U.S. patent application number 11/417429 was filed with the patent office on 2006-08-31 for compression and compressed inversion of interaction data.
Invention is credited to Francis X. Canning.
Application Number | 20060195306 11/417429 |
Document ID | / |
Family ID | 27390540 |
Filed Date | 2006-08-31 |
United States Patent
Application |
20060195306 |
Kind Code |
A1 |
Canning; Francis X. |
August 31, 2006 |
Compression and compressed inversion of interaction data
Abstract
A compression technique compresses interaction data. A fast
method processes the compressed data without the need to first
decompress the data. In one embodiment, the compression technique
is used to compress data in an interaction matrix. The interaction
matrix (such as a moment method impedance matrix) contains
interaction data between sources (e.g., basis functions or
expansion functions) and testers (e.g., testing functions). The
sources are collected into groups of sources according to specified
criteria. One useful criteria is based on grouping sources
relatively close to one another. For each group of sources, a
composite source is calculated. The testers are also collected into
groups and composite testers are calculated. The use of composite
sources and composite testers to compute couplings when the source
and tester are not close to each other allows the interaction
matrix to be computed as a sparse matrix with a block format.
Inventors: |
Canning; Francis X.;
(Newbury Park, CA) |
Correspondence
Address: |
KNOBBE MARTENS OLSON & BEAR LLP
2040 MAIN STREET
FOURTEENTH FLOOR
IRVINE
CA
92614
US
|
Family ID: |
27390540 |
Appl. No.: |
11/417429 |
Filed: |
May 3, 2006 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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09676727 |
Sep 29, 2000 |
|
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11417429 |
May 3, 2006 |
|
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60175454 |
Jan 10, 2000 |
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60201149 |
May 2, 2000 |
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Current U.S.
Class: |
703/2 |
Current CPC
Class: |
H03M 7/30 20130101 |
Class at
Publication: |
703/002 |
International
Class: |
G06F 17/10 20060101
G06F017/10 |
Claims
1. A method, comprising: identifying first interaction data between
a plurality of first sources and a plurality of first testers;
computing at least one composite source from one or more of said
first sources; computing at least one composite tester from one or
more of said first testers; transforming said first interaction
data to second interaction data comprising said at least one
composite source and further comprising said at least one composite
tester, wherein said second interaction data comprises N sources
and M testers; identifying P sources out of said N sources;
identifying Q testers out of said M testers, wherein P and Q are
greater than zero and either P is substantially less than N or Q is
substantially less than M; and applying a matrix transform to
produce third interaction data from said second interaction data,
wherein said third interaction data comprises relatively fewer
interaction values than said second interaction data, and wherein
the effect of any of said Q testers in exciting any of said P
sources can be computed, at least in part, using said third
interaction data.
2. The method of claim 1, wherein said third interaction data is in
the form of a product comprising a substantially lower triangular
matrix and a substantially upper triangular matrix.
3. The method of claim 2, further comprising the step of computing
the excitation of said P sources by at least one of said Q
testers.
4. The method of claim 1, further comprising computing a
factorization of said second interaction data wherein said
factorization comprises a matrix with a significant triangular
structure and using only a portion of said matrix with significant
triangular structure to produce said third interaction data.
5. The method of claim 4, further comprising the step of computing
the excitation of said P sources by at least one of said Q
testers.
6. The method of claim 1, wherein said at least one composite
tester comprises one or more composite testers that receive weakly
from distant sources.
7. The method of claim 6, further comprising the step of computing
the excitation of said P sources by at least one of said Q
testers.
8. The method of claim 6, wherein said at least one composite
source comprises one or more composite sources that broadcast
weakly to distant locations.
9. The method of claim 8, further comprising the step of computing
the excitation of said P sources by at least one of said Q
testers.
10. The method of claim 1, wherein said at least one composite
tester comprises one or more composite testers testing a localized
excitation.
11. The method of claim 10, further comprising the step of
computing the excitation of said P sources by at least one of said
Q testers.
12. The method of claim 1, further comprising computing an LU
factorization of said second interaction data and using only the
bottom right portion of said LU factorization to produce said third
interaction data.
13. The method of claim 12, further comprising the step of
computing the excitation of said P sources for an excitation
wherein a number of said N sources describing said excitation have
a strength that based on a desired accuracy may be approximated by
zero.
14. A method of data compression for a first system of linear
equations comprising: computing two or more composite basis
functions from two or more original basis functions in said first
system of linear equations; computing two or more composite testing
functions from two or more original testing functions in said first
system of linear equations; and transforming at least a portion of
said first system of linear equations to a second system of linear
equations using a second set of basis functions and a second set of
testing functions, wherein said second set of basis functions
comprises said two or more composite basis functions and said
second set of testing functions comprises said two or more
composite testing functions, wherein based on a desired accuracy
relatively more elements of said second system of equations than
elements of said first system of linear equations may be
approximated by zero, and wherein at least one of said composite
basis functions is spatially relatively close to at least one of
said composite testers.
