Allan Pinkus

Modulated Fourier Expansions for Continuous and Discrete Oscillatory Systems

This article reviews some of the phenomena and theoretical results on the long-time energy behaviour of continuous and discretized oscillatory systems that can be explained by modulated Fourier expansions: longtime preservation of total and oscillatory energies in oscillatory Hamiltonian systems and their numerical discretisations, near-conservation of energy and angular momentum of symmetric multistep methods for celestial mechanics, metastable energy strata in nonlinear wave equations. We describe what modulated Fourier expansions are and what they are good for.

Foundations of computational mathematics : Santander 2005

1. Non universal polynomial equation solving C. Beltran and L. M. Pardo 2. Toward accurate polynomial evaluation in rounded arithmetic James Demmel, Ioana Dumitriu and Olga Holtz 3. Sparse grids for higher dimensional problems Michael Griebel 4. Long-time energy conservation Liviu I. Ignat and Enrique Zuazua 5. Dispersive properties of numerical schemes for NSE Ernst Hairer 6. Eigenvalues and nonsmooth optimization Adrian Lewis 7. Discrete noether theorems E. L. Mansfield 8. Hyperbolic 3-manifolds and their computational aspects G. Robert Meyerhoff 9. Smoothed analysis of algorithms and heuristics A. Spielman and Shang-Hua Teng 10. High-dimensional transport-dominated diffusion problems Endre Suli 11. Greedy approximations V. N. Temlyakov.

Foundations of Computational Mathematics, Hong Kong 2008

This volume is a collection of articles based on the plenary talks presented at the 2008 Society for the Foundations of Computational Mathematics meeting in Hong Kong. The talks were given by some of the foremost world authorities in computational mathematics. The topics covered reflect the breadth of research within the area as well as the richness and fertility of interactions between seemingly unrelated branches of pure and applied mathematics. As a result, this volume will be of interest to researchers in the field of computational mathematics and also to non-experts who wish to gain some insight into the state of the art in this active and significant field.

Through the Kaleidoscope: Symmetries, Groups and Chebyshev-Approximations from a Computational Point of View

In this paper we survey parts of group theory, with emphasis on structures that are important in design and analysis of numerical algorithms and in software design. In particular, we provide an extensive introduction to Fourier analysis on locally compact abelian groups, and point towards applications of this theory in computational mathematics. Fourier analysis on non-commutative groups, with applications, is discussed more briefly. In the final part of the paper we provide an introduction to multivariate Chebyshev polynomials. These are constructed by a kaleidoscope of mirrors acting upon an abelian group, and have recently been applied in numerical Clenshaw–Curtis type numerical quadrature and in spectral element solution of partial differential equations, based on triangular and simplicial subdivisions of the domain.

Foundations of computational mathematics, Budapest 2011

Preface List of contributors 1. The state of the art in Smales 7th problem C. Beltran 2. The shape of data Gunnar Carlsson 3. Upwinding in finite element systems of differential forms S. H. Christiansen 4. On the complexity of computing quadrature formulas for SDEs S. Dereich, T. Muller-Gronbach and K. Ritter 5. The quantum walk of F. Riesz F. A. Grunbaum and L. Velazquez 6. Modulated Fourier expansions for continuous and discrete oscillatory systems E. Hairer and Ch. Lubich 7. The dual role of convection in 3D Navier-Stokes equations T. Y. Hou, Z. Shi and S. Wang 8. Algebraic and differential invariants Evelyne Hubert 9. Through the kaleidoscope: symmetries, groups and Chebyshev-approximations from a computational point of view H. Munthe-Kaas, M. Nome and B. N. Ryland 10. Sage: creating a viable free open source alternative to Magma, Maple, Mathematica and MATLAB W. Stein.

The Shape of Data

Many interesting problems in the study of data can be interpreted as questions about the “shape” of the data. For example, the existence of a cluster decomposition of a data set can be viewed as an aspect of its shape, as can the presence of loops and higher dimensional features. These shape theoretic aspects are important in identifying conceptually coherent groups within a data set, or perhaps the presence of periodic or recurrent behavior. Topology can be characterized as the study of shape, including both questions about how to represent shape efficiently as well as how to measure it, in a suitable sense. Over the last decade, there has been an effort to adapt the methods of topology to the study of data, so that one can become more precise about the shape theoretic aspects. I will talk about some of these developments, with examples. For further information and inquiries about building access for persons with disabilities, please contact Dan Moreau at 773.702.8333 or send him an email at dmoreau@galton.uchicago.edu. If you wish to subscribe to our email list, please visit the following website: https://lists.uchicago.edu/web/arc/statseminars.