigtlaender, Catholic University of Eichstätt-Ingolstadt
Date, Time	6 March 2019, 16:15-17:15
Location	Y27 H25
Abstract	We present a systematic approach towards understanding the sparsity properties of different frame constructions like Gabor systems, wavelets, shearlets, and curvelets. We use the following terminology: Analysis sparsity means that the frame coefficients are sparse (in an \ell^p sense), while synthesis sparsity means that the function can be written as a linear combination of the frame elements using sparse coefficients. While these two notions are completely distinct for general frames, we show that if the frame in question is sufficiently nice, then both forms of sparsity of a function are equivalent to membership of the function in a certain decomposition space. These decomposition spaces are a common generalization of Besov spaces and modulation spaces. While Besov spaces can be defined using a dyadic partition of unity on the Fourier domain, modulation spaces employ a uniform partition of unity, and general decomposition spaces use an (almost) arbitrary partition of unity on the Fourier domain. To each decomposition space, there is an associated frame construction: Given a generator, the resulting frame consists of certain translated, modulated and dilated versions of the generator. These are chosen so that the frequency concentration of the frame is similar to the frequency partition of the decomposition space. For instance, Besov spaces yield wavelet systems, while modulation spaces yield Gabor systems. We give conditions on the (possibly compactly supported!) generator of the frame which ensure that analysis sparsity and synthesis sparsity of a function are both equivalent to membership of the function in the decomposition space.

Understanding sparsity properties of frames using decomposition spacesread_more

Y27 H25

13 March 2019
16:15-17:15

Prof. Dr. Ulrik Fjordholm
University of Oslo

Event Details

Zurich Colloquium in Applied and Computational Mathematics

Title	Convergence rates of numerical approximations of hyperbolic conservation laws
Speaker, Affiliation	Prof. Dr. Ulrik Fjordholm, University of Oslo
Date, Time	13 March 2019, 16:15-17:15
Location	Y27 H25
Abstract	Hyperbolic conservation laws abound in the physical and engineering sciences, but their nonlinear and discontinuous nature necessitate numerical approximations. While the stability, compactness and convergence theory for these PDE is at a mature state, the available theory on convergence _rates_ is at best sub-optimal, and in many cases altogether lacking. In this talk I will describe some recent developments in this field. In particular, we argue that the Wasserstein distance is a natural metric in which to measure convergence. This is joint work with Susanne Solem.

Convergence rates of numerical approximations of hyperbolic conservation lawsread_more

Y27 H25

27 March 2019
16:15-17:15

Prof. Dr. Robert Scheichl
IWR, Uni Heidelberg

Event Details

Zurich Colloquium in Applied and Computational Mathematics

Title	Multilevel Uncertainty Quantification with Sample-Adaptive Model Hierarchies
Speaker, Affiliation	Prof. Dr. Robert Scheichl, IWR, Uni Heidelberg
Date, Time	27 March 2019, 16:15-17:15
Location	Y27 H 25
Abstract	Sample-based multilevel uncertainty quantification tools, such as multilevel Monte Carlo, multilevel quasi-Monte Carlo or multilevel stochastic collocation, have recently gained huge popularity due to their potential to efficiently compute robust estimates of quantities of interest (QoI) derived from PDE models that are subject to uncertainties in the input data (coefficients, boundary conditions, geometry, etc). Especially for problems with low regularity, they are asymptotically optimal in that they can provide statistics about such QoIs at (asymptotically) the same cost as it takes to compute one sample to the target accuracy. However, when the data uncertainty is localised at random locations, such as for manufacturing defects in composite materials, the cost per sample can be reduced significantly by adapting the spatial discretisation individually for each sample. Moreover, the adaptive process typically produces coarser approximations that can be used directly for the multilevel uncertainty quantification. In this talk, I will present two novel developments that aim to exploit these ideas. In the first part I will present Continuous Level Monte Carlo (CLMC), a generalisation of multilevel Monte Carlo (MLMC) to a continuous framework where the level parameter is a continuous variable. This provides a natural framework to use sample-wise adaptive refinement strategy, with a goal-oriented error estimator as our new level parameter. We introduce a practical CLMC estimator (and algorithm) and prove a complexity theorem showing the same rate of complexity as for MLMC. Also, we show that it is possible to make the CLMC estimator unbiased with respect to the true quantity of interest. Finally, we provide two numerical experiments which test the CLMC framework alongside a sample-wise adaptive refinement strategy, showing clear gains over a standard MLMC approach with uniform grid hierarchies. In the second part, I will show how to extend the sample-adaptive strategy to multilevel stochastic collocation (MLSC) methods providing a complexity estimate and numerical experiments for a MLSC method that is fully adaptive in the dimension, in the polynomial degrees and in the spatial discretisation. This is joint work with Gianluca Detommaso (Bath), Tim Dodwell (Exeter) and Jens Lang (Darmstadt).

