Fast solvers for frequency-domain wave-scattering problems and applications

Lead Research Organisation: University of Bath
Department Name: Mathematical Sciences

Abstract

The computation of wave phenomena is widely needed in many application areas, for example models of radar and telecommunications devices require the computation of electromagnetic waves while the implementations of seismic and medical imaging algorithms use acoustic, elastic, and electromagnetic waves to obtain information about the earth's subsurface and the human body respectively.

Computer models of the propagation of waves arise naturally in the design and implementation of these technologies. Medical imaging technicians use computer models of how the material composition of the human body scatters incoming electromagnetic waves in order to solve the "inverse problem'' of reconstructing the internal makeup of a human being from an observed scattered wave field. Similarly, seismologists use computer models of how the material properties of the earth's subsurface affects the transmission of elastic waves in order to reconstruct the earth's subsurface properties from observed echoes of elastic waves

This technology is hugely useful, for example in the medical context it means we can often diagnose health problems without a need for more invasive techniques. In the seismology case it makes something seemingly impossible become possible - since it is never physically possible to explore all of the earth's subsurface properties by simply boring holes.

However the fast and accurate computer modelling of such wave phenomena is complicated and costly (in terms of computer time), principally (but not solely) because of the highly oscillatory nature of the waves and the complicated media through which they pass. Thus there is a strong need for new methods that speed up such models and that task is a principal focus of this research.

This project will devise and mathematically justify new families of fast methods for implementing these computer wave models, and will make the new methods available through two software platforms which are accessible to a wide range of scientists as well as in an additional specialist high performance computing library.

As well as devising new methods for modelling (which work well on today's multiprocessor computers), the project will also involve direct collaboration with two companies - Schlumberger (a Project Partner, interested in seismology) and ABB (interested in electromagnetic computations) - as well as two academic groups, one in geosciences and one in electromagnetics.

Planned Impact

WHAT IMPACT?

* New knowledge on the formulation, analysis and implementation of fast solvers for frequency-domain wave problems.
* New public-domain software implementing the new algorithms.
* Training the next generation of researchers at the interface of numerical analysis/computational PDEs/wave modelling.
* Applications of this new knowledge in seismic inversion, acoustics, and electromagnetics.

WHO MIGHT BENEFIT FROM THE RESEARCH?

* The large community of academic beneficiaries listed above.
* Schlumberger Gould Research, our Project Partner.
* ABB, a company with whom we collaborate.
* Geoscientists working in forward modelling and in seismic inversion.
* Electrical engineers working in forward and inverse problems in electromagnetics.
* Members of the UK acoustics network interested in computational acoustics.
* Specialists in medical imaging solving inverse problems in electromagnetics.

HOW MIGHT THEY BENEFIT FROM THIS RESEARCH?

* The benefits for the academic community are described above.
* The public-domain implementations of the new methods will be available globally.
* A specific problem in optimising placement of sensors in seismic inversion of particular interest to Schlumberger will be solved.
* New methods for solving Maxwell's equations in highly heterogeneous media (of particular interest to ABB) will be developed.
 
Description 1. We have done a thorough analysis of one-level domain decomposition methods for homogeneous and heterogeneous wave propagetion problems in the high frequency regime.
2. We have done a complete analysis of the analytical proprties of the heterogeneous Helmholtz equation.
3. We have analysed solvers for Maxwells equations in the high frequency regime and tested the methods on several challenging test problem showing efficiency on thousands of processors.
4. We have formulated new regularized algorithms for sensor optimization in seismic imaging algorithms in the context of bilevel learning.
Exploitation Route The work on solvers for high frequency problems will be used by software developers within the modern systems Freefem++, HPDDM and Firedrake
Sensor optimisation algorithms will be available to Schlumberger Cambridge Research for consideration for use in their codes.
Sectors Electronics,Energy

 
Description Analysis of GENEO for indefinite problems 
Organisation Heidelberg University
Country Germany 
Sector Academic/University 
PI Contribution We have engaged in substantial discussions about the performance of the GENEO preconditioner for indefinite and non-self-adjoint problems. The team of I.G. Graham and E. Spence (Bath), V. Dolean and N. Bootland(Strathclyde) and R. Scheichl and C. Ma (Heidelberg) are actively collaborating on this topic. The initial discussions took place at CIRM in Luminy at a conference whose organisers included Dr V Dolean (Strathclyde), in September 2019.
Collaborator Contribution Strathclyde and Heidelberg are collaborating as above. Dr C. Ma will visit Bath to progress the research in March 2020.
Impact None as yet
Start Year 2019
 
