Tuesday 30 June - Friday 3 July 2009

Capstone Conference

Organiser: Andrew Stuart

ABSTRACTS of Plenary Talks

Franco Brezzi (IMATI-CNR and IUSS-Pavia) Recent perspectives on Discontinuous Galerkin methods

Roughly speaking, Discontinuous Galerkin methods aim at finding the approximate solution of a boundary value problem for PDE looking, a priori, among piecewise polynomial functions that might be discontinuous in the passage from one element to the neighboring one.

Recently DG methods have been applied to several types of problems (from the "original" conservation laws to continuous mechanics to electromagnetic problems) and analyzed from several perspectives (from hybridizable methods to weighted residuals).

The talk will start with a brief glance at the whole panorama, and then concentrate on pros and cons of the least squares approach in several applications.

Emmanuel Candes (CalTech) Fast Algorithms for the Computation of Oscillatory Integrals

This talk introduces a novel multiscale algorithm for the numerical evaluation of Fourier integral operators whose overall structure follows that of the butterfly algorithm. This is of interest for such fundamental computations are connected with the problem of finding numerical solutions to wave equations, and also frequently arise in many applications including reflection seismology, curvilinear tomography and others. This algorithm is accurate, is highly efficient in terms of memory requirement, and runs in O(N log N) flops where N is the number of input/output components, thus enjoying near-optimal computational complexity.

This is joint work with Laurent Demanet and Lexing Ying.

Roland Freund (UC Davis) Krylov subspace-based dimension reduction of large-scale linear dynamical systems

Krylov subspace methods can be used to generate Pade approximants of the transfer functions of linear dynamical systems. Although this Krylov-Pade connection has been known for a long time, somewhat surprisingly, it was not until the 1990s that this connection was exploited to devise practical numerical procedures for dimension reduction of large-scale linear dynamical systems.

In this talk, we first describe a problem arising in the simulation of electronic circuits that had triggered the interest in Krylov subspace-based dimension reduction in the 1990s. We then present an overview of the current state-of-the art of this class of dimension reduction algorithms. In particular, we focus on recent advances in structure-preserving reduction techniques. Finally, we describe a number of open problems in Krylov subspace-based dimension reduction.

Tom Hou (CalTech) The interplay between computation and analysis in the study of 3D incompressible flows

Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems for both computation and analysis. We review some recent theoretical and computational studies of the 3D Euler equations which show that there is a subtle dynamic depletion of nonlinear vortex stretching due to local geometric regularity of vortex filaments. The local geometric regularity of vortex filaments can lead to tremendous cancellation of nonlinear vortex stretching. We also investigate the stabilizing effect of convection in 3D incompressible Euler and Navier-Stokes equations. The convection term is the main source of nonlinearity for these equations. It is often considered destabilizing although it conserves energy due to the incompressibility condition. Here we reveal a surprising nonlinear stabilizing effect that the convection term plays in regularizing the solution. Finally, we present a new class of solutions for the 3D Euler and Navier-Stokes equations, which exhibit very interesting dynamic growth property. By exploiting the special structure of the solution and the cancellation between the convection term and the vortex stretching term, we prove nonlinear stability and the global regularity of this class of solutions.

Rupert Klein (Berlin) Challenges in modelling and computing planetary scale atmospheric flows at high resolution

Current production General Circulation Models (GCMs) for global atmospheric flow simulations are based on the Hydrostatic Primitive Equations (HPEs). These result from the full three-dimensional compressible flow equations in the limit of large horizontal-to-vertical scale ratios. While this asymptotic limit suppresses vertically propagating sound waves, it does support long-wave horizontally travelling acoustic modes, which are called "Lamb waves" and are considered important.

In contrast, current computational simulations of small-scale atmospheric processes, such as cloud formation, local storms, or pollutant transport on city-scales, are based on analogues of the classical incompressible flow equations, i.e., they are "sound-proof".

Modern high-performance computing hardware is beginning to allow atmospheric flow modellers to use grids with horizontal spacing in the range of merely a few kilometres. At such high resolution, the hydrostatic approximation breaks down, and one enters the scale range of sound-proof model applications.

In this lecture, I will elucidate why simply resorting to solving the full compressible flow equations without approximation on high-resolution grids is not as straight- forward as it may seem, I will explain numerical techniques designed to address the ensuing issues, and I will summarize recently proposed sets of multiscale model equations that are designed to capture both regimes at the continuum level while maintaining the advantages of the established single-scale equation sets.

T. LeLievre (ENPS) Sampling problems in molecular dynamics

One aim of molecular dynamics is to compute some macroscopic quantities from microscopic models, through means of some relevant observables in an appropriate thermodynamic ensemble. These calculations are typically difficult because the measure to sample is highly metastable, which makes classical Markov Chain Monte Carlo algorithms very slow to converge. We review some classical methods which are used in molecular dynamics to deal with this problem, with an emphasis on adaptive importance sampling methods.Applications of the method to other field (such as Bayesian statistics) are mentioned.

Caroline Lasser (Berlin) Semiclassical approximations for quantum molecular dynamics

In almost every molecular system electrons and nuclei interact considerably, and the mathematics of molecular dynamics has to start from the basic model of non-relativistic quantum mechanics, the time-dependent linear Schrödinger equation. We review semiclassical approximation schemes for the high-frequency regime, emphasizing the phase space point of view.