15. The method of claim 14 further comprising: transforming an
excitation described by said original testing functions into an
excitation described by said second set of testing functions; using
said second system of equations and said excitation as described by
said second set of testing functions computing the associated
source as described by said second set of basis functions; and
transforming said source described by said basis functions from
said second set of basis functions into a source described by said
original basis functions.
16. A method of data compression for a first interaction data based
on a set of basis functions, comprising: identifying a matrix of
transmitted disturbances comprising transmitted disturbances for a
plurality of spherical angles for said set of basis functions;
reducing a rank of said matrix of transmitted disturbances to yield
a set of composite basis functions; and transforming said first
matrix of interaction data to yield a second matrix of interaction
data comprising data for said set of composite basis functions,
wherein based on a desired accuracy relatively more elements of
said second matrix of interaction data may be approximated by zero
than of said first matrix of interaction data, and wherein said
first matrix of interaction data is not entirely contained within
said matrix of transmitted disturbances.
17. A method of data compression for a first interaction data based
on a set of basis functions, comprising: identifying a matrix of
transmitted disturbances comprising transmitted disturbances for a
plurality of spherical angles for said set of basis functions;
reducing a rank of said matrix of transmitted disturbances to yield
a set of composite basis functions; identifying a matrix of
received disturbances comprising received disturbances for a
plurality of spherical angles for said set of testing functions;
reducing a rank of said matrix of transmitted disturbances to yield
a set of composite testing functions; transforming said first
matrix of interaction data to yield a second matrix of interaction
data comprising data for said set of composite basis functions and
said set of composite testing functions, wherein based on a desired
accuracy relatively more elements of said second matrix of
interaction data may be approximated by zero than of said first
matrix of interaction data; and further wherein said first matrix
of interaction data is not entirely contained within one of said
matrix of transmitted disturbances and said matrix of received
disturbances.
18. The method of claim 17, wherein said first matrix of
interaction data is not entirely contained within said matrix of
transmitted disturbances and said first matrix of interaction data
is not entirely contained within said matrix of received
disturbances.
Description
REFERENCE TO RELATED APPLICATION
[0001] The present application is a divisional application of U.S.
application No. 09/676,727 titled "COMPRESSION AND COMPRESSED
INVERSION OF INTERACTION DATA", which was filed on Sep. 29, 2000
which claims priority under 35 U.S.C. 119(e) from U.S. Provisional
Application No. 60/175,454, filed on Jan. 10, 2000, titled
"COMPRESSION AND COMPRESSED INVERSION OF INTERACTION DATA" and from
U.S. Provisional Application No. 60/201,149, filed on May 2, 2000,
titled "COMPRESSION AND COMPRESSED INVERSION OF INTERACTION DATA,"
the contents of which are hereby incorporated by reference in their
entirety.
BACKGROUND OF THE INVENTION
[0002] 1. Field of the Invention
[0003] The invention relates to methods for compressing the stored
data, and methods for manipulating the compressed data, in
numerical solutions involving numerous mutual interactions,
especially when the nature of these interactions approaches an
asymptotic form for large distances, such as, for example, antenna
problems solved using the method of moments.
[0004] 2. Description of the Related Art
[0005] Many numerical techniques are based on a "divide and
conquer" strategy wherein a complex structure or a complex problem
is broken up into a number of smaller, more easily solved problems.
Such strategies are particularly useful for solving integral
equation problems involving radiation, heat transfer, scattering,
mechanical stress, vibration, and the like. In a typical solution,
a larger structure is broken up into a number of smaller
structures, called elements, and the coupling or interaction
between each element and every other element is calculated. For
example, if a structure is broken up into 16 elements, then the
inter-element mutual interaction (or coupling) between each element
and every other element can be expressed as a 16 by 16 interaction
matrix.
[0006] As computers become more powerful, such element-based
numerical techniques are becoming increasingly important. However,
when it is necessary to simultaneously keep track of many, or all,
mutual interactions, the number of such interactions grows very
quickly. The size of the interaction matrix often becomes so large
that data compression schemes are desirable or even essential.
Also, the number of computer operations necessary to process the
data stored in the interaction matrix can become excessive. The
speed of the compression scheme is also important, especially if
the data in the interaction matrix has to be decompressed before it
can be used.
[0007] Typically, especially with radiation-type problems involving
sound, vibration, stress, temperature, electromagnetic radiation,
and the like, elements that are physically close to one another
produce strong interactions. Elements that are relatively far apart
(usually where distance is expressed in terms of a size,
wavelength, or other similar metric) will usually couple less
strongly. For example, when describing the sound emanating from a
loudspeaker, the sound will change in character relatively quickly
in the vicinity of that speaker. If a person standing very near the
speaker moves one foot closer, the sound may get noticeably louder.
However, if that person is sitting at the other end of a room, and
moves one foot closer, then the change in volume of the sound will
be relatively small. This is an example of a general property of
many physical systems. Often, in describing the interaction of two
nearby objects, relatively more detail is needed for an accurate
description, while relatively less detail is needed when the two
objects are further apart.