Description Analysis of GENEO for indefinite problems 
Organisation University of Strathclyde
Country United Kingdom 
Sector Academic/University 
PI Contribution We have engaged in substantial discussions about the performance of the GENEO preconditioner for indefinite and non-self-adjoint problems. The team of I.G. Graham and E. Spence (Bath), V. Dolean and N. Bootland(Strathclyde) and R. Scheichl and C. Ma (Heidelberg) are actively collaborating on this topic. The initial discussions took place at CIRM in Luminy at a conference whose organisers included Dr V Dolean (Strathclyde), in September 2019.
Collaborator Contribution Strathclyde and Heidelberg are collaborating as above. Dr C. Ma will visit Bath to progress the research in March 2020.
Impact None as yet
Start Year 2019
 
Description Multilevel strategies for Helmholtz equation 
Organisation Charles University
Country Czech Republic 
Sector Academic/University 
PI Contribution I. Graham and E.A. Spence (Bath) have formulated several recursive strategies for domain deocmposition methods for Helmholtz problems, with preliminary theory, especially for problems with cavities.
Collaborator Contribution S. Congreve (Charles) and P.H. Tournier (Sorbonne) have done extensive tests. We expect to turn this collaboration into a publication in due course.
Impact None as yet
Start Year 2019
 
Description Multilevel strategies for Helmholtz equation 
Organisation Sorbonne University
Country France 
Sector Academic/University 
PI Contribution I. Graham and E.A. Spence (Bath) have formulated several recursive strategies for domain deocmposition methods for Helmholtz problems, with preliminary theory, especially for problems with cavities.
Collaborator Contribution S. Congreve (Charles) and P.H. Tournier (Sorbonne) have done extensive tests. We expect to turn this collaboration into a publication in due course.
Impact None as yet
Start Year 2019
 
Description Sensor optimization in FWI 
Organisation Schlumberger Limited
Department Schlumberger Cambridge Research
Country United Kingdom 
Sector Academic/University 
PI Contribution E.A. Spence, S. Gazzola and I.G. Graham: Supervision of research student S. Downing, working on problems of direct relevance to Schlumberger
Collaborator Contribution During the lifetime of this Award: Staff from Schlumberger have attended supervision sessions for the student and also the stakeholder's meeting for the EPSRC Grant funding this project which was held on October 7 2019.
Impact Funded by previous projects, jointly with the same company: Elizabeth Arter PhD: Sweeping Preconditioners for Helmholtz problems using absorption , 2019. Supported by Schlumberger Gould Research. Douglas Shanks PhD: Robust solvers for large indefinite systems in seismic inversion , supported by Schlumberger Gould Research, 2015
Start Year 2009
 
Description Theory of domain decomposition for highly indefinite problems 
Organisation Chinese University of Hong Kong
Country Hong Kong 
Sector Academic/University 
PI Contribution During the lifetime of this award: I.G. Graham and S. Gong (Bath) visited J. Zou (CUHK) for during December 2019 for discussions and planning the next publication.
Collaborator Contribution J. Zou and his research group joined the research discussions. We have planned a future research direction. The Chinese University supported the travel and subsistence costs of I.G. Graham and S. Gong.
Impact Domain Decomposition with local impedance conditions for the Helmholtz equation I.G. Graham, E.A. Spence, J. Zou arXiv:1806.03731
Start Year 2014
 
Description Wave equation solvers in Full Waveform Inversion 
Organisation University of Côte d'Azur
Country France 
Sector Academic/University 
PI Contribution We travelled to Nice in October 2019 to meet the team of Geophysicists headed by Prof S. Operto. We had intensive discussions about fast solvers and novel inverse solvers in the context of FWI. The team from Bath was I.G. Graham, S. Gazzola, E.A. Spence and S. Downing. Graham and Spence gave an invited talk.
Collaborator Contribution Professor Operto's team (Nice) provided a 2 day discussion session on FWI and related problems. V. Dolean provided expertise on fast solvers in the discussion.
Impact None as yet
Start Year 2019
 
Description Wave equation solvers in Full Waveform Inversion 
Organisation University of Strathclyde
Country United Kingdom 
Sector Academic/University 
PI Contribution We travelled to Nice in October 2019 to meet the team of Geophysicists headed by Prof S. Operto. We had intensive discussions about fast solvers and novel inverse solvers in the context of FWI. The team from Bath was I.G. Graham, S. Gazzola, E.A. Spence and S. Downing. Graham and Spence gave an invited talk.
Collaborator Contribution Professor Operto's team (Nice) provided a 2 day discussion session on FWI and related problems. V. Dolean provided expertise on fast solvers in the discussion.
Impact None as yet
Start Year 2019
 
Description Stakeholders meeting for this project 
Form Of Engagement Activity A formal working group, expert panel or dialogue
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Industry/Business
Results and Impact On October 7 2019 at Bath we held the first Stakeholders meeting for our grant. Attendees included industrialists from Schlumberger and ABB, software developers from Imperial College and Toulouse as well as the academic members, PDRAs and research students from our teams.
Year(s) Of Engagement Activity 2019