At the interface between semiclassical analysis and numerical analysis of wave propagation problems
Lead Research Organisation:
University of Bath
Department Name: Mathematical Sciences
Abstract
Our understanding of wave propagation underpins several important technologies in everyday use. For example, WiFi and mobile phones use electromagnetic waves to transmit information, and the technologies in seismic and medical imaging use acoustic, elastic, and electromagnetic waves to obtain information about the ocean floor and the human body.
Over the last few years, the investigator has worked with, on the one hand, researchers interested in wave phenomena from a purely mathematical point of view (without any thought of applications), and, on the other hand, researchers interested primarily in improving how wave phenomena are simulated in applications (with this understanding then feeding into new technologies). This experience has uncovered huge untapped potential in the relationship between the theoretical aspects of wave propagation and the more-practical aspects, and this fellowship seeks to exploit this.
The overall goals are to (i) prove fundamental theoretical results about wave propagation, motivated by applications, and (ii) use these theoretical results to prove fundamental results about how wave propagation is simulated using computers, addressing long-standing open problems and developing new numerical methods that have the potential to change the technologies used in the huge variety of practical applications of wave propagation.
Over the last few years, the investigator has worked with, on the one hand, researchers interested in wave phenomena from a purely mathematical point of view (without any thought of applications), and, on the other hand, researchers interested primarily in improving how wave phenomena are simulated in applications (with this understanding then feeding into new technologies). This experience has uncovered huge untapped potential in the relationship between the theoretical aspects of wave propagation and the more-practical aspects, and this fellowship seeks to exploit this.
The overall goals are to (i) prove fundamental theoretical results about wave propagation, motivated by applications, and (ii) use these theoretical results to prove fundamental results about how wave propagation is simulated using computers, addressing long-standing open problems and developing new numerical methods that have the potential to change the technologies used in the huge variety of practical applications of wave propagation.
Planned Impact
Our understanding of acoustic, electromagnetic and elastic wave propagation is used in a plethora of technologies upon which our society depends, for example: radar, sonar, mobile phone technology, ultrasound, noise barriers on motorways, optical fibres, seismic imaging. The results of this Fellowship will enhance our understanding of wave phenomena, and thus ultimately contribute to improved technologies which will benefit the general public. In the short term, there are potential pathways to certain industrial applications (listed below) from several results this Fellowship seeks to obtain. In the longer term, results from this Fellowship will form the basis of future investigations into wave propagation problems.
Examples of short-term potential industrial impacts of this Fellowship include the following.
- Medical imaging: ultrasound. Boundary integral equations are currently used in the simulation of high-frequency acoustic waves used in ultrasound. The new integral-equation formulations of the Fellowship therefore have the potential to make these high-frequency ultrasound simulations faster and more reliable.
- Medical imaging: microwaves. A particular example of the use of electromagnetic waves in medical imaging is the use of microwaves to diagnose strokes. The new methods of the Fellowship for the fast numerical solution of the electromagnetic waves will be able to simulate microwaves travelling through the brain, and so they have the potential to make this diagnosis process faster and more reliable.
-Seismic imaging. The seismic imaging community currently use time-domain solvers to simulate the elastic waves used in imaging technologies, but there is an ongoing "quest" for an optimal frequency-domain solver. There is strong precedent for extending Helmholtz solvers to frequency-domain elastic waves, and thus the methods of the Fellowship for the Helmholtz equation with variable wave speed will generate huge interest in this area.
Examples of short-term potential industrial impacts of this Fellowship include the following.
- Medical imaging: ultrasound. Boundary integral equations are currently used in the simulation of high-frequency acoustic waves used in ultrasound. The new integral-equation formulations of the Fellowship therefore have the potential to make these high-frequency ultrasound simulations faster and more reliable.
- Medical imaging: microwaves. A particular example of the use of electromagnetic waves in medical imaging is the use of microwaves to diagnose strokes. The new methods of the Fellowship for the fast numerical solution of the electromagnetic waves will be able to simulate microwaves travelling through the brain, and so they have the potential to make this diagnosis process faster and more reliable.
-Seismic imaging. The seismic imaging community currently use time-domain solvers to simulate the elastic waves used in imaging technologies, but there is an ongoing "quest" for an optimal frequency-domain solver. There is strong precedent for extending Helmholtz solvers to frequency-domain elastic waves, and thus the methods of the Fellowship for the Helmholtz equation with variable wave speed will generate huge interest in this area.
