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:

Machine learning and hybrid methods for scientific computing

Differential equations are cornerstone of modern scientific modelling and exploration. The last several decades have seen the development of increasingly successful and sophisticated techniques for the approximation of solutions to differential equations, i.e. for scientific computing. In recent years machine learning techniques have shown significant promise to help support these developments, by providing efficient and versatile tools for simulation. In my recent work, I am interested in studying the use of such modern techniques in the simulation of differential equations, for example in the context of adaptive meshing and fast solvers for time-evolution equations.

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.

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.

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.

Highly oscillatory quadrature and 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.

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.

Dimension-independent backward error analysis for the midpoint rule applied to the nonlinear Schrödinger equation.
Faou, E., Maierhofer, G., Schratz, K.
In preparation.

Explicit symmetric low-regularity integrators for nonlinear Schrödinger equations.
Feng, Y., Maierhofer, G., Wang, C.
In preparation.

A fast neural hybrid Newton solver adapted to implicit methods for nonlinear dynamics.
Jin, T., Maierhofer, G., Schratz, K., Xiang, Y.
Under review.
Download: preprint

G-Adaptive mesh refinement - leveraging graph neural networks and differentiable finite element solvers.
Rowbottom, J., Maierhofer, G., Deveney, T., Schratz, K., Lio, P., Schönlieb, C.-B., Budd, C.
Under review.
Download: preprint

An accelerated Levin–Clenshaw–Curtis method for the evaluation of highly oscillatory integrals.
Maierhofer, G., Iserles, A.
Under review.
Download: preprint

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.
Mathematics of Computation.
Download: published version | preprint

Numerical integration of Schrödinger maps via the Hasimoto transform.
Banica, V., Maierhofer, G., Schratz, K.
SIAM Journal on Numerical Analysis.
Download: published version | 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

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.

G-Adaptivity

In collaboration with James Rowbottom, Dr Teo Deveney, Prof. Katharina Schratz, Prof. Pietro Liò, Prof. Carola-Bibiane Schönlieb and Prof. Chris Budd we investigated the use of graph neural networks for adaptive meshing in finite element methods. The results of this ongoing research can be found in a recent preprint which is listed here.

The associated code can be downloaded here: GitHub repository

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