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Autumn Semester 2016

Date / Time Speaker Title Location
28 September 2016
16:15-17:15 		Prof. Dr. John Butcher
University of Auckland, New Zealand
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Zurich Colloquium in Applied and Computational Mathematics

Title The construction of high order G-symplectic methods
Speaker, Affiliation Prof. Dr. John Butcher, University of Auckland, New Zealand
Date, Time 28 September 2016, 16:15-17:15
Location HG E 1.2
Abstract A general linear method is "G-symplectic" if a certain algebraic condition is satisfied. Many examples of these methods are known and their behaviour is now well understood, both theoretically and experimentally. The focus is now on deriving high order methods in the anticipation that they will provide accurate and efficient integration schemes for mechanical and other problems. One of the starting points is an analysis of the order conditions (Butcher, J., Imran, G., Order conditions for G-symplectic methods, BIT, 55 (2015), 927-948. It was shown that the order conditions are related to unrooted trees in a similar way to what is known for symplectic Runge-Kutta methods (Sanz-Serna J. M., Abia L., Order conditions for canonical Runge-Kutta schemes, SIAM J. Numer. Anal. 28, 1081-1096 (1991)). Starting from the order conditions, simplifications can be made by assuming time-reversal symmetry and enhanced stage order. Even after high order methods have been found, the construction of suitable starting schemes is usually an essential step before working algorithms can be built. But sometimes this can be avoided.
The construction of high order G-symplectic methodsread_more
HG E 1.2
5 October 2016
16:15-17:15 		Prof. Dr. Martin Burger
Universität Münster
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Zurich Colloquium in Applied and Computational Mathematics

Title Uncertainty Quantification in the Variational Regularization of Inverse Problems
Speaker, Affiliation Prof. Dr. Martin Burger, Universität Münster
Date, Time 5 October 2016, 16:15-17:15
Location HG E 1.2
Abstract Variational techniques in the regularization of inverse problems have evolved to become a standard tool in the field. In particular in image reconstruction, the use of nonquadratic regularization functionals (and possibly data fidelities) such as sparsity-promoting l1-minimization or edge-enhancing techniques based on total variation made enormous impact. Consequently novel questions with respect to the quantification of uncertainties were raised, questions of particular relevance being the relation to appropriate Bayesian prior and posterior models on the one hand and the estimation of errors caused by random noise in the data. In this talk we will discuss several recent developments in these problems in important case of convex regularization functionals (log-concave priors in the Bayesian setup), based on the use of Bregman distances and other dual error measures. We discuss a quite general approach to estimate errors in the solutions of the variational regularization methods, which is based on duality techniques for convex optimization and can treat large (unbounded) noise. As a direct consequence we obtain estimates on the expected error for a setup with white noise in the data. Moreover, we discuss the characterization of the minimizer as a maximum a-posteriori probability (MAP) estimate for an appropriate model. For this sake we introduce the novel concept of weak MAP estimates and relate those to minimizers of a natural Bayes cost. This talk is based on joint work with Tapio Helin (Helsinki), Felix Lucka (UCL) and Hanne Kekkonen (Warwick)
Uncertainty Quantification in the Variational Regularization of Inverse Problemsread_more
HG E 1.2
12 October 2016
16:15-17:15 		Prof. Dr. Gianluigi Rozza
SISSA, Triest
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Zurich Colloquium in Applied and Computational Mathematics