Multilevel Uncertainty Quantification with Sample-Adaptive Model Hierarchiesread_more

Y27 H 25

3 April 2019
16:15-17:15

Dr. Victorita Dolean Maini
University of Strathclyde Glasgow

Event Details

Zurich Colloquium in Applied and Computational Mathematics

Title	Robust preconditioners for time harmonic wave propagation problems
Speaker, Affiliation	Dr. Victorita Dolean Maini, University of Strathclyde Glasgow
Date, Time	3 April 2019, 16:15-17:15
Location	Y27 H 25
Abstract	Solving wave propagation problems in harmonic regime is a very challenging task because of their indefinite nature and highly oscillatory solution when the wavenumber k is high. Although there have been different attempts to solve them efficiently, we believe that there is no established and robust preconditioner, whose behaviour is independent of k, for general decompositions into subdomains. Several attempts have been made in the literature, e.g. by Conen et al (2014) for Helmholtz problems with heterogeneous coefficients or Graham et al. (2017) for Helmholtz equations and later on extended to Maxwell’s equations with specific boundary conditions but the mechanism and the limits of applicability of such methods are far from being fully understood. In this talk we will present a few recent methods and illustrate their application on several difficult examples.

Robust preconditioners for time harmonic wave propagation problemsread_more

Y27 H 25

10 April 2019
16:15-17:15

Prof. Dr. Praveen Chandrasekhar
TIFR Bangalore, India

Event Details

Zurich Colloquium in Applied and Computational Mathematics

Title	Divergence-free discontinuous Galerkin method for MHD and Maxwell's equations
Speaker, Affiliation	Prof. Dr. Praveen Chandrasekhar, TIFR Bangalore, India
Date, Time	10 April 2019, 16:15-17:15
Location	Y27 H25
Abstract	Some PDE models like MHD and Maxwell's equations contain magnetic field as a dependent variable which must be divergence-free due to the non-existence of magnetic monopoles. This is an inherent constraint satisfied by the induction equation due to its curl structure. Numerical schemes may not preserve this structure unless they are specifically designed for this purpose. A staggered storage of variables is useful to satisfy such constraints by a numerical scheme. In this talk, I will describe two approaches to construct high order numerical approximations based on discontinuous Galerkin method that are constraint preserving. In the first approach, we perform a divergence-free reconstruction of the magnetic field while in the second approach, the divergence constraint is automatically satisfied by the numerical scheme due to the use of H(div) elements. The numerical flux used in such DG methods must satisfy a consistency condition between the 1-D and multi-D Riemann solvers, and we construct HLL-type schemes for MHD that exhibit such consistency. These methods are useful in applications where explicit time stepping schemes can be used and I will show some results for MHD and Maxwell's equations.

Divergence-free discontinuous Galerkin method for MHD and Maxwell's equationsread_more