[0008] As another example, consider a speaker producing sound
inside a room. To determine the sound intensity throughout that
room, one can calculate the movement (vibration) of the walls and
objects in the room. Typically such calculation will involve
choosing a large number of evenly spaced locations in the room, and
determining how each location vibrates. The vibration at any one
location will be a source of sound, which will typically react with
every other location in the room. The number of such interactions
would be very large and the associated storage needed to describe
such interactions can become prohibitively large. Moreover, the
computational effort needed to solve the matrix of interactions can
become prohibitive.
SUMMARY OF THE INVENTION
[0009] The present invention solves these and other problems by
providing a compression scheme for interaction data and an
efficient method for processing the compressed data without the
need to first decompress the data. In other words, the data can be
numerically manipulated in its compressed state.
[0010] Given a first region containing sources relatively near to
each other, and a second region containing sources relatively near
to each other, but removed from the first region; one embodiment
provides a simplified description of the possible interactions
between these two regions. That is, the first region can contain a
relatively large number of sources and a relatively large amount of
data to describe mutual interactions between sources within the
first region. In one embodiment, a reduced amount of information
about the sources in the first region is sufficient to describe how
the first region interacts with the second region. One embodiment
includes a way to find these reduced interactions with relatively
less computational effort than in the prior art.
[0011] For example, one embodiment includes a first region of
sources in one part of a problem space, and a second region of
sources in a portion of the problem space that is removed from the
first region. Original sources in the first region are modeled as
composite sources (with relatively fewer composite sources than
original sources). In one embodiment, the composite sources are
described by linear combinations of the original sources. The
composite sources are reacted with composite testers to compute
interactions between the composite sources and composite testers in
the two regions. The use of composite sources and composite testers
allows reactions in the room (between regions that are removed from
each other) to be described using fewer matrix elements than if the
reactions were described using the original sources and testers.
While an interaction matrix based on the original sources and
testers is typically not a sparse matrix, the interaction matrix
based on the composite sources and testers is typically a sparse
matrix having a block structure.
[0012] One embodiment is compatible with computer programs that
store large arrays of mutual interaction data. This is useful since
it can be readily used in connection with existing computer
programs. In one embodiment, the reduced features found for a first
interaction group are sufficient to calculate interactions with a
second interaction group or with several interaction groups. In one
embodiment, the reduced features for the first group are sufficient
for use in evaluating interactions with other interaction groups
some distance away from the first group. This permits the
processing of interaction data more quickly even while the data
remains in a compressed format. The ability to perform numerical
operations using compressed data allows fast processing of data
using multilevel and recursive methods, as well as using
single-level methods.
BRIEF DESCRIPTION OF THE DRAWINGS
[0013] The advantages and features of the disclosed invention will
readily be appreciated by persons skilled in the art from the
following detailed description when read in conjunction with the
drawings listed below.
[0014] FIG. 1A illustrates a wire or rod having a physical property
(e.g., a current, a temperature, a vibration, stress, etc.) I(l)
along its length, where the shape of I(l) is unknown.
[0015] FIG. 1B illustrates the wire from FIG. 1A, broken up into
four segments, where the function I(l) has been approximated by
three known basis functions f.sub.i(l), and where each basis
function is multiplied by an unknown constant I.sub.i.
[0016] FIG. 1C illustrates a piecewise linear approximation to the
function I(l) after the constants I.sub.i have been determined.
[0017] FIG. 2 is a flowchart showing the process steps used to
generate a compressed (block sparse) interaction matrix.
[0018] FIG. 3 illustrates partitioning a body into regions.
[0019] FIG. 4 shows an example of an interaction matrix (before
transformation) for a body partitioned into five differently sized
regions.
[0020] FIG. 5 shows an example of an interaction matrix after
transformation (but before reordering) for a body partitioned into
five regions of uniform size.
[0021] FIG. 6 shows an example of an interaction matrix after
transformation and reordering for a body partitioned into five
regions of uniform size.
[0022] FIG. 7 illustrates the block diagonal matrix D.sup.R.
[0023] FIG. 8 is a plot showing the digits of accuracy obtained
after truncating the basis functions for a block of the entire
interaction matrix, with a block size of 67 by 93.
[0024] FIG. 9 is a plot showing the digits of accuracy obtained
after truncating the basis functions for a block of the entire
interaction matrix, with a block size of 483 by 487.
[0025] FIG. 10, consisting of FIGS. 10A and 10B, is a flowchart
showing the process of generating a compressed (block sparse)
impedance matrix in connection with a conventional moment-method
computer program.
[0026] FIG. 11 is a three-dimensional plot showing magnitudes of
the entries in a 67 by 93 element block of the interaction matrix
(before transformation) for a wire grid model using the method of
moments.
[0027] FIG. 12 is a three-dimensional plot showing magnitudes of
the entries of the interaction matrix from FIG. 11 after
transformation.
[0028] In the drawings, the first digit of any three-digit number
generally indicates the number of the figure in which the element
first appears. Where four-digit reference numbers are used, the
first two digits indicate the figure number.
DETAILED DESCRIPTION
[0029] Many physical phenomena involve sources that generate a
disturbance, such as an electromagnetic field; electromagnetic
wave, a sound wave, vibration, a static field (e.g., electrostatic
field, magnetostatic field, gravity field, etc) and the like.