Michael Overton (New York) Characterization and construction of the nearest defective matrix via coalescence of pseudospectra (coauthors are R. Alam, S. Bora and R. Byers)

( pdf of Abstract)

Mike Shelley (Courant) Microscale instability and mixing in driven and active complex fluids

Complex fluids are fluids whose suspended microstructure feeds back and influences the macroscopic flow. A well-known example is a polymer suspension and a novel one is a bacterial bath wherein many swimming micro-organisms interact with each other through the surrounding fluid. In either case, these systems can display very rich dynamics even at system sizes where inertia is negligible, and both systems have important applications to micro-fluidic mixing and transport. They are also very difficult to model and understand, as the micro- and macro-scales are intimately coupled, and making progress can require approaches that span the particle and continuum scales. I will discuss two examples motivated by recent experimental observations. In the first, I discuss numerical studies using classical nonlinear PDE models of viscoelastic flow at low Reynolds number. Using the extensional flow geometry of a four-roll mill, we have found symmetry breaking instabilities that give rise to multiple frequencies of flow oscillation, the appearance of coherent structures, and fluid mixing driven by small-scale vortex creation and destruction. In the second example, I will discuss recent modeling and simulations of active suspensions made up of many swimming particles. We find that such systems can stable or unstable at large-scales, depending upon the micro-mechanical swimming mechanism, and if unstable the flows have coherent structures whose stretch-fold dynamics yields rapid mixing.

Anna-Karin Tornberg (Stockholm) Model reduction for efficient simulation of fiber suspensions

There is a strong anisotropy in the motion of slender rigid fibers. This anisotropy contributes to the very rich and complex dynamical behavior of fiber suspensions. The forming of "clusters" or "flocs" are purely three dimensional phenomena, and the direct simulation of these problems require simulations with many fibers for long times.

Earlier, we have developed a numerical algorithm to simulate the sedimentaion of fiber suspensions, considering a Stokes flow, for which boundary integral formulations are applicable. The algorithm is based on a non-local slender body approximation that yields a system of coupled integral equations, relating the forces exerted on the fibers to their velocities, which takes into account the hydrodynamic interactions of the fluid and the fibers. Even though there is a great gain in reducing a three-dimensional problem to a system of one-dimensional integral equations, the simulations are still computationally expensive, and the code has been parallelized and run on large computers to allow for more fibers and longer simulationtimes.

In this talk, I will present a model where approximations have been made to reduce the computational cost. Modifications have mainly been made concerning computation of long range interactions of fibers. The cost is substantially reduced by e.g. adaptively truncated force expansions and the use of multipole expansions combined with analytical quadrature.

I will present results from various simulations and discuss the accuracy of the new model as I compare these results to results from large parallel simulations with the full model. A substantial reduction of the computational effort is normally attained, and the computational cost may comprise only a small fraction of the cost of the full model. This is however affected by parameters of the problem, such as the geometry and the fiber concentration, as will be discussed.

Eric Vanden Eijnden (Courant) Modeling and simulation of reactive events

The dynamics of a wide range of systems involve reactive events, aka activated processes, such as conformation changes of macromolecules, nucleation events during first-order phase transitions, chemical reactions, bistable behavior of genetic switches, or regime changes in climate. The occurrence of these events is related to the presence of dynamical bottlenecks of energetic and/or entropic origin which effectively partition the phase-space of the dynamical system into metastable basins. The system spends most of its time fluctuating within these long-lived metastable states and only rarely makes transitions between them. The reactive events often determine the long-time evolution of the system one is primarily interested in. Unfortunately, computing up to the time scale at which these events occur represent an enormous challenge and so there is an urgent need for developing new numerical tools for such computations. One possibility is to build these numerical tools based on the Freidlin-Wentzell theory of large deviations which provides a complete picture of how and when rare events occurs in certain classes of systems. However, large deviation theory is valid in a parameter range where the random noise affecting the system is very small, which is often an inadequate assumption in complex systems. In addition, it becomes cumbersome to build numerical tools directly on Freidlin-Wentzell large deviation theory when the rare reactive events involves intermediates states, multiple pathways, etc. which is also the typical situation in high-dimensional systems. In this talk, I will explain why and describe a framework which allows to go beyond large deviation theory and can be used to identify the pathways and rate of rare reactive events in situations where the noise is not necessarily small, there are multiple pathways, etc. I will also describe numerical tools that can be built on this framework. Finally I will illustrate these tools on a selection of examples from molecular dynamics and material sciences.

D.J. Wilkinson (Newcastle) Stochastic modelling and Bayesian inference for biochemical network dynamics

This talk will provide an overview of computationally intensive methods for stochastic modelling and Bayesian inference for problems in computational systems biology. Particular emphasis will be placed on the problem of inferring the rate constants of mechanistic stochastic biochemical network models using high-resolution time course data, such as that obtained from single-cell fluorescence microscopy studies. The computational difficulties associated with "exact" methods make approximate techniques attractive. There are many possible approaches to approximation, including methods based on diffusion approximations, and methods exploiting stochastic model "emulators". Other important inferential problems will also be considered, such as inferring dynamic network connectivity from time course microarray data.

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See also:
Mathematics Research Centre
Mathematical Interdisciplinary Research at Warwick (MIR@W)
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