Y27 H25

17 April 2019
16:15-17:15

Prof. Dr. Paul Ledger
College of Engineering, Swansea University

Event Details

Zurich Colloquium in Applied and Computational Mathematics

Title	Characterisation of multiple inhomogeneous conducting objects in metal detection using magnetic polarizability tensors
Speaker, Affiliation	Prof. Dr. Paul Ledger, College of Engineering, Swansea University
Date, Time	17 April 2019, 16:15-17:15
Location	Y27 H 25
Abstract	Locating and identifying hidden conducting objects has a range of important applications in metal detection including searching for buried treasure, identifying land mines and in the early detection of concealed terrorist threats. There is a need to distinguish between multiple objects, for example, in benign situations, such as coins and keys accidentally left in a pocket during a security search or a treasure hunter becoming lucky and discovering a hoard of Roman coins, as well as in threat situations, where the risks need to be clearly identified from the background clutter. Furthermore, objects are also often inhomogeneous and made up of several different metals. For instance, the barrel of a gun is invariably steel while the frame could be a lighter alloy, jacketed bullets have a lead shot and a brass jacket and modern coins often consist of a cheaper metal encased in nickel or brass alloy. Thus, in practical metal detection applications, it is important to be able to characterise both multiple objects and inhomogeneous objects. Traditional approaches to the metal detection 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, an alternative approach in which prior information about the form of the conducting permeable object has been introduced will be discussed. Ammari, Chen, Chen, Garnier and Volkov [1] have obtained the leading order term in an asymptotic expansion of the perturbed magnetic field, due to the presence of a homogeneous conducting permeable object, as the object size tends to zero. This expansion separates the object's position from its shape and material description, offering considerable advantages in case of isolated objects. We have shown that this leading order term simplifies for orthonormal coordinates and results in a characterisation of a conducting permeable object by a complex symmetric rank 2 magnetic polarizability tensor (MPT) for the eddy current case [2]. Interestingly, the MPT is different to the symmetric rank 2 Poyla-Szegö polarizability tensor (also known as the Poyla-Szegö polarisation tensor) that is known to characterise small permeable objects in magnetostatics and small conducting objects in electrical impedance tomography [3]. For instance, computing the coefficients of the MPT rely on the solution of vectorial curl-curl transmission problems while the latter on simpler scalar transmission problems. The topology ofan object plays an interesting role in the MPT coefficients. Including more terms in the asymptotic expansion increases the accuracy of the representation of the perturbed field. The higher order terms contain higher order tensors, which provide more information about the shape and material parameters of an object, and can help to improve object identification. Complete asymptotic expansions of the perturbed field for the electrical impedance tomography problem have been obtained by Ammari and Kang [3] and provide acomplete characterisation of an object by generalised polarisation tensors. For the eddy current case, we have extended the leading order term obtained in [1] to a complete expansion of the perturbed magnetic field. The higher order terms in our expansion characterise a conducting permeable object in terms of a new class of generalised magnetic polarizability tensors [4], of which the rank 2 MPT is the simplest case. Practical metal detection problems contain multiple and inhomogeneous objects. We have also provided an extension of [1, 2] to characterise multiple inhomogeneous objects, including objectsthat are closely spaced, in terms of MPTs [5]. The talk will review recent work on MPTs and explore the interesting properties exhibited by these tensors for shapes and topologies of objects. The role that a dictionary of MPTs for different shaped objects can play in object classification will also be described. References: [1] H. Ammari, J. Chen, Z. Chen, J. Garnier and D. Volkov, Target detection and characterizationfrom electromagnetic induction data, J. Math. Pure Appl. 2014: 101: 54-75. [2] P.D. Ledger and W.R.B. Lionheart, Characterising the shape and material properties of hidden targets from magnetic induction data, IMA J. Appl. Math. 2015: 80: 1776-1798. [3] H. Ammari and H. Kang, Polarization and Moment Tensors: With Applications to Inverse Problems and Effective Medium Theory, New-York, Springer, 2007. [4] P.D. Ledger and W.R.B. Lionheart, Generalised magnetic polarizability tensors, Math. Meth. Appl. Sci., 2018: 41: 3175-3196. [5] P.D. Ledger, W.R.B. Lionheart and A.A.S. Amad, Characterisation of multiple conducting permeable objects in metal detection by polarizability tensors, Math. Meth. Appl. Sci., 2019:42: 830-860.

Characterisation of multiple inhomogeneous conducting objects in metal detection using magnetic polarizability tensors read_more

Y27 H 25

24 April 2019
16:15-17:15

Prof. Dr. Giovanni S. Alberti
University of Genova

Event Details

Zurich Colloquium in Applied and Computational Mathematics

Title	Infinite-dimensional inverse problems with finite measurements
Speaker, Affiliation	Prof. Dr. Giovanni S. Alberti, University of Genova
Date, Time	24 April 2019, 16:15-17:15
Location	Y27 H 25
Abstract	In this talk I will discuss how ideas from applied harmonic analysis, in particular sampling theory and compressed sensing (CS), may be applied to inverse problems in PDEs. The focus will be on inverse boundary value problems for the conductivity and the Schrodinger equations, and I will give uniqueness and stability results, both in the linearized and in the nonlinear case. These results make use of a recent general theory of infinite-dimensional CS for deterministic and non-isometric operators, which will be briefly surveyed. This is joint work with Matteo Santacesaria.

