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Topic: Multipole expansion

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	K.-T. Lim Thesis - Chapter 3
		The key feature of the CMM and other multipole methods [3,4,5,6] is that they replace the effects of atoms with multipole expansions representing the fields due to those atoms.
		The multipole expansions are computed for each cell at each level within the tree, starting from the leaves, or smallest cells, and working upwards in the tree to the root, which represents the entire system.
		Once the leaf cell multipoles have been computed, the expansions may be translated to the centers of the next higher-level ("parent") cells and combined.
	www.wag.caltech.edu /publications/theses/ktl/Chapter3.html (1664 words)

	Multipole expansion - Wikipedia, the free encyclopedia
		A multipole expansion is a series expansion of a potential, usually in powers (or inverse powers) of the distance to the origin, as well as functions describing the angular dependence.
		Multipole expansions are widely used in problems involving gravitational fields of systems of masses, electric and magnetic fields of charge and current distributions, and the propagation of electromagnetic waves.
		Multipole expansions are also useful in numerical simulations, and form the basis of the fast multipole method [1] of Greengard and Rokhlin, a general technique for efficient computation of energies and forces in systems of interacting particles.
	en.wikipedia.org /wiki/Multipole_expansion (679 words)

	Multipole moments - Wikipedia, the free encyclopedia
		Multipole moments are the coefficients of a series expansion of a potential, usually involving powers (or inverse powers) of the distance to the origin, as well as some angular dependence.
		The zeroth-order term in the expansion is called the monopole moment, the first-order term is denoted as the dipole moment, and the third, fourth, etc. terms are denoted as quadrupole, octupole, etc. moments.
		Multipole moments in mathematics and mathematical physics form an orthogonal basis for the decomposition of a function, based on the response of a field to point sources that are brought infinitely close to each other.
	en.wikipedia.org /wiki/Multipole_moment (328 words)

	1.1 Gravitational-wave generation formalisms
		The multipole expansion is one of the most useful tools of physics, but its use in general relativity is difficult because of the non-linearity of the theory and the tensorial character of the gravitational interaction.
		In the stationary case, the multipole moments are determined by the expansion of the metric at spatial infinity, while, in the case of non-stationary fields, the moments, starting with the quadrupole, are defined at future null infinity.
		The explicit expressions of the multipole moments (for instance in STF guise) as integrals over the source, valid in the linearized theory but irrespective of a slow motion hypothesis, are completely known [101, 39, 38, 57].
	www.univie.ac.at /EMIS/journals/LRG/Articles/lrr-2002-3/articlesu1.html (2380 words)

	The Fast Multipole Algorithm (Site not responding. Last check: 2007-10-18)
		A truncated multipole expansion is then calculated for each sub-cell at the finest refinement level which expresses the effect of all particles in that sub-cell on distant particles.
		To calculate the multipole expansion at the lowest level of spatial decomposition equations (30) and (31) are used which represent the aggregate potential due to the k particles.
		Given a charge-neutral system, all multipole and local expansions at all refinement levels of the basic unit cell are calculated and the potential due to all particles in the unit cell is obtained.
	www.ee.duke.edu /~ayt/ewaldpaper/node19.html (809 words)

	BioMagnetic Research and Technology \| Full text \| Simultaneous suppression of disturbing fields and localization of ... (Site not responding. Last check: 2007-10-18)
		Multipole expansions are used also to model spatially distributed biological sources such as brain currents [7,8].
		The application of multipoles for the localization of magnetic dipoles is described in [9,10], and is used in other technical areas without disturbing field suppression [11].
		The multipole localization is an effective algorithm because it unites a method for the suppression of disturbing fields with a localization method.
	www.biomagres.com /content/2/1/6 (2733 words)

	Example Application Codes
		These multipole expansions are combined in the upper levels, until the root of the oct-tree contains a multipole expansion.
		Then local series expansions of the influence of sufficiently separated multipole expansions on cells are formed.
		The local expansion about the center of a cell is found by adding up all the influences from the cells of the interaction set.
	godel.ph.utexas.edu /Members/parashar/sc96/node4.html (1537 words)

	Definition of Multipole expansion
		Hence the multipole expansion is a practical tool for the approximation of fields; far away from a given configuration of charges, the first few terms are typically dominant.
		The n=1 term is the dipole term; applying Stokes' theorem we recover its usual form in terms of the area of the loop.
		The fast multipole algorithm of Greengard and Rokhlin is a general technique for accelerating computer simulations of particle dynamics and electrostatics.
	www.wordiq.com /definition/Multipole_expansion (479 words)

	5.5 Post-Newtonian field in the near zone
		Indeed the post-Newtonian expansion replaces the resolution of a hyperbolic-like d’Alembertian equation by a perturbatively equivalent hierarchy of elliptic-like Poisson equations.
		Rapidly it is found during the post-Newtonian iteration that the right-hand side of the Poisson equations acquires a non-compact support (it is distributed over all space), and that as a result the standard Poisson integral diverges at the bound of the integral at spatial infinity, i.e.
		denotes the multipole expansion of the pseudo-tensor (in the sense of Equation (62
	relativity.livingreviews.org /Articles/lrr-2006-4/articlesu12.html (995 words)