Examples of sources include a moving object (such as a loudspeaker
that excites sound waves in air) and an electrical current (that
excites electric and magnetic fields), etc. For example, the
electric currents moving on an antenna produce electromagnetic
waves. Many sources produce disturbances both near the source and
at a distance from the source.
[0030] Sometimes it is convenient to consider disturbances as being
created by an equivalent source (e.g., a fictitious source) rather
than a real physical source. For example, in most regions of space
(a volume of matter for example) there are a large number of
positive electric charges and a large number of negative electric
charges. These positive and negative charges nearly exactly cancel
each other out. It is customary to perform calculations using a
fictitious charge, which is the net difference between the positive
and negative charge, averaged over the region of space. This
fictitious charge usually cannot be identified with any specific
positive or negative particle.
[0031] A magnetic current is another example of a fictitious source
that is often used. It is generally assumed that magnetic monopoles
and magnetic currents do not exist (while electric monopoles and
electric currents do exist). Nevertheless, it is known how to
mathematically relate electric currents to equivalent magnetic
currents to produce the same electromagnetic waves. The use of
magnetic sources is widely accepted, and has proven very useful for
certain types of calculations. Sometimes, it is convenient to use a
source that is a particular combination of electric and magnetic
sources. A distribution of sources over some region of space can
also be used as a source. The terms "sources" and "physical
sources" are used herein to include all types of actual and/or
fictitious sources.
[0032] A physical source at one location typically produces a
disturbance that propagates to a sensor (or tester) at another
location. Mathematically, the interaction between a source and a
tester is often expressed as a coupling coefficient (usually as a
complex number having a real part and an imaginary part). The
coupling coefficients between a number of sources and a number of
testers is usually expressed as an array (or matrix) of complex
numbers. Embodiments of this invention includes efficient methods
for the computation of these complex numbers, for the storing of
these complex numbers, and for computations using these complex
numbers.
[0033] The so-called Method of Moments (MoM) is an example of
numerical analysis procedure that uses interactions between source
functions and testing functions to numerically solve a problem that
involves finding an unknown function (that is, where the solution
requires the determination of a function of one or more variables).
The MoM is used herein by way of example and not as a limitation.
One skilled in the art will recognize that the MoM is one of many
types of numerical techniques used to solve problems, such as
differential equations and integral equations, where one of the
unknowns is a function. The MoM is an example of a class of
solution techniques wherein a more difficult or unsolvable problem
is broken up into one or more interrelated but simpler problems.
Another example of this class of solution techniques is Nystrom's
method. The simpler problems are solved, in view of the known
interrelations between the simpler problems, and the solutions are
combined to produce an approximate solution to the original, more
difficult, problem.
[0034] For example, FIG. 1A shows a wire or rod 100 having a
physical property (e.g., a current, a temperature, a stress, a
voltage, a vibration, a displacement, etc.) along its length. An
expression for the physical property is shown as an unknown
function I(l). The problem is to calculate I(l) using the MoM or a
similar "divide and conquer" type of technique. By way of example,
in many physical problems involving temperature, vibration, or
electrical properties, etc. I(l) will be described by an integral
equation of the form: E({overscore (R)})=.intg.I(l)G(l, {overscore
(R)})dl
[0035] Where G(l, {overscore (R)}) is known everywhere and
E({overscore (R)}) is known for certain values of {overscore (R)}.
In many circumstances, G(l, {overscore (R)}) is a Green's function,
based on the underlying physics of the problem, and the value of
E({overscore (R)}) is known only at boundaries (because of known
boundary conditions). The above equation is usually not easily
solved because I(l) is not known, and thus the integration cannot
be performed. The above integral equation can be turned into a
differential equation (by taking the derivative of both sides), but
that will not directly provide a solution. Regardless of whether
the above equation is expressed as an integral equation or a
differential equation, the equation can be numerically solved for
I(l) by creating a set of simpler but interrelated problems as
described below (provided that G(l, {overscore (R)}) possesses
certain mathematical properties known to those of skill in the
art).
[0036] As shown in FIG. 1B, in order to compute a numerical
approximation for I(l), the wire 100 is first divided up into four
segments 101-104, and basis function f.sub.1(l), f.sub.2(l), and
f.sub.3(l) are selected. In FIG. 1B the basis functions are shown
as triangular-shaped functions that extend over pairs of segments.
The unknown function I(l) can then be approximated as:
I(l).apprxeq.I.sub.1f.sub.1(l)+I.sub.2f.sub.2(l)+I.sub.3f.sub.3(l)
[0037] where I.sub.1, I.sub.2, and I.sub.3 are unknown complex
constants. Approximating I(l) in this manner transforms the
original problem from one of finding an unknown function, to a
problem of finding three unknown constants. The above approximation
for I(l) is inserted into the original integral equation above to
yield: E({overscore (R)})=.intg.I.sub.1f.sub.1(l)G(l, {overscore
(R)})dl+.intg.I.sub.2f.sub.2(l)G(l, {overscore
(R)})dl+.intg.I.sub.3f.sub.3(l)G(l, {overscore (R)})dl
[0038] The above integrals can now be performed because the
functional form of the integrands are all known (G(l, {overscore
(R)}) was determined by the problem being solved, the functions
f.sub.i(l) were selected, and the constants I.sub.1, I.sub.2 and
I.sub.3 can be moved outside the integrals). However, this does not
yet solve the problem because the values of I.sub.1, I.sub.2 and
I.sub.3 are still unknown.
