Georg Maierhofer
Henslow Research Fellow

About Me

I am a Henslow Research Fellow at Clare Hall within the University of Cambridge. I am affiliated with the Applied and Computational Analysis research group in the Department of Applied Mathematics and Theoretical Physics (DAMTP). My research interests cover a range of topics in applied and numerical analysis from wave scattering, over highly-oscillatory phenomena to low-regularity and geometric integration of time evolution equations. For more details please see my research page.

In addition, I am also one of the group coordinators for the special interest group on Mathematical Analysis in Acoustics through the UK Acoustics Network Plus.

Previously I was a Marie Skłodowska-Curie postdoctoral fellow at Sorbonne Université (formerly Pierre and Marie Curie University) in Prof. Katharina Schratz's research group. My fellowship project was entitled GLIMPSE and concerned the development and study of Geometric and Low-regularity Integrators for the Matching and Preservation of Structure in the computation of dispersive Equations. I completed my PhD as a research scholar at Trinity College within the University of Cambridge under the supervision of Prof. Nigel Peake and Prof. Arieh Iserles.

Resume

A more detailed CV is available upon request.

  • Employment

  • Henslow Research Fellow, Clare Hall

    Nov 2023 - present

    University of Cambridge, UK

  • Marie Skłodowska-Curie Fellow, Laboratoire Jacques-Louis Lions

    May 2022 - Oct 2023

    Sorbonne Université, France

  • Postdoctoral Researcher, Laboratoire Jacques-Louis Lions

    Sep 2021 - April 2022

    Sorbonne Université, France

  • Education

  • University of Cambridge, Trinity College, UK

    2017 - 2021

    PhD student in the Cambridge Centre for Analysis (CCA)

  • University of Cambridge, Trinity College, UK

    2013 - 2017

    MMath (Part III in Mathematics) & BA (hons) in Mathematics

  • Honors and Awards

  • SIAM CS&E Hackathon Challenge Winner

    Feb 2023

    SIAM Conference on Computational Science and Engineering, Netherlands

  • Marie Skłodowska-Curie Postdoctoral Fellowship

    May 2022 - Oct 2023

    Marie Skłodowska-Curie Actions, European Commission

  • MathInGreaterParis Fellowship

    Awarded Jan 2022

    Fondation Sciences Mathematiques de Paris & European Commission

  • Junior Research Leader at the Simons Semester 'Around transport and diffusion phenomena'

    Dec 2021

    Institute for Mathematics, Polish Academy of Sciences & Simons Foundation

  • SIAM CS&E BGCE Prize Finalist

    Mar 2021

    Bavarian Graduate School of Computational Engineering, Germany

  • Rouse Ball Travelling Studentship in Mathematics

    Mar 2020

    Trinity College, University of Cambridge, UK

  • Smith-Knight & Rayleigh-Knight Prize

    Mar 2019

    Faculty of Mathematics, University of Cambridge, UK

  • Election to Research Scholar

    Oct 2017

    Trinity College, University of Cambridge, UK

  • Election to Senior Scholar

    Oct 2016

    Trinity College, University of Cambridge, UK

  • Undergraduate Research Bursary

    June 2016

    London Mathematical Society, UK

  • Winner TakeAIM competition

    Nov 2015

    Smith Institute, UK

  • Election to Junior Scholar

    Oct 2014

    Trinity College, University of Cambridge, UK

  • High School Student Award

    Sept 2013

    Austrian Mathematical Society, Austria

Research

My main research interests focus on problems in computational mathematics for partial differential equations and the study of waves. In more detail I have worked on the following topics:

Numerical methods for evolution equations

A large part of my current ongoing research concerns the development and study of efficient numerical methods for time evolution equations. I have a particular interest in dispersive equations which arise for instance in the study of water waves, ferromagnetism and relativistic wave equations in particle physics and their approximation in low-regularity regimes. In this context I work with splitting methods, resonance-based methods and exponential integrators, and seek to better understand their properties including convergence rates for low-regularity data, cost and their efficient numerical implementation.

Gaussian beam scattering
Figure 1: Evolution of a low-regularity vortex filament in an ideal fluid (more details)

Geometric numerical integration

A particular focus of my recent research lies in the development of structure preserving algorithms in the context of dispersive Hamiltonian PDEs and resonance-based methods. In this work I aim to further develop theoretical guarantees on the behaviour of such methods including exact preservation of first integrals and long-time behaviour.

