Numerical analysis | Numerical differential equations

Fast multipole method

The fast multipole method (FMM) is a numerical technique that was developed to speed up the calculation of long-ranged forces in the n-body problem. It does this by expanding the system Green's function using a multipole expansion, which allows one to group sources that lie close together and treat them as if they are a single source. The FMM has also been applied in accelerating the iterative solver in the method of moments (MOM) as applied to computational electromagnetics problems. The FMM was first introduced in this manner by Leslie Greengard and Vladimir Rokhlin Jr. and is based on the multipole expansion of the vector Helmholtz equation. By treating the interactions between far-away basis functions using the FMM, the corresponding matrix elements do not need to be explicitly stored, resulting in a significant reduction in required memory. If the FMM is then applied in a hierarchical manner, it can improve the complexity of matrix-vector products in an iterative solver from to in finite arithmetic, i.e., given a tolerance , the matrix-vector product is guaranteed to be within a tolerance The dependence of the complexity on the tolerance is , i.e., the complexity of FMM is . This has expanded the area of applicability of the MOM to far greater problems than were previously possible. The FMM, introduced by Rokhlin Jr. and Greengard has been said to be one of the top ten algorithms of the 20th century. The FMM algorithm reduces the complexity of matrix-vector multiplication involving a certain type of dense matrix which can arise out of many physical systems. The FMM has also been applied for efficiently treating the Coulomb interaction in the Hartree–Fock method and density functional theory calculations in quantum chemistry. (Wikipedia).

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Multimeter Review / DMM Review / buyers guide / tutorial

A list of my multimeters can be purchased here: http://astore.amazon.com/m0711-20?_encoding=UTF8&node=5 In this video I do a review of several digital multimeters. I compare features and functionality. I explain safety features, number of digits, display count, accuracy and resolution. Th

From playlist Multimeter reviews, buyers guide and comparisons.

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Visit my website for more Tips, Videos, DIY projects and more: http://www.mjlorton.com/ --------------------- Click "Show more" ------------------------------- Multimeter tutorial - This series of videos is a tutorial on how to use a digital multimeter for beginners. It will also explain

From playlist How to use a multimeter for beginners

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From playlist Solve Multi-Step Equations......Help!

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From playlist Summer School on Gravitational Wave Astronomy -2019

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From playlist 2013 IEEE Cluster Conference plenary presentations

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From playlist Numerical Analysis and Scientific Computing

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FFT based spectral Ewald methods as an alternative to multipole methods – A.-K. Tornberg – ICM2018

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From playlist Course | Computer Systems Laboratory Colloquium (2006-2007)

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From playlist How to Solve Multi Step Equations with Variables on Both Sides

Related pages

N-body problem | Boundary element method | Numerical analysis | Density functional theory | Barnes–Hut simulation | Multipole expansion | Algorithm | Hartree–Fock method | Computational electromagnetics | Helmholtz equation