[0039] Fortunately, as indicated above, the value of E({overscore
(R)}) is usually known at various specific locations (e.g., at
boundaries). Thus, three equations can be written by selecting
three locations {overscore (R)}.sub.1, {overscore (R)}.sub.2,
{overscore (R)}.sub.3, where the value of E({overscore (R)}) is
known. Using these three selected locations, the above equation can
be written three times as follows: E({overscore
(R)}.sub.1)=I.sub.1.intg.f .sub.1(l)G(l, {overscore
(R)}.sub.1)dl+I.sub.2.intg.f.sub.2(l)G(l, {overscore (R)}.sub.1)
dl+I.sub.3.intg.f.sub.3(l)G(l, {overscore (R)}.sub.1)dl
E({overscore (R)}.sub.2)=I.sub.1.intg.f .sub.1(l)G(l, {overscore
(R)}.sub.2)dl+I.sub.2.intg.f.sub.2(l)G(l, {overscore (R)}.sub.2)
dl+I.sub.3.intg.f.sub.3(l)G(l, {overscore (R)}.sub.2)dl
E({overscore (R)}.sub.3)=I.sub.1.intg.f .sub.1(l)G(l, {overscore
(R)}.sub.3)dl+I.sub.2.intg.f.sub.2(l)G(l, {overscore (R)}.sub.3)
dl+I.sub.3.intg.f.sub.3(l)G(l, {overscore (R)}.sub.3)dl
[0040] Rather than selecting three specific locations for
E({overscore (R)}), it is known that the accuracy of the solution
is often improved by integrating known values of E({overscore (R)})
using a weighting function over the region of integration. For
example, assuming that E({overscore (R)}) is known along the
surface of the wire 100, then choosing three weighting functions
g.sub.1(l), g.sub.2(l), and g.sub.3(l), the desired three equations
in three unknowns can be written as follows (by multiplying both
sides of the equation by g.sub.i(l) and integrating): .intg. E
.times. .times. ( ' ) .times. g 1 .function. ( ' ) .times. .times.
d ' = I 1 .times. .intg. .intg. f 1 .function. ( ) .times. g 1
.function. ( ' ) .times. .times. G .times. .times. ( , ' ) .times.
d .times. d ' + I 2 .times. .intg. .intg. f 2 .function. ( )
.times. .times. g 1 .function. ( ' ) .times. .times. G .times.
.times. ( , ' ) .times. .times. d .times. d ' + I 3 .times. .intg.
.intg. f 3 .function. ( ) .times. .times. g 1 .function. ( ' )
.times. .times. G .times. .times. ( , l ' ) .times. .times. d
.times. d ' ##EQU1## .intg. E .times. .times. ( ' ) .times. .times.
g 2 .function. ( ' ) .times. .times. d ' = I 1 .times. .intg.
.intg. f 1 .function. ( ) .times. .times. g 2 .function. ( ' )
.times. .times. G .times. .times. ( , ' ) .times. .times. d .times.
d ' + I 2 .times. .intg. .intg. f 2 .function. ( ) .times. .times.
g 2 .function. ( ' ) .times. .times. G .times. .times. ( , ' )
.times. .times. d .times. d ' + I 3 .times. .intg. .intg. f 3
.function. ( ) .times. g 2 .function. ( ' ) .times. .times. G
.times. .times. ( , ' ) .times. .times. d .times. d ' ##EQU1.2##
.intg. E .times. .times. ( ' ) .times. .times. g 3 .function. ( ' )
.times. .times. d ' .times. = I 1 .times. .intg. .intg. f 1
.function. ( ) .times. .times. g 3 .function. ( ' ) .times. .times.
G .times. .times. ( , ' ) .times. .times. d .times. d ' + I 2
.times. .intg. .intg. f 2 .function. ( ) .times. .times. g 3
.function. ( ' ) .times. .times. G .times. .times. ( , ' ) .times.
.times. d .times. d ' + I 3 .times. .intg. .intg. f 3 .function. (
) .times. g 3 .function. ( ' ) .times. .times. G .times. .times. (
, ' ) .times. .times. d .times. d ' ##EQU1.3##
[0041] Note that the above double-integral equations reduce to the
single-integral forms if the weighting functions g.sub.i(l) are
replaced with delta functions.
