**Scottish Numerical Methods Network 2020**

**Second (virtual) workshop: 12th June 2020**

Multiscale & Multilevel Methods

The meeting will take place via Zoom, hosted by ICMS .

Please register
here and we
will provide you with the meeting link on the morning
of the event.

**9:45-10:00 Meeting opens and opening remarks **

**10:00-10:45 Daniel Peterseim** (Augsburg) *Sparse Compression of Expected Solution Operators* [abstract]

**10:45-11:30 Jennifer Pestana ** (Strathclyde)

*Preconditioned iterative methods for multilevel Toeplitz matrices* [abstract]

**11:30-13:30** Break

**13:30-14:15 Lubomir Banas** (Bielefeld) *Finite element approximation of the stochastic total variation flow.* [abstract]

**14:15-14:45 Timo Sprekeler** (Oxford) *Numerical Homogenization of Nondivergence-Form Equations* [abstract]

**14:45-15:15 Jonna Roden** (Edinburgh) *PDE-Constrained Optimization for Multiscale Particle Dynamics* [abstract]

Discussion

**Abstracts:**

**Daniel Peterseim, Sparse Compression of Expected Solution Operators**

We show that the expected solution operator of prototypical linear
elliptic partial differential equations with random coefficients is well
approximated by a computable sparse matrix. This result is based on a
random localized orthogonal multiresolution decomposition of the
solution space that allows both the sparse approximate inversion of the
random operator represented in this basis as well as its stochastic
averaging. The approximate expected solution operator can be interpreted
in terms of classical Haar wavelets. When combined with a suitable
sampling approach for the expectation, this construction leads to an
efficient method for computing a sparse representation of the expected
solution operator which is relevant for stochastic homogenization and
uncertainty quantification.

**Jennifer Pestana, Preconditioned iterative methods for multilevel Toeplitz matrices**

Linear systems involving multilevel Toeplitz matrices arise in a number of
applications, e.g., the discretization of certain 2D and 3D partial
differential or fractional differential equations, and image debarring
problems. Krylov subspace methods are effective solvers for such problems,
and for symmetric multilevel Toeplitz matrices theoretical results and
efficient methods enable fast solvers with guaranteed convergence rates.
Here we describe how to obtain similar results for nonsymmetric multilevel
Toeplitz problems by employing a simple symmetrization. We discuss
theoretical properties of these symmetrized matrices and propose effective
preconditioners.

**Lubomir Banas, Finite element approximation of the stochastic total variation flow.**

We propose an energy preserving fully discrete finite element
approximation of the regularized stochastic total variation flow (STVF)
equation.
The problem can be interpreted a (regularized) stochastically perturbed
gradient flow of the total variation energy functional,
which is used, e.g., for image denoising and the modelling of damage
evolution. Due to its singular character the solution
of the stochastic total variation flow has to be formulated as a
stochastic variational inequality (SVI).
We show that the numerical solution converges to the SVI solution of the
STVF equation for vanishing discretization and regularization
parameters.
We also present numerical experiments to demonstrate the practicability
of the proposed numerical scheme.

**Timo Sprekeler, Numerical Homogenization of Nondivergence-Form Equations**

In this talk, we will discuss periodic homogenization problems of the form
subject to a homogeneous Dirichlet boundary condition. We derive
corrector estimates and propose, and
rigorously analyze, a numerical scheme based on finite element
approximations for such problems. If time allows,
we extend the results to the case of nonuniformly oscillating coefficients.
This is joint work with Yves Capdeboscq
(Université de Paris, CNRS, Sorbonne Université, LJLL) and Endre Süli
(University of Oxford).

**Jonna Roden, PDE-Constrained Optimization for Multiscale Particle Dynamics**

There are many industrial and biological processes, such as beer
brewing, nano-separation and bird flocking, which can be described by
integro-PDEs. These PDEs define the dynamics of a particle density
within a fluid bath, under the influence of diffusion, external forces
and particle interactions, and often include complex, nonlocal
boundary conditions.
A key challenge is to optimize these types of processes. For example,
in nano-separation, it is of interest to determine the optimal inflow
rate of particles (the control), which leads to high separation of the
particles (the target), at a minimal financial cost.
Mathematically, this requires tools from PDE-constrained optimization.
A standard technique is to derive a system of optimality conditions
and solve it numerically. Due to the nonlinear, nonlocal nature of the
governing PDE and boundary conditions, the optimization of multiscale
particle dynamics problems requires the development of new theoretical
and numerical methods.
I will present the system of nonlinear, nonlocal integro-PDEs that
describe the optimality conditions for such an optimization problem.
Furthermore, I will introduce a numerical method, which combines
pseudospectral methods with an optimization algorithm. This provides a
tool for the fast and accurate solution of these optimality systems.
Finally, some examples of future industrial applications will be
given. This is joint work with Ben Goddard and John Pearson.

Local organisers:

Lehel Banjai

Lyonell Boulton

Mariya Ptashnyk