	The Fast Multipole Method
		The multipole expansion of each lowest level box is then translated to the center of its parent box via one of three special FMM operators.
		The second of the FMMs operators is used for each box to transform the multipole expansions of all well-separated boxes (those that are not nearest neighbours) into Taylor expansions about the center of the current box.
		However, only those multipole expansions from boxes which are well-separated at the present level and not well-separated at the parent level are interacted.
	www-theor.ch.cam.ac.uk /people/ross/thesis/node97.html (383 words)

	Example Application Codes
		These multipole expansions are combined in the upper levels, until the root of the oct-tree contains a multipole expansion.
		Then local series expansions of the influence of sufficiently separated multipole expansions on cells are formed.
		The local expansion about the center of a cell is found by adding up all the influences from the cells of the interaction set.
	wwwrel.ph.utexas.edu /Members/parashar/sc96/node4.html (1537 words)

	Macroscopic Multipole Method (Site not responding. Last check: 2007-10-18)
		The method is based on the observation, also realized by Schmidt and Lee [44], that once the multipole expansion of the unit cell is computed, translates of this unit cell (i.e.
		In a point charge system, given p (the number of multipole terms) and k (lattice size), the macroscopic multipole method commences by dividing the simulation cell recursively into smaller sub-cells as in the FMA method.
		The multipole expansion of each sub-cell at the finest level of refinement is calculated and subcells are grouped into bigger structures up to the unit cell in what is known as the upward pass.
	www.ee.duke.edu /~ayt/ewaldpaper/node22.html (404 words)

	Generalized Multipole Expansion
		In our case we are interested in an expansion not of the inverse distance, but rather of its fourth power.
		In the case n=-1, ie the Laplace expansion (27), a
		This expansion can be applied to Equation (3) by putting n=-4.
	www.cs.princeton.edu /~ah/publications/multipole/html/node31.html (211 words)

	A PARALLEL ADAPTIVE FAST MULTIPOLE ALGORITHM FOR N-BODY PROBLEMS (Site not responding. Last check: 2007-10-18)
		In the FMA, the effect due to a well-separated (sufficiently far away) group of particles is approximated by a multipole expansion, which is a refined formalization of the center-of-mass, leading to provable error bounds.
		In the AFMA, sending multipole expansions instead of particles has the advantage that a single multipole expansion is usually much less voluminous than the particle data itself, for cells containing tens or hundreds of particles.
		The number of multipole expansion terms is 8, corresponding to the high-accuracy simulations in the work by Board [].
	charm.cs.uiuc.edu /papers/FastMultipoleICPP95.www (3680 words)

	12.5.2 Fast Algorithms
		It would be possible to build a fast algorithm by evaluating the multipole expansion at the location of particles that do not belong to the group.
		In this case, the multipole expansion is transformed into a Taylor series around the center of the second group, where the influence of the first one is sought.
		The expansions provide an accurate representation of the velocity field when the distance between the groups is large compared to their radii.
	www.netlib.org /utk/lsi/pcwLSI/text/node286.html (609 words)

	Paper Citing NAMD - Abstract
		Secondly, the expansion order is given in terms of the ratio of the distance of a field point to a source box to the box size, which reflects the relative error of the expansion.
		With that, a large portion of the sources has been evaluated by the multipole expansion at low levels of the source tree, which is around two-thirds of sources at the first two levels of the tree averagely.
		The algorithm reduces the CPU time dependency on expansion order p from O(p(2)) of the classical treecode to be lower than a linear dependency in p(max), where p(max) is the maximum expansion order used in the variable order expansion.
	www.ks.uiuc.edu /Research/namd/citations/abstracts/ISI:000224284500010.html (361 words)

	On the Use of Multipole Expansion of the Coulomb Potential in Quantum Chemistry (Site not responding. Last check: 2007-10-18)
		Some features of the multipole expansion of the Coulomb potential V for a system of point charges are studied.
		It is shown that multipole expansion is convergent both locally in L2(R3) and weakly on some classes of functions.
		One-particle Hamiltonians H(n) = H(0) + V(n) where H(0) is the kinetic energy operator and V(n) is the n-th partial sum of the multipole expansion of V, are discussed, and the convergence of their eigenvalues to those of H = H(0) + V with increasing n is proved.
	www.chem.msu.su /eng/publ/moleq4.html (142 words)

	Course Outline
		This course will introduce you to the fast multipole method, a numerical algorithm that is extremely promising for achieving fast solutions of many applied problems in science, engineering, biology, computer vision and statistics.
		The course is directed at beginning graduate students who have had an introductory numerical analysis course, and have some knowledge of linear algebra.
		What are multipole methods and what is this course about.
	www.umiacs.umd.edu /~ramani/cmsc878R/outline.html (442 words)

	Introduction
		The coefficients that appear in this multipole expansion are the static moments.
		The torque triggers a measurable precession of the nuclear spin.
		Only the few first orders of the multipole expansion are usually kept.
	www.physics.rutgers.edu /~tmertzi/Thesis/node3.html (326 words)