[0042] The three equations in three unknowns can be expressed in
matrix form as: V=ZI or [ V 1 V 2 V 3 ] = [ Z 11 Z 12 Z 13 Z 21 Z
22 Z 23 Z 31 Z 32 Z 33 ] .function. [ I 1 I 2 I 3 ] ##EQU2## where
V.sub.i=.intg.E(l')g.sub.i(l')dl' and
Z.sub.ij=.intg..intg.f.sub.j(l)g.sub.i(l')G(l, l')dldl'
[0043] Solving the matrix equation yields the values of I.sub.1,
I.sub.2, and I.sub.3. The values I.sub.1, I.sub.2, and I.sub.3 can
then be inserted into the equation
I(l).apprxeq.I.sub.1f.sub.1(l)+I.sub.2f.sub.2(l)+I.sub.3f.sub.3(l)
to give an approximation for I(l). If the basis functions are
triangular functions as shown in FIG. 1B, then the resulting
approximation for I(l) is a piecewise linear approximation as shown
in FIG. 1C. The I.sub.i are the unknowns and the V.sub.i are the
conditions (tpyically, the V.sub.i are knowns). Often there are the
same number of conditions as unknowns. In other cases, there are
more conditions than unknowns or less conditions than unknown.
[0044] The accuracy of the solution is largely determined by the
shape of the basis functions, by the shape of the weighting
functions, and by the number of unknowns (the number of unknowns
usually corresponds to the number of basis functions).
[0045] Unlike the Moment Method described above, some techniques do
not use explicit basis functions, but, rather, use implicit basis
functions or basis-like functions. For example, Nystrom's method
produces a numerical value for an integral using values of the
integrand at discrete points and a quadrature rule. Although
Nystrom's method does not explicitly use an expansion in terms of
explicit basis functions, nevertheless, in a physical sense, basis
functions are still being used (even if the use is implicit). That
is, the excitation of one unknown produces some reaction throughout
space. Even if the computational method does not explicitly use a
basis function, there is some physical excitation that produces
approximately the same reactions. All of these techniques are
similar, and one skilled in the art will recognize that such
techniques can be used with the present invention. Accordingly, the
term "basis function" will be used herein to include such
implicitly used basis functions. Similarly, the testers may be
implicitly used.
[0046] When solving most physical problems (e.g., current, voltage,
temperature, vibration, force, etc), the basis functions tend to be
mathematical descriptions of the source of some physical
disturbance. Thus, the term "source" is often used to refer to a
basis function. Similarly, in physical problems, the weighting
functions are often associated with a receiver or sensor of the
disturbance, and, thus, the term "tester" is often used to refer to
the weighting functions.
[0047] As described above in connection with FIGS. 1A-1C, in
numerical solutions, it is often convenient to partition a physical
structure or a volume of space into a number of smaller pieces and
associate the pieces with one or more sources and testers. In one
embodiment, it is also convenient to partition the structure of (or
volume) into regions, where each region contains a group of the
smaller pieces. Within a given region, some number of sources is
chosen to describe with sufficient detail local interactions
between sources and testers within that region. A similar or
somewhat smaller number of sources in a given region is generally
sufficient to describe interactions between sources in the source
region and testers in the regions relatively close by. When the
appropriate sources are used, an even smaller number of sources is
often sufficient to describe interactions between the source region
and testers in regions that are not relatively close by (i.e.,
regions that are relatively far from the source region).
[0048] Embodiments of the present invention include methods and
techniques for finding composite sources. Composite sources are
used in place of the original sources in a region such that a
reduced number of composite sources is needed to calculate the
interactions with a desired accuracy.
[0049] In one embodiment, the composite sources for a first region
are the same regardless of whether the composite sources in the
first region are interacting with a second region, a third region,
or other regions. The use of the same composite sources throughout
leads to efficient methods for factoring and solving the
interaction matrix.
[0050] Considering the sources in the first region, one type of
source is the so-called multipole, as used in a multipole
expansion. Sources like wavelets are also useful. In some cases
wavelets allow a reduced number of composite sources to be used to
describe interactions with distant regions. However, there are
disadvantages to wavelet and multipole approaches. Wavelets are
often difficult to use, and their use often requires extensive
modifications to existing or proposed computer programs. Wavelets
are difficult to implement on non-smooth and non-planar bodies.
[0051] Multipole expansions have stability problems for slender
regions. Also, while a multipole expansion can be used for
describing interactions with remote regions, there are severe
problems with using multipoles for describing interactions within a
region or between spatially close regions. This makes a
factorization of the interaction matrix difficult. It can be very
difficult to determine how to translate information in an
interaction matrix into a wavelet or multipole representation.
[0052] FIG. 2 is a flowchart that illustrates a compression
technique 200 for compressing an interaction matrix by combining
groups of sources and groups of testers into composite sources and
testers. The use of composite sources and composite testers allows
the original interaction matrix to be transformed into a block
sparse matrix having certain desirable properties.
[0053] Embodiments of the present invention include a technique for
computing and using composite sources to provide compression of an
interaction matrix by transforming the interaction matrix into a
block sparse matrix. The present technique is compatible with
existing and proposed computer programs. It works well even for
rough surfaces and irregular grids of locations. For a given
region, the composite sources allow computation of a disturbance
(e.g., radiation) produced by the source throughout a desired
volume of space. A reduced number of these composite sources is
sufficient to calculate (with a desired accuracy) disturbances at
other relatively distant regions. This method of compressing
interaction data can be used with a variety of computational
methods, such as, for example, an LU (Lower Triangular Upper
triangular) factorization of a matrix or as a preconditioned
conjugate gradient iteration. In many cases, the computations can
be done while using the compressed storage format.