Infinite-dimensional inverse problems with finite measurementsread_more

Y27 H 25

15 May 2019
16:15-17:15

Prof. Dr. Arnulf Jentzen
ETH Zurich

Event Details

Zurich Colloquium in Applied and Computational Mathematics

Title	Overcoming the curse of dimensionality: from nonlinear Monte Carlo to deep artificial neural networks
Speaker, Affiliation	Prof. Dr. Arnulf Jentzen, ETH Zurich
Date, Time	15 May 2019, 16:15-17:15
Location	Y27 H 25
Abstract	Partial differential equations (PDEs) are among the most universal tools used in modelling problems in nature and man-made complex systems. For example, stochastic PDEs are a fundamental ingredient in models for nonlinear filtering problems in chemical engineering and weather forecasting, deterministic Schroedinger PDEs describe the wave function in a quantum physical system, deterministic Hamiltonian-Jacobi-Bellman PDEs are employed in operations research to describe optimal control problems where companys aim to minimise their costs, and deterministic Black-Scholes-type PDEs are highly employed in portfolio optimization models as well as in state-of-the-art pricing and hedging models for financial derivatives. The PDEs appearing in such models are often high-dimensional as the number of dimensions, roughly speaking, corresponds to the number of all involved interacting substances, particles, resources, agents, or assets in the model. For instance, in the case of the above mentioned financial engineering models the dimensionality of the PDE often corresponds to the number of financial assets in the involved hedging portfolio. Such PDEs can typically not be solved explicitly and it is one of the most challenging tasks in applied mathematics to develop approximation algorithms which are able to approximatively compute solutions of high-dimensional PDEs. Nearly all approximation algorithms for PDEs in the literature suffer from the so-called "curse of dimensionality" in the sense that the number of required computational operations of the approximation algorithm to achieve a given approximation accuracy grows exponentially in the dimension of the considered PDE. With such algorithms it is impossible to approximatively compute solutions of high-dimensional PDEs even when the fastest currently available computers are used. In the case of linear parabolic PDEs and approximations at a fixed space-time point, the curse of dimensionality can be overcome by means of Monte Carlo approximation algorithms and the Feynman-Kac formula. In this talk we introduce a new nonlinear Monte Carlo algorithm for high-dimensional nonlinear PDEs. We prove that this algorithm does indeed overcome the curse of dimensionality in the case of a general class of semilinear parabolic PDEs and we thereby prove, for the first time, that a general semilinear parabolic PDE with a nonlinearity depending on the PDE solution can be solved approximatively without the curse of dimensionality.

Overcoming the curse of dimensionality: from nonlinear Monte Carlo to deep artificial neural networksread_more

Y27 H 25

22 May 2019
16:15-17:15

Prof. Dr. Stefan Steinerberger
Yale University

Event Details

Zurich Colloquium in Applied and Computational Mathematics

Title	Sampling on Manifolds and Finite Graphs
Speaker, Affiliation	Prof. Dr. Stefan Steinerberger, Yale University
Date, Time	22 May 2019, 16:15-17:15
Location	Y27 H25
Abstract	I will discuss the classical problem of finding good sampling points (say, for the purpose of numerical integration, approximation or interpolation) on Riemannian manifolds (with special focus on the sphere and the torus) and finite graphs. This has beautiful connections to classical Harmonic Analysis, Analytic Number Theory and Combinatorics. On finite Graphs we use an abstract spectral definition, one that can be used to construct the Platonic solids in IR^3, to recover special subsets of vertices ("the Platonic Bodies inside a Graph"). These objects are quite mysterious but very beautiful (many pictures provided) and I will discuss some open problems associated with them.

Sampling on Manifolds and Finite Graphsread_more

Y27 H25

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