Gaussian beam scattering
Figure 2: Long-time behaviour of actions at non-resonant and resonant time steps (more details)

Analysis of collocation methods for Fredholm integral equations

In collaboration with Prof. Daan Huybrechs at KU Leuven, we are studying the theoretical properties of collocation methods for integral equations that arise in wave scattering. In particular we are interested in understanding the effect of the location and the number of collocation points on the error of the approximate solution. This work is motivated by results from signal analysis and approximation theory, where it was found that oversampling allows for improved robustness to redundancies in the approximation spaces. The understanding of the strengths and limitations of collocation methods is especially relevant to practical applications since in collocation methods are often easier to implement and cheaper to compute than their traditionally stabler counterpart, the Galerkin method.

Oversampled collocation methods
Figure 3: Improved convergence rates in oversampled collocation methods (more details)

Computational high-frequency wave propagation

High-frequency wave scattering problems are beyond the reach of most conventional methods for numerical wave propagation due to the infeasible computational cost that large frequencies would require. As a result a significant amount of recent research has focussed on the development of hybrid numerical-asymptotic (HNA) methods, which are robust even for large frequencies and achieve in many cases a near frequency independent solution time. The underlying idea is to incorporate physical knowledge into numerical schemes to enhance quality and efficiency in the high-frequency regime, while preserving the precision and flexibility of a grid based approach. An essential step in boundary HNA methods is the numerical evaluation of highly oscillatory integrals. My current work focusses on the development of (provably) efficient quadrature methods for the specific types of oscillators and singularities encountered in boundary integral methods for computational wave scattering.

Gaussian beam scattering
Figure 4: Scattering of a high-frequency Gaussian beam (more details)

Wiener-Hopf method and applications

The Wiener-Hopf method is a very successful technique for solving certain boundary value problems which are relevant in acoustics, electromagnetic theory, hydrodynamics and elasticity. Typically the method exploits analyticity properties of Fourier half-line transforms combined with Liouville’s theorem to arrive at an explicit form of solution. I am currently applying this method to a specific model problem which arises from the need to understand noise development and propagation in turbofan engines.

The Wiener-Hopf method is a very successful technique for solving certain boundary value problems which are relevant in acoustics, electromagnetic theory, hydrodynamics and elasticity. Typically the method exploits analyticity properties of Fourier half-line transforms combined with Liouville’s theorem to arrive at an explicit form of solution. I am currently applying this method to a specific model problem which arises from the need to understand noise development and propagation in turbofan engines.

Gaussian beam scattering
Figure 5: Total outgoing acoustic power for cascade scattering (more details)

Publications

An accelerated Levin–Clenshaw–Curtis method for the evaluation of highly oscillatory integrals.
Maierhofer, G., Iserles, A.
In preparation.

Symmetric resonance based integrators and forest formulae.
Alama Bronsard, Y., Bruned, Y., Maierhofer, G., Schratz, K.
Under review.
Download: preprint

Bridging the gap: symplecticity and low regularity in Runge-Kutta resonance-based schemes.
Maierhofer, G., Schratz, K.
Under review.
Download: preprint

Long-time error bounds of low-regularity integrators for nonlinear Schrödinger equations.
Feng, Y., Maierhofer, G., Schratz, K.
Accepted to Mathematics of Computation.
Download: preprint

Numerical integration of Schrödinger maps via the Hasimoto transform.
Banica, V., Maierhofer, G., Schratz, K.
Accepted to SIAM Journal on Numerical Analysis.
Download: preprint

Recursive moment computation in Filon methods and application to high-frequency wave scattering in two dimensions
Maierhofer, G., Iserles, A., Peake, N.
IMA Journal of Numerical Analysis.
Download: published version | preprint

An analysis of least-squares oversampled collocation methods for compactly perturbed boundary integral equations in two dimensions.
Maierhofer, G., Huybrechs, D.
Journal of Computational and Applied Mathematics (2022).
Download: published version | preprint

Convergence analysis of oversampled collocation boundary element methods in 2D
Maierhofer, G., Huybrechs, D.
Advances in Computational Mathematics (2022).
Download: published version | preprint

Acoustic and hydrodynamic power of wave scattering by an infinite cascade of plates in mean flow
Maierhofer, G., Peake, N.
Journal of Sound and Vibration (2021).
Download: published version | preprint

Wave scattering by an infinite cascade of non-overlapping blades
Maierhofer, G., Peake, N.
Journal of Sound and Vibration (2020).
Download: published version | preprint

Learning the Sampling Pattern for MRI
Sherry, F., Benning, M., De los Reyes, J. C., Graves, M. J., Maierhofer, G., Williams, G., Schönlieb, C.-B. and Ehrhardt, M.
IEEE Transactions on Medical Imaging (2020).
Download: published version | preprint

Mirror, Mirror, on the Wall, Who’s Got the Clearest Image of Them All? — A Tailored Approach to Single Image Reflection Removal
Heydecker, D., Maierhofer, G., Aviles-Rivero, A. I., Fan, Q., Chen, D., Schönlieb, C.-B. and Süsstrunk, S.
IEEE Transactions on Image Processing (2019).
Download: published version | preprint