Organisations
Publications
Lafontaine D
(2021)
A sharp relative-error bound for the Helmholtz h-FEM at high frequency
in Numerische Mathematik
Spence E
(2023)
A simple proof that the hp-FEM does not suffer from the pollution effect for the constant-coefficient full-space Helmholtz equation
in Advances in Computational Mathematics
Moiola A
(2019)
Acoustic transmission problems: Wavenumber-explicit bounds and resonance-free regions
in Mathematical Models and Methods in Applied Sciences
Graham I
(2021)
Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification
in Advances in Computational Mathematics
Marchand P
(2022)
Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency?
in Advances in Computational Mathematics
Diwan G
(2019)
Can coercive formulations lead to fast and accurate solution of the Helmholtz equation?
in Journal of Computational and Applied Mathematics
Chandler-Wilde S
(2021)
Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains
in Numerische Mathematik
Gong S
(2022)
Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation
in Numerische Mathematik
Gong S
(2022)
Convergence of restricted additive Schwarz with impedance transmission conditions for discretised Helmholtz problems
in Mathematics of Computation
Galkowski J
(2023)
Does the Helmholtz Boundary Element Method Suffer from the Pollution Effect?
in SIAM Review
| Description | This fellowship focused on the following fundamental problem involving high-frequency acoustic and electromagnetic waves: given an incident field and obstacle, find the scattered wave produced when the incident field interacts with the obstacle. Only in very simple situations (e.g., scattering by a ball-shaped obstacle) can one "write down the solution" for the scattered wave. Therefore, to obtain information about the scattered wave for applications, one usually has to calculate an approximation to the scattered wave using a computer. The area of mathematics called "numerical analysis" can tackle this problem of calculating approximations to the scattered wave. In this setting of high-frequency waves, the overall goal of numerical analysis is to compute the scattered wave at arbitrarily-high frequency to arbitrary accuracy. Achieving this goal is difficult for several reasons, including the following two: (a) Approximating a high-frequency function by piecewise polynomials requires a large number of degrees of freedom: think of sampling a highly-oscillating function at fixed points, and then "joining the dots" (with straight lines) to approximate the function - the more oscillatory the function, the closer together the sample locations need to be to get an accurate approximation. (b) When the obstacle traps geometric-optic rays (e.g., a bottle), at certain frequencies (dictated by the obstacle) the scattered wave is very sensitive to changes in the incident wave, and this makes it difficult to accurately compute the scattered wave. This feature is related to the classic phenomenon of how an opera singer can break a wine glass (an obstacle that traps rays) by singing the right note, i.e., creating waves at the right frequency. The idea of the fellowship was to tackle the numerical-analysis goal above using the tools and techniques developed by mathematicians interested in high-frequency waves from a purely theoretical point of view. These tools and techniques come from the area of maths called "semiclassical analysis". This approach has settled several long-standing open problems in numerical-analysis of high-frequency scattering, including the following. 1. Regarding the difficulty (a) above: we obtained results showing how the number of degrees of freedom must grow with frequency to maintain accuracy for a wide variety of acoustic scattering problems, solved using either the so-called "finite element method" or the "boundary element method" (both very popular methods for solving these wave scattering problems in the applied sciences and engineering). 2. Regarding the difficulty (b) above: we showed rigorously that this extremely-bad behaviour is extremely rare. 3. Some numerical methods (such as the finite element method) truncate the unbounded domain of propagation (exterior to the scatterer). One then needs to impose a condition on this artificial boundary, roughly speaking, saying that the scattered wave travel "outwards". Although designing and studying such conditions has been an active area of research since the 1970s, there were no rigorous results about how the error incurred in imposing this artificial boundary condition depends on the frequency of the waves. We proved such results, with numerical experiments showing agreement of the theory with what happens in practice. |
| Exploitation Route | The numerical methods studied in the fellowship, namely the finite-element and boundary-element methods, are widely used to compute approximations to wave scattering problems in the applied sciences and engineering. The information about these methods obtained by the fellowship will ultimately allow these end users to select the appropriate methods for their particular problems, and be confident of obtaining accurate solutions efficiently. |
| Sectors | Aerospace, Defence and Marine,Digital/Communication/Information Technologies (including Software) |
| URL | https://people.bath.ac.uk/eas25/papers2.html |