Title Reduced Order Methods: state of the art and perspectives: focus on Computational Fluid Dynamics
Speaker, Affiliation Prof. Dr. Gianluigi Rozza, SISSA, Triest
Date, Time 12 October 2016, 16:15-17:15
Location HG E 1.2
Abstract In this talk we deal with the state of the art of Reduced Order Methods (ROM) for parametric Partial Differential Equations (PDEs) and we provide some perspectives in their current trends and developments, with a special interest in Computational Fluid Dynamics (CFD) parametric problems. Systems modelled by PDEs are depending by several complex parameters in need of being reduced, even before the computational phase in a pre-processing step. Efficient parametrizations (random inputs, geometry, physics) are very important to be able to properly address an offline-online decoupling of the computational procedures and to allow competitive computational performances. Current ROM developments include: a better use of high fidelity methods, also spectral element method, enhancing the quality of the reduced model too; more efficient sampling techniques to reduce the number of the basis functions, retained as snapshots, and the dimension of online systems; the improvements of the certification of accuracy based on residual based error bounds and stability factors; for nonlinear system also investigations on bifurcations of parametric solutions is crucial and it may be obtained thanks to a reduced eigenvalue analysis. All the previous aspects are very important in CFD problems in order to be able to study complex industrial and biomedical flow problems in real time, and to couple viscous flows -velocity, pressure, and also thermal field - with a structural field or a porous medium. This last task requires also an efficient reduced parametric treatment of interfaces.
Reduced Order Methods: state of the art and perspectives: focus on Computational Fluid Dynamicsread_more
HG E 1.2
19 October 2016
16:15-17:15 		Dr. Mario Hefter
University of Kaiserslautern, Germany
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Zurich Colloquium in Applied and Computational Mathematics

Title Strong Convergence Rates for Cox-Ingersoll-Ross Processes: Full Parameter Range
Speaker, Affiliation Dr. Mario Hefter, University of Kaiserslautern, Germany
Date, Time 19 October 2016, 16:15-17:15
Location HG E 1.2
Abstract In recent years, strong (pathwise) approximation of stochastic differential equations (SDEs) has intensively been studied for SDEs of the form \begin{align*} \mathrm{d} X_t = (a-bX_t)\mathrm{d} t + \sigma\sqrt{X_t}\mathrm{d} W_t, \end{align*} with a scalar Brownian motion $W$ and parameters $a,\sigma>0$, $b\in\mathbb{R}$. These SDEs are, e.g., used to describe the volatility in the Heston model and the interest rate in the Cox-Ingersoll-Ross model. In the particular case of $b=0$ and $\sigma=2$ the solution is a squared Bessel process of dimension $a$. We propose a tamed Milstein scheme $Y^N$, which uses $N\in\mathbb N$ values of the driving Brownian motion $W$, and prove positive polynomial convergence rates for all parameter constellations. More precisely, we show that for every $1\leq p<\infty$ and every $\varepsilon>0$ there exists a constant $C>0$ such that \begin{align*} \sup_{0\leq t\leq 1}\left( E{ |X_t-Y_t^N|^p } \right)^{1/p} \leq C\cdot \frac{1}{N^{\min(1,\delta)/(2p)-\varepsilon}} \end{align*} for all $N\in\mathbb N$, where $\delta = {4a}/{\sigma^2}$. This is joint work with Andr� Herzwurm.
Strong Convergence Rates for Cox-Ingersoll-Ross Processes: Full Parameter Rangeread_more
HG E 1.2
2 November 2016
16:15-17:15 		Dr. Andrea Moiola
University of Reading
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Zurich Colloquium in Applied and Computational Mathematics

Title Sobolev spaces on non-Lipschitz sets with application to boundary integral equations on fractal screens
Speaker, Affiliation Dr. Andrea Moiola, University of Reading
Date, Time 2 November 2016, 16:15-17:15
Location HG E 1.2
Abstract The scattering of a time-harmonic acoustic wave by a planar screen with Lipschitz boundary is classically modelled by boundary integral equations (BIEs). If the screen is not Lipschitz, e.g. has fractal boundary, the correct Sobolev space setting to pose the problem is not obvious, because many of the relations between the standard definitions of Sobolev spaces on subsets of Euclidean space (e.g. restriction, completion of spaces of smooth functions, interpolation...) that hold in the Lipschitz case, fail to hold in general. To extend the BIE framework to general screens, we study properties of the classical fractional Sobolev spaces (or Bessel potential spaces)on general non-Lipschitz subsets of Rn. In particular, we extend results about duality, s-nullity (whether a set with empty interior can support distributions with given Sobolev regularity), and about the equivalence or not between alternative space definitions, providing several examples. An interesting application is the approximation of variational problems posed on fractal sets by problems posed on prefractal approximations. This is a joint work with S.N. Chandler-Wilde (Reading) and D.P. Hewett (UCL).
Sobolev spaces on non-Lipschitz sets with application to boundary integral equations on fractal screensread_more
HG E 1.2
9 November 2016
16:15-17:15 		Prof. Dr. Philippe Ciarlet
City University Hong Kong
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Zurich Colloquium in Applied and Computational Mathematics