Peekaboo - Where are the Objects? Structure Adjusting Superpixels.
Maierhofer, G., Heydecker, D., Aviles-Rivero, A. I., Alsaleh, S. M. and Schönlieb, C.-B.
25th IEEE International Conference on Image Processing (2018).
Download: published version | preprint

An extension of standard Latent Dirichlet Allocation to multiple corpora
Foster, A., Li, H., Maierhofer, G., Shearer, M.
SIAM Undergraduate Research Online Volume 9 (2016).
Download: published version

Geometric Measure of Arens Irregularity
Hernandez Palomares, R., Hu, E., Maierhofer, G. A., Rao, P.
Fields Institute for Research in Mathematical Sciences (2015).
Download: published version

Code

On this page you can find links to code associated with some of my research. If you have any questions regarding this, would like a demonstration or for suggestions of improvement and collaboration please feel free to get in touch.

Geometric low-regularity integrators (GLIMPSE)

This is part of my ongoing interest in the study of structure preserving low-regularity integrators for dispersive nonlinear equations which commenced with the Marie Skłodowska-Curie postdoctoral fellowship GLIMPSE. The theoretical results of this research are detailed in several publications which can be found here.

The associated code can be downloaded here: GitHub repository

Oversampling in boundary integral equations

In collaboration with Prof. Daan Huybrechs we have studied the effects of oversampling in collocation methods for boundary integral equations. A theoretical analysis describing the favorable properties of oversampling in this context can be found in two recent publications which are available here.

The associated code can be downloaded here: GitHub repository

Selected talks

  • Invited research talks

  • Structure-preserving low-regularity integrators for dispersive nonlinear equations

    September 2023

    Workshop on Numerical Analysis of Partial Differential Equations, PolyU, Hong Kong

  • Computing Schrödinger maps using the Hasimoto transform

    August 2023

    Minisymposium on Numerical methods and analysis for dispersive PDEs, ICOSAHOM, South Korea

  • Analysis of oversampled collocation methods for wave scattering problems

    February 2023

    Canonical scattering problems workshop, Isaac Newton Insitute, UK

  • A structure preserving low-regularity integrator for the Korteweg–De Vries equation

    September 2022 | Invited minisymposium speaker

    CMAM 2022, Vienna University of Technology, Austria

  • Structure preserving low-regularity integrators for the Korteweg–De Vries and the nonlinear Schrödinger equations

    July 2022

    30th Birthday of Acta Numerica, Będlewo, Poland

  • Highly oscillatory quadrature and low-regularity integrators for nonlinear evolution equations

    June 2022

    AMAC Seminar, Université Grenoble Alpes, France

  • Structure preserving low-regularity integrators for dispersive nonlinear partial differential equations

    May 2022

    NUMA Seminar, KU Leuven, Belgium

  • Highly oscillatory quadrature and low-regularity integrators for nonlinear dispersive equations

    April 2022

    Numerical Analysis Seminar, University of Bath, UK

  • Numerical methods in acoustics and nonlinear dispersive partial differential equations

    March 2022

    Waves group meetings, University of Cambridge, UK

  • Analysis of oversampled collocation methods for Friedholm integral equations

    February 2022

    Workshop on Nonlinear Waves and Hamiltonian PDE’s, La Thuile, Italy

  • How many observations are enough? - Convergence analysis of least-squares oversampled collocation methods for Fredholm integral equations

    December 2021

    Institute for Mathematics, Polish Academy of Sciences

    October 2021

    Workshop – EPSRC Project on Transfer Operator Methods, University of Nottingham, UK | Link to recording

  • Convergence analysis of least-squares oversampled collocation for boundary element methods

    August 2021

    Numerical Analysis Workshop with Applications in Acoustics, Dorset, UK

  • A general method for moment computation in Filon methods

    March 2021

    8th BGCE Student Paper Prize Finalist Symposium, SIAM CS&E Conference

  • Highly oscillatory quadrature for integrals with singularities

    December 2019

    Cambridge-Imperial Computational PhD Seminar, Imperial College London

  • Outreach talks

  • How Mathematics can help you sleep at night... - Overview of research in the Mathematical Analysis in Acoustics SIG

    December 2021 | Part of organising committee

    First UKAN+ Connecting SIGs event, virtual

  • Maths makes waves - My experience with TakeAIM and Applied Mathematics

    February 2020 | Invited talk

    Smith Institute TakeAIM awards ceremony, Imperial College London

Contact Information

  • Dr Georg Maierhofer
    Clare Hall
    Herschel Road
    CB3 9AL Cambridge
    United Kingdom
  • E-mail:
    gam37[at]cam.ac.uk

  • Phone:
    +44 1223 332360

Contact Map