	Multipole Expansion
		The multipole expansion is widely used in electromagnetism and celestial mechanics which study phenomena whose behavior depends on powers of the reciprocal distance.
		It is natural to take advantage of a mature set of techniques and extend its application to radiometry.
		The details of the expansion are relegated to appendix A.2.
	www.cs.princeton.edu /~ah/publications/multipole/html/node2.html (482 words)

	1 Introduction (Site not responding. Last check: 2007-10-18)
		As in the case of the fast multipole method (FMM) [6], it is possible to implement higher order multipole expansion to achieve high accuracy.
		However, the translation formulae for multipole expansion are rather complex and difficult to program.
		In the multipole expansion, we approximate the potential field generated by a clump of particles by multipole expansion.
	grape.c.u-tokyo.ac.jp /~makino/papers/ppmmpreprint/node1.html (321 words)

	CS267: Notes for Lecture 27, Apr 25 1995
		A "multipole expansion" of the potential phi(x,y) is a kind of Taylor expansion which is accurate when x^2+y^2 is large.
		The corresponding outer expansion is accurate outside the blue dashed square centered at z1, so it is certainly accurate outside the fl dashed square centered at z2.
		We need only "add" the outer expansions in the sense of adding the masses M (since the total mass of the particles in n is the sum of the masses of its children), and adding the coefficients alpha(i) (since the potential of a collection of particles is the sum of the potentials of each particle).
	www.cs.berkeley.edu /~demmel/cs267-1995/lecture27/lecture27.html (4493 words)

	CS267: Notes for Lecture 25, Apr 16 1996
		For ease of exposition, we will sometimes refer to the expansion associated with Outer(n) as an outer expansion, since it is to be evaluated outside n, and to contrast it with the different expansion discussed in the next section.
		We need only "add" the outer expansions in the sense of adding the masses M (since the total mass of the particles in n is the sum of the masses of its children), and adding the coefficients alpha
		Adding inner expansions means just adding their coefficients beta(i), which is valid, as before, because the potential of a set of particles is the sum of potentials of each particle alone.
	www.cs.berkeley.edu /~demmel/cs267/lecture27/lecture27.html (4317 words)

	Publications (Site not responding. Last check: 2007-10-18)
		The method uses the Helmholtz vector decomposition theorem and the differential algebraic framework of COSY INFINITY to determine a solution to the Laplace equation.
		The solution is locally expressed as a Taylor expansion of the field which can be computed to arbitrary order.
		It provides a natural multipole decomposition of the field which is required for the computation of transfer maps, and also allows to obtain very accurate finite element representations with very small numbers of cells.
	www.bt.pa.msu.edu /cgi-bin/display.pl?name=LapCAP04 (143 words)

	5.2 General expression of the multipole expansion
		denotes the post-Newtonian expansion of the stress-energy pseudo-tensor defined by Eq.
		can be obtained from the known formula for the multipole expansion of the retarded solution of a wave equation with compact-support source.
		This formula, given in Appendix B of Ref. [16], yields the second term in Eq.
	www.univie.ac.at /EMIS/journals/LRG/Articles/lrr-2002-3/articlesu9.html (748 words)

	N-Body Problem.
		A method, called the fast multipole method, based on multipole expansion was developed in [65] and refined by introducing computational-geometric ideas in distance geometry [25].
		To apply the fast multipole method, one must first construct a tree decomposition of the set of particles, along with a set of pairs of nodes in the tree.
		The large constant in the fast multipole method arises both from geometry and from the need to manipulate series expansions, the size of which grows with the desired output precision.
	people.csail.mit.edu /seth/pubs/taskforce/paragraph3_12_0_0_3.html (501 words)

	Multipole expansion of the scalar wave equation
		The solutions of this equation in free space are conveniently written as an expansion in orthogonal spherical waves.
		This expansion is known as the multipole expansion.
		This is a particularly useful result, as we shall discover, since it easily allows us to express the general solution of the inhomogeneous wave equation as a multipole expansion.
	farside.ph.utexas.edu /~rfitzp/teaching/jk1/lectures/node97.html (346 words)

	Resume (Site not responding. Last check: 2007-10-18)
		Treecodes based on multipole expansions deal with mutual interactions among set of bodies in space.
		In the case of the Coulomb potential, there are multipole based treecodes such as the Fast Multipole Method (FMM) that can rapidly evaluate the potentials (complexity linear on the number of particles).
		This algorithm is based on the multipole expansion of the force vector and it is combined with a tree data structure.
	students.cs.tamu.edu /kasthuri/resume.html (917 words)

	Multipole expansion of the vector wave equation
		Thus, magnetic multipole fields are sometimes termed transverse electric (TE) multipole fields.
		It follows that the fields of an electric multipole are given by
		The two sets of multipole fields (7.48), (7.49), and (7.51), form a complete set of vector solutions to Maxwell's equations in free space.
	farside.ph.utexas.edu /teaching/jk1/lectures/node98.html (278 words)

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