[0054] FIG. 2 is a flowchart 200 illustrating the steps of solving
a numerical problem using composite sources. The flowchart 200
begins in a step 201 where a number of original sources and
original testers are collected into groups, each group
corresponding to a region. Each element of the interaction matrix
describes an interaction (a coupling) between a source and a
tester. The source and tester are usually defined, in part, by
their locations in space. The sources and testers are grouped
according to their locations in space. In one embodiment, a number
of regions of space are defined. A reference point is chosen for
each region. Typically the reference point will lie near the center
of the region. The sources and testers are grouped into the regions
by comparing the location of the source or tester to the reference
point for each region. Each source or tester is considered to be in
the region associated with the reference point closest to the
location. (For convenience, the term "location" is used hereinafter
to refer to the location of a source or a tester.)
[0055] Other methods for grouping the sources and testers (that is,
associating locations with regions) can also be used. The process
of defining the regions is problem-dependent, and in some cases the
problem itself will suggest a suitable set of regions. For example,
if the sources and testers are located on the surface of a sphere,
then curvilinear-square regions are suggested. If the sources and
testers are located in a volume of space, then cubic regions are
often useful. If the sources and testers are located on a complex
three-dimensional surface, then triangular patch-type regions are
often useful.
[0056] Generally the way in which the regions are defined is not
critical, and the process used to define the regions will be based
largely on convenience. However, it is usually preferable to define
the regions such that the locations of any region are relatively
close to each other, and such that there are relatively few
locations from other regions close to a given region. In other
words, efficiency of the compression algorithm is generally
improved if the regions are as isolated from one another as
reasonably possible. Of course, adjacent regions are often
unavoidable, and when regions are adjacent to one another,
locations near the edge of one region will also be close to some
locations in an adjacent region. Nevertheless, the compression will
generally be improved if, to the extent reasonably possible,
regions are defined such that they are not slender, intertwining,
or adjacent to one another. For example, FIG. 3 illustrates a
volume of space partitioned into a rectangular box 300 having
eleven regions A through K corresponding to reference points
301-311.
[0057] As shown in FIG. 2, after the step 201 the process advances
to a step 202. In the step 202, the unknowns are renumbered, either
explicitly or implicitly, so that locations within the same region
are numbered consecutively. It is simpler to continue this
description as if the renumbering has actually been done
explicitly. However, the following analysis can also be performed
without explicit renumbering.
[0058] The term "spherical angles" is used herein to denote these
angles. One skilled in the art will recognize that if a
two-dimensional problem is being solved, then the spherical angles
reduces to a planar angle. Similarly, one skilled in the art will
recognize that if a higher-dimensional problem is being solved
(such as, for example, a four dimensional space having three
dimensions for position and one dimension for time) then the term
spherical angle denotes the generalization of the three-dimensional
angle into four-dimensional space. Thus, in general, the term
spherical angle is used herein to denote the notion of a
"space-filling" angle for the physical problem being solved.
[0059] After renumbering, the process advances to a block 203 where
one or more composite sources for each region are determined. If
there are p independent sources within a region, then q composite
sources can be constructed (where q.ltoreq.p). The construction of
composite sources begins by determining a relatively dense set of
far-field patterns (usually described in a spherical coordinate
system) at relatively large distances from the region. As used
herein, far-field refers to the field in a region where the field
can be approximated in terms of an asymptotic behavior. For
example, in one embodiment, the far-field of an antenna or other
electromagnetic radiator includes the field at some distance from
the antenna, where the distance is relatively larger than the
electrical size of the antenna.
[0060] A far-field pattern is constructed for each independent
source. In the present context, dense means to avoid having any
overly-large gaps in the spherical angles used to calculate the set
of disturbances. Dense also means that if the disturbance is
represented by a vector, then each vector component is represented.
For example, for a scalar problem, one might choose p spherical
angles. These angles are typically substantially equally spaced,
and the ranges of angles include the interaction angles occurring
in the original interaction matrix (if all of the interactions
described in the original matrix lie within a plane, then one can
choose directions only within that plane rather than over a
complete sphere).
[0061] The far-field data is stored in a matrix s having p columns
(one column for each source location within the region), and rows
associated with angles. While each source is logically associated
with a location in a given region, these sources are not
necessarily located entirely within that region. While each source
corresponds to a location (and each location is assigned to a
region), sources that have a physical extent can extend over more
than one region. The entries in the matrix s can be, for example,
the field quantity or quantities that emanate from each source. It
is desirable that the field quantity is chosen such that when it
(or they) are zero at some angle then, to a desired approximation,
all radiated quantities are zero at that angle. While it is
typically desirable that the angles be relatively equally spaced,
large deviations from equal spacing can be acceptable.
[0062] These composite sources are in the nature of equivalent
sources. A smaller number of composite sources, compared to the
number of sources they replace, can produce similar disturbances
for regions of space removed from the region occupied by these
sources.
