Scottish Computational
Mathematics Symposium

Speakers:

Titles and Abstracts:

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We present convergence results for discontinuous collocation, Galerkin and quadrature Galerkin approximations for first kind convolution Volterra integral equations with a smooth kernel.

In the 1970s Brunner proved that discontinuous collocation schemes for these equations converge iff a parameter (which depends on the collocation points) lies in the interval [-1,1], and schemes for which the parameter is zero exhibit local superconvergence of one order higher.

However, it is now known that local superconvergence is only possible for discontinuous Galerkin approximations when a particular derivative of the exact solution is zero at time t = 0. New results enable us to explain the connection between these seemingly unrelated superconvergence properties.

This is joint work with Hermann Brunner and Dugald Duncan.

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In convection-diffusion problems, approximations obtained by standard numerical methods are known to be polluted with spurious (unphysical) oscillations when diffusion is small and grids are not sufficiently (unpractically) fine. Among the most successful techniques to overcome this situation is the use of Shishkin meshes. However, they present some drawbacks such as the difficulty to design them on domains with nontrivial boundaries. In the present work, by analyzing the numerics of Shishkin meshes we devise a method to simulate them that can be applied on complex domains. Numerical results show the new technique obtains approximations which, for practical purposes, are oscillation-free. This is the case even on very irregular grids completely unrelated to Shishkin meshes.

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The digital revolution is generating novel large scale examples of connectivity patterns that change over time; this scenario may be formalized as a graph with a fixed set of nodes whose edges switch on and off. For example, we may have networks of interacting Facebookers or Tweeters. To understand and quantify the key properties of such evolving networks, we can extend classical graph theoretical notions like degree and pathlength, and more general network science concepts such as centrality, to the dynamic setting. In this talk I will focus on linear algebra-based ideas and show that computations based on matrix products can capture various aspects of information flow around an evolving network. Illustrative examples will be given for both synthetic and real-world data sets.

This is joint work with Ernesto Estrada and Peter Grindrod.

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Euler's numerical integration method is still frequently used for pedagogical purposes. For instance Dave Griffiths and I chose it as our first example in a 1986 paper on modified equations which contributed to revitalise an idea that had been around since the 1950's and was to become commonplace in the 1990's. In this talk I shall focus on the question: did Euler know very much about Euler's method?

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The necessity for reliable and efficient incompressible flow solvers is widely recognised. We will focus on two components of such a solver: the error control used for self-adaptive time stepping; and the linear solver used at each time level. Conventional codes typically use semi-implicit time integration leading to a Poisson or Stokes-type system at every time step, but with a stability restriction on the time step. Our alternative approach is a stable version of the TR-AB2 smart integrator originally developed by Gresho in the 1980's. Such fully-implicit time integration methods have no restriction on the time step, but have only become feasible in the last five years because of developments in solution techniques for the linear (or linearized) Oseen systems that arise at each time level. To this end, the preconditioning framework that we propose offers the possibility of fast convergence that is robust with respect to the problem parameters (namely; the mesh size, the time step and the Reynolds number). Our preconditioning framework can be readily extended to multi-physics models that include the Navier-Stokes equations as a component. We illustrate this by extending the methodology to the case of the Boussinesq model of buoyancy driven flow.

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The hybrid Monte Carlo (HMC) method uses Hamiltonian mechanics, with randomized momentum coordinates, as the basis for algorithms designed to sample a pdf, in the position coordinates. I will overview recent work concerning the design and analysis of HMC methods for the sampling of high or infinite dimensional probability measures. In particular I will show
(i) how the standard HMC method should be scaled for a wide class of high dimensional problems; and
(ii) how to circumvent these scaling issues for problems where the target measure has a density with respect to a Gaussian measure.

The work is in collaboration with Alex Beskos, Natesh Pillai, Frank Pinski, Gareth Roberts and Chus Sanz-Serna. For details see:

• Optimal tuning of the Hybrid Monte-Carlo Algorithm
• Hybrid Monte-Carlo on Hilbert Spaces

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The numerical approximation of Partial Differential Equation (PDE) problems leads typically to large dimensional linear or linearised systems of equations. For problems where such PDEs provide only a constraint on an Optimization problem (so-called PDE-constrained Optimization problems), the systems are many times larger in dimension.

We will discuss the solution of such problems by preconditioned iterative techniques in particular where the PDEs in question arise from models describing incompressible fluid flow.