Title Continuity of a surface as a function of its fundamental forms: recent advances and applications
Speaker, Affiliation Prof. Dr. Philippe Ciarlet, City University Hong Kong
Date, Time 9 November 2016, 16:15-17:15
Location HG E 1.2
Abstract The fundamental theorem of surface theory asserts that a surface can be recovered from the knowledge of its two fundamental forms if these two forms satisfy the Gauss and Codazzi-Mainardi equations on a simply-connected open subset of the plane, in which case it is uniquely defined up to translations and rotations. While this well-known existence and uniqueness result is classically established in spaces of continuously differentiable functions, it has been recently extended to other function spaces, in particular to Sobolev spaces. A related question is to examine whether such a recovered surface is a continuous function of its fundamental forms. A first positive answer was given by the author when the spaces are those of continuously differentiable functions equipped with their Fr�chet topologies. More recently, it was shown in various works, by Liliana Gratie, Maria Malin, Cristinel Mardare, and the author, that such a continuity result holds as well when the immersions defining the surfaces belong to ad hoc Sobolev spaces. In this talk, we will briefly review these recent advances and also consider some of their potential applications, for instance to the intrinsic approach to nonlinearly elastic shell theory, where the fundamental forms of the unknown deformed middle surface of a shell are taken as the new unknowns.
Continuity of a surface as a function of its fundamental forms: recent advances and applicationsread_more
HG E 1.2
16 November 2016
16:15-17:15 		Prof. Dr. Istvan Gyongy
University of Edinburgh
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Zurich Colloquium in Applied and Computational Mathematics

Title On numerical solutions of parabolic stochastic PDEs given on the whole space
Speaker, Affiliation Prof. Dr. Istvan Gyongy, University of Edinburgh
Date, Time 16 November 2016, 16:15-17:15
Location HG E 1.2
Abstract Stochastic PDEs of nonlinear filtering are considered. These equations are given on the whole Euclidean space in the spatial variable. To solve them numerically we localise the equations onto large balls. This localisation reduces various kinds of spatial approximation schemes to finite dimensional problems. The error is shown to be exponentially small. As an application, a numerical scheme is presented which combines the localisation and the space and time discretisation, and thus is fully implementable. The talk is based on a joint work with Mate Gerencser.
On numerical solutions of parabolic stochastic PDEs given on the whole space read_more
HG E 1.2
23 November 2016
16:15-17:15 		Prof. Dr. Christiane Tretter
Universität Bern
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Zurich Colloquium in Applied and Computational Mathematics

Title Variational principles for eigenvalues in spectral gaps and applications
Speaker, Affiliation Prof. Dr. Christiane Tretter, Universität Bern
Date, Time 23 November 2016, 16:15-17:15
Location HG E 1.2
Abstract In this talk variational principles for eigenvalues in gaps of the essential spectrum are presented which are used to derive two-sided eigenvalue bounds. Applications of the results include the Klein-Gordon equation, even when complex eigenvalues occur, and spectral problems arising in the analysis of 2D photonic crystals.
Variational principles for eigenvalues in spectral gaps and applicationsread_more
HG E 1.2
30 November 2016
16:15-17:15 		Prof. Dr. Jinchao Xu
PennState University
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Zurich Colloquium in Applied and Computational Mathematics

