**Scottish Numerical Methods Network 2019**

**Second workshop: ICMS, 16 May 2019**

Numerical analysis and modelling of non-local phenomena

16 May, 10:00 - 13:00

Note that ICMS has moved from its previous location to the new Bayes Centre

**09:30-09:45 Registration **

**09:45-10:00 Opening **

**10:00-10:45 Bangti Jin** (UCL) *Time-stepping schemes for fractional diffusion* [abstract]

**10:45-11:15 Katie Baker** (Heriot-Watt) *Numerical Methods for fractionally damped wave equations* [abstract]

**11:15-11:45** Coffee break

**11:45-12:30 Markus Melenk** (TU Vienna) *AFEM for the fractional Laplacian* [abstract]

**12:30-13:00 Gissell Estrada Rodriguez** (Heriot-Watt) *Nonlocal macroscopic limits of kinetic equations in biological and robotic systems* [abstract]

**13:00-14:30** Lunch break

16 May, 14:30 - 16:00

Note that ICMS has moved from its previous location to the new Bayes Centre

**14:30-15:15 Penny Davies** (Strathclyde) *The MRE inverse problem for the elastic shear modulus* [abstract]

**15:15-16:00 Ben Goddard** (Edinburgh)

* Pseudospectral methods for integro-PDEs* [abstract]

Discussion

This is an open meeting. If you are interested in attending this meeting, please email Lehel Banjai or Mariya Ptashnyk for more information about possible support for travel and accommodation.

Travel information is here.

**Abstracts:**

**Bangti Jin, Numerical analysis and modelling of non-local phenomena**

Overall the last decade, a large number of time stepping schemes have been developed for time-fractional di usion problems. These schemes can be generally divided into: finite difference type, convolution quadrature type and discontinuous Galerkin methods. Many of these methods are developed by assuming that the solution is suciently smooth, which however is generally not true. In this talk, I will describe our recent works in analyzing and developing robust numerical schemes that do not assume solution regularity directly, but only data regularity. The algorithms will be illustrated by numerical experiments.

**Katie Baker, Numerical Methods for fractionally damped wave
equations**

We study numerical quadrature methods that are used to model fractional wave
equations with improved efficiency and reduced computational complexity. The PDE we consider models high
intensity focused ultrasound (HIFU) and is given by a wave equation with a damping term defined by a
nonlocal fractional derivative. The numerical scheme that we use to generate solutions is derived from
finite elements in space and a combination of 2nd order central differences and
convolution quadrature (CQ) to discretize time. We present a fully discrete method and its analysis. We
also describe a fast implementation method and illustrate the methods by comparing
numerical results of the standard CQ scheme and the fast CQ.

**Markus Melenk, AFEM for the fractional Laplacian **

For the discretization of the integral fractional Laplacian \((-\Delta)^s\), \(0 <
s < 1\), based on piecewise linear functions, we present and analyze
a reliable weighted residual * a posteriori* error estimator.
In order to compensate for a lack of \(L^2\)-regularity of the residual
in the regime \(3/4 < s < 1\), this weighted residual error estimator includes
as an additional weight a power of the distance from the mesh skeleton.
We prove optimal convergence rates for an \(h\)-adaptive algorithm driven
by this error estimator in the framework of [Carstensen, Feischl, Page, Praetorius, axioms of adaptivity, CAMWA 2014]. Key to the analysis of the adaptive algorithm are novel local inverse estimates for the fractional Laplacian. These local inverse estimates have further applications. For example, they underlie the proof that multilevel diagonal scaling leads to uniformly bounded condition numbers even in the presence of locally refined meshes.

This is joint work with M. Faustmann and D. Praetorius.

**Gissell Estrada Rodriguez, Nonlocal macroscopic limits of kinetic equations in biological and robotic systems.**

This talk addresses the derivation of nonlocal partial differential equations from individual movement. More specifically, my main results concern how fractional diffusion and swarming behaviour arise as a limit of microscopic kinetic models for interacting particles. Applications of these results to biology and swarm robotic systems are discussed.

**Penny Davies, Deep Gaussian processes and applications in regression.**

Magnetic resonance elastography (MRE) is a powerful technique for noninvasive determination of the biomechanical properties of tissue, with important applications in disease diagnosis. A typical experimental scenario is to induce waves in the tissue by time-harmonic external mechanical osciillation and then measure the tissue's displacement at fixed spatial positions 8 times during a complete time-period, extracting the dominant frequency signal from the discrete Fourier transform in time. Accurate reconstruction of the tissue's elastic moduli from MRE data is a challenging inverse problem, and I will describe a new approach based on combining approximations at different frequencies into a single overdetermined system.

This is joint work with Eric Barnhill and Ingolf Sack (Department of Radiology, Charité-Universitätsmedizin, Berlin)

**Ben Goddard, Deep Gaussian processes and applications in regression.**

Motivated by applications in statistical mechanics and fluid dynamics,
we propose a novel, efficient pseudospectral collocation scheme for
non-local, non-linear, integro-PDEs. In particular, we compute the
non-local, integral terms in real space with the help of a specialised
Gauss quadrature. Due to the exponential accuracy of the quadrature, and
a convenient choice of collocation points near interfaces, we can use
grids with a significantly lower number of nodes than most other
reported methods. In addition, the method can be directly applied to
both (non-periodic) bounded and unbounded domains. We demonstrate the
effectiveness of the scheme by applying it to examples from soft matter
and complex fluids.

Local organisers:

Lehel Banjai,
Mariya Ptashnyk