Multilevel fast multipole method
{{Short description|Computational mathematics}}
The multilevel fast multipole method (MLFMM) is used along with method of moments (MoM) a numerical computational method of solving linear partial differential equations which have been formulated as integral equations of large objects almost faster without loss in accuracy.{{cite web | url=http://www.kfs.oeaw.ac.at/index.php?option=com_content&view=article&id=361:multilevel-fast-multipole-method-mlfmm&catid=104&Itemid=696&lang=en | title=Multilevel Fast Multipole Method (MLFMM) | publisher=Austrian Academy of Sciences – Acoustics Research Institute | accessdate=20 April 2014}} This method is an alternative formulation of the technology behind the MoM and is applicable to much larger structures like radar cross-section (RCS) analysis, antenna integration on large structures, reflector antenna design, finite size antenna arrays, etc., making full-wave current-based solutions of such structures a possibility.{{cite web | url=https://www.feko.info/product-detail/numerical_methods/mlfmm/mlfmm | title=Multilevel Fast Multipole Method (MLFMM) | publisher=Feko | accessdate=20 April 2014}}{{cite web | url=http://www.efieldsolutions.com/mlfmm.php | title=Multilevel Fast Multipole Method (MLFMM) | publisher=E field | accessdate=20 April 2014| date=2013-04-30 }}
Method
The MLFMM is based on the Method of Moments (MoM), but reduces the memory complexity from to , and the solving complexity from to , where represents the number of unknowns and the number of iterations in the solver. This method subdivides the Boundary Element mesh into different clusters and if two clusters are in each other's far field, all calculations that would have to be made for every pair of nodes can be reduced to the midpoints of the clusters with almost no loss of accuracy. For clusters not in the far field, the traditional BEM has to be applied. That is MLFMM introduces different levels of clustering (clusters made out of smaller clusters) to additionally enhance computation speed.{{cite journal | journal=Progress in Electromagnetics Research | volume=82 | pages=51–63 | publisher=PIER |author1=P.-L. Rui | title=Schwarz-Krylov Subspace Method for MLFMM Analysis of Electromagnetic Wave Scattering Problems |author2=R.-S. Chen |author3=Z.-W. Liu |author4=Y.-N. Gan |name-list-style=amp | doi=10.2528/PIER08013003 | year=2008 | doi-access=free }}{{cite book | chapter=Theory and application of an MLFMM/FEM hybrid framework in FEKO | date=7–9 Nov 2011 |author1=Bingle, M Burger, E. | title=2011 IEEE International Conference on Microwaves, Communications, Antennas and Electronic Systems (COMCAS 2011) | pages=1–3 |author2=Jakobus, U. |author3=van Tonder, J.J. | doi=10.1109/COMCAS.2011.6105819 | isbn=978-1-4577-1694-2 | s2cid=39160247 }}{{cite book | chapter=Parallel computation methods for enhanced MOM and MLFMM performance | date=11 May 2009 |author1=D'Ambrosio, K. | title=2009 IEEE Long Island Systems, Applications and Technology Conference | pages=1–4 |author2=Pirich, R. |author3=Kaufman, A. |author4=Mesecher, D. | doi=10.1109/LISAT.2009.5031571 | isbn=978-1-4244-2347-7 | s2cid=18786124 }}{{cite web | url=http://www.emss.de/downloads/pdffiles/jak08c_p.pdf | title=Advanced EMC Modeling by Means of a Parallel MLFMM and Coupling with Network Theory | publisher=EMSS | accessdate=20 April 2014 |author1=Ulrich Jakobus |author2=Johann van Tonder |author3=Marlize Schoeman |name-list-style=amp }}{{cite web | url=https://www.cst.com/Content/Events/UGM2009/6-1-3-Electrically-Large-Applications-and-Integral-Equation-solver.pdf | title=(Electrically) Large Applications & Integral Equation solver | publisher=CST | accessdate=20 April 2014}}{{cite web | url=https://www.esi-group.com/software-services/virtual-environment/electromagnetics/cem-solutions/multilevel-fast-multipole | title=Multilevel Fast Multipole Method (MLFMM) | publisher=ESI | accessdate=20 April 2014 | archive-url=https://archive.today/20140420071724/https://www.esi-group.com/software-services/virtual-environment/electromagnetics/cem-solutions/multilevel-fast-multipole | archive-date=20 April 2014 | url-status=dead }}