Convolution Quadrature

This is one of two conversations which Gudrun Thäter recorded
alongside the conference Women in PDEs which took place at our
Department in Karlsruhe on 27-28 April 2017.


Maria Lopez-Fernandez from the University La Sapienza in Rome was
one of the seven invited speakers. She got her university degree
at the University of Valladolid in Spain and worked as an
academic researcher in Madrid and at the University of Zürich.


Her field of research is numerical analyis and in particular the
robust and efficient approximation of convolutions. The
conversation is mainly focussed on its applications to wave
scattering problems. The important questions for the numerical
tools are: Consistency, stability and convergence analysis. The
methods proposed by Maria are Convolution Quadrature type methods
for the time discretization coupled with the boundary integral
methods for the spatial discretization. Convolution Quadrature
methods are based on Laplace transformation and numerical
integration. They were initially mostly developed for parabolic
problems and are now adapted to serve in the context of
(hyperbolic) wave equations. Convolution quadrature methods
introduce artificial dissipation in the computation, which
stabilzes the numerics. However it would be physically more
meaningful to work instead with schemes which conserve mass.


She is mainly interested in


fast algorithms with

reduced memory requirements and

adaptivity in time and space.



The motivational example for her talk was the observation of
severe acoustic problems inside a new building at the University
of Zürich. Any conversation in the atrium made a lot of noise and
if someone was speaking loud it was hard to understand by the
others. An improvement was provided by specialised engineers who
installed absorbing panels. From the mathematical point of view
this is an nice application of the modelling and numerics of wave
scattering problems. Of course, it would make a lot of sense to
simulate the acoustic situation for such spaces before building
them - if stable fast software for the distribution of acoustic
pressure or the transport of signals was available.


The mathematical challenges are high computational costs, high
storage requirements and and stability problems. Due to the
nonlocal nature of the equations it is also really hard to make
the calculations in parallel to run faster. In addition
time-adaptive methods for these types of problems were missing
completely in the mathematical literature. In creating them one
has to control the numerical errors with the help of a priori and
a posteriori estimates which due to Maria's and others work
during the last years is in principle known now but still very
complicated. Also one easily runs into stability problems when
changing the time step size.


The acoustic pressure distribution for the new building in Zürich
has been sucessfully simulated by co-workers in Zürich and Graz
by using these results together with knowledge about the
sound-source and deriving heuristic measures from that in order
to find a sequence of time steps which keeps the problem stable
and adapt to the computations effectively.


There is a lot of hope to improve the performance of these tools
by representing the required boundary element matrices by
approximations with much sparser matrices.
References

M. López Fernández, S. Sauter: Generalized Convolution
Quadrature with Variable Time Stepping. Part II: Algorithm and
Numerical Results. Applied Numerical Mathematics, 94, pp. 88 -
105 (2015)

M. López Fernández, S. Sauter: Generalized Convolution
Quadrature based on Runge-Kutta Methods. Numerische Mathematik,
133 (4), pp. 734 - 779 (2016)

S. Sauter, M. Schanz: Convolution Quadrature for the Wave
Equation with Impedance Boundary Conditions. Journal of
Computational Physics, Vol 334, pp. 442 - 459 (2017)




Podcasts

T. Arens: Lärmschutz, Gespräch mit S. Ritterbusch im
Modellansatz Podcast, Folge 16, Fakultät für Mathematik,
Karlsruher Institut für Technologie (KIT), 2014.

F. Sayas: Acoustic Scattering, Conversation with G. Thäter in
the Modellansatz Podcast, Episode 58, Department of Mathematics,
Karlsruhe Institute of Technology (KIT), 2016.