Title A unified approach to the design and analysis of AMG
Speaker, Affiliation Prof. Dr. Jinchao Xu, PennState University
Date, Time 30 November 2016, 16:15-17:15
Location HG E 1.2
Abstract In this talk, I will present a general framework for the design and analysis of Algebraic or Abstract Multi-Grid (AMG) methods. Given a smoother, such as Gauss-Seidel or Jacobi, we provide a general approach to the construction of a quasi-optimal coarse space and we prove that under appropriate assumptions the resulting two-level AMG method for the underlying linear system converges uniformly with respect to the size of the problem, the coefficient variation, and the anisotropy. Our theory applies to most existing multigrid methods, including the standard geometric multigrid method, the classic AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG, and spectral AMGe.
A unified approach to the design and analysis of AMGread_more
HG E 1.2
7 December 2016
16:15-17:15 		Prof. Dr. Jens Markus Melenk
University of Technology Vienna
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Zurich Colloquium in Applied and Computational Mathematics

Title directional H^2-matrices for Helmholtz integral operators
Speaker, Affiliation Prof. Dr. Jens Markus Melenk, University of Technology Vienna
Date, Time 7 December 2016, 16:15-17:15
Location HG E 1.2
Abstract Boundary Element Methods (BEM) are an important tool for the numerical solution of acoustic and electromagnetic scattering problems. These BEM matrices are fully populated so that data-sparse approximations are required to reduce the complexity from quadratic to log-linear. For the high-frequency case of large wavenumber, standard blockwise low-rank approaches are insufficient. One possible data-sparse matrix format for this problem class that can lead to log-linear complexity are directional H^2-matrices. We present a full analysis of a specific incarnation of this approach. Directional H^2-matrices are blockwise low rank matrices, where the block structure is determined by the so-called parabolic admissibility condition. In order to achieve log-linear complexity with this admissibility condition, a nested multilevel structure is essential that provides a data-sparse connection between clusters of source and target points on different levels. We present a particular variant of directional H^2-matrices in which all pertinent objects are obtained by polynomial interpolation. This allows us to rigorously establish exponential convergence in the block rank in conjunction with log-linear complexity. The work presented here is joint with S.~B\"orm (Kiel).
directional H^2-matrices for Helmholtz integral operatorsread_more
HG E 1.2
14 December 2016
16:15-17:15 		Prof. Dr. Stefan Volkwein
Universität Konstanz
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Zurich Colloquium in Applied and Computational Mathematics

Title POD-Based Multicriterial Optimal Control by the Reference Point Method (joint work with S. Banholzer and D. Beermann)
Speaker, Affiliation Prof. Dr. Stefan Volkwein, Universität Konstanz
Date, Time 14 December 2016, 16:15-17:15
Location HG E 1.2
Abstract In the talk bicriterial optimal control problem governed by a parabolic partial differential equation (PDE) and bilateral control constraints is considered. For the numerical optimization the reference point method is utilized. The PDE is discretized by a Galerkin approximation utilizing the method of proper orthogonal decomposition (POD). POD is a powerful approach to derive reduced-order approximations for evolution problems. Numerical examples illustrate the efficiency of the proposed strategy.
POD-Based Multicriterial Optimal Control by the Reference Point Method (joint work with S. Banholzer and D. Beermann)read_more
HG E 1.2
15 December 2016
17:15-18:15 		Dr. Konstantinos Dareiotis
Uppsala University, Sweden
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Zurich Colloquium in Applied and Computational Mathematics

Title L^{infty}-estimates for Stochastic PDEs of parabolic type and applications
Speaker, Affiliation Dr. Konstantinos Dareiotis, Uppsala University, Sweden
Date, Time 15 December 2016, 17:15-18:15
Location HG E 1.2
Abstract We will present L^{\infty}-estimates for solutions of Stochastic PDEs (SPDEs) of parabolic type obtained by techniques motivated by the works of De Giorgi and Moser in the deterministic setting. The global estimates will then be applied in order to prove solvability for a class of semilinear SPDEs, while the local estimates will be used in order to obtain a weak Harnack-type inequality for solutions of linear equations, which is in turn used to deduce information about the oscillation of the solutions. The results are from joint work with Mate Gerencser (IST, Austria).
L^{infty}-estimates for Stochastic PDEs of parabolic type and applicationsread_more
HG E 1.2

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