[0063] As described above, sources are collected into groups of
sources, each group being associated with a region. For each group
of sources, a group of composite sources is calculated. The
composite source is in the nature of an equivalent source that, in
regions of space removed from the region occupied by the group in
replaces, produces a far-field (disturbance) similar to the field
produced by the group it replaces. Thus, a composite source (or
combination of composite sources) efficiently produces the same
approximate effects as the group of original sources at desired
spherical angles and at a relatively large distance. To achieve a
relatively large distance, is it often useful to use a limiting
form as the disturbance goes relatively far from its source.
[0064] Each composite source is typically a linear combination of
one or more of the original sources. A matrix method is used to
find composite sources that broadcast strongly and to find
composite sources that broadcast weakly. These composite sources
are constructed from the original sources. The matrix method used
to find composite sources can be a rank-revealing factorization
such as singular value decomposition. For a singular value
decomposition, the unitary transformation associated with the
sources gives the composite sources as a linear combination of
sources.
[0065] Variations of the above are possible. For example, one can
apply the singular value decomposition to the transpose of the s
matrix. One can employ a Lanczos Bidiagonalization, or related
matrix methods, rather than a singular value decomposition. There
are other known methods for computing a low rank approximation to a
matrix. Some examples of the use of Lanczos Bidiagonalization are
given in Francis Canning and Kevin Rogovin, "Fast Direct Solution
of Standard Moment-Method Matrices," IEEE AP Magazine, Vol. 40, No.
3, June 1998, pp. 15-26.
[0066] There are many known methods for computing a reduced rank
approximation to a matrix. A reduced rank approximation to a matrix
is also a matrix. A reduced rank matrix with m columns can be
multiplied by any vector of length m. Composite sources that
broadcast weakly are generally associated with the space of vectors
for which that product is relatively small (e.g., in one
embodiment, the product is zero or close to zero). Composite
sources that broadcast strongly are generally associated with the
space of vectors for which that product is not necessarily
small.
[0067] Composite sources can extend over more than one region. In
one embodiment, this is achieved by using the technique used with
Malvar wavelets (also called local cosines) to extend Fourier
transforms on disjoint intervals to overlapping orthogonal
functions.
[0068] Persons of ordinary skill in the art know how near-field
results are related to far-field results. A relationship between
near-field and far-field can be used in a straightforward way to
transform the method described above using far-field data into a
method using near-field data. Note that, the "far-field" as used
herein is not required to correspond to the traditional
2d.sup.2/.lamda. far-field approximation. Distances closer than
2d.sup.2/.lamda. can be used (although closer distances will
typically need more composite sources to achieve a desired
accuracy). A distance corresponding to the distance to other
physical regions is usually far enough, and even shorter distances
can be acceptable.
[0069] Once composite sources are found, the process advances to a
step 204 where composite testers are found. Composite testers are
found in a manner analogous to the way that composite sources are
found. Recall that composite sources are found using the way in
which sources of the interaction matrix "broadcast" to distant
locations. Composite testers are found using the way in which the
testers of the interaction matrix "receive" from a dense group of
directions for a distant disturbance. It is helpful if the received
quantity or quantities which are used include relatively all field
quantities, except (optionally) those which are very weakly
received. For example, when receiving electromagnetic radiation
from a distant source, the longitudinal component is approximately
zero and can often be neglected. A matrix R describing how these
testers receive is formed. A matrix method is used to construct
composite testers that receive strongly and testers that receive
weakly. The matrix method can be a rank-revealing factorization
such as singular value decomposition. A singular value
decomposition gives the composite testers as a linear combination
of the testers which had been used in the original matrix
description.
[0070] Once composite sources and testers have been found, the
process advances to a step 205 or to a step 215 where the
interaction matrix is transformed to use composite sources and
testers. The steps 205 and 215 are alternatives. FIG. 4 shows an
example of an interaction matrix 400 having 28 unknowns (28 sources
and 28 testers) grouped into five physical regions (labeled I-V).
The shaded block 401 of the matrix 400 represents the interaction
for sources in the fourth region (region IV) and testers in the
second region (region II). The interaction of a pair of regions
describes a block in the interaction matrix 400. The blocks of the
transformed matrix can be computed at any time after the composite
functions for their source and tester regions are both found. That
is, the block 401 can be computed after composite sources for
region IV and testers for region II are found.
[0071] The step 215 of FIG. 2 shows one method for computing all of
the blocks in the matrix 400 by computing the entries for these
blocks using the original sources and testers. Then, the process
advances to an optional step 216 where these blocks are transformed
into a description in terms of the composite sources and composite
testers.
[0072] One advantage of using composite sources and testers is that
many entries in the transformed matrix will be zero. Therefore,
rather than transforming into a description using composite modes,
the step 205 shows calculation of the transformed block directly
using the composite sources and composite testers (without first
calculating the block using the original sources and testers). In
other words, the composite sources are used as basis functions, and
the composite testers are used as weighting functions. Within each
block, entries that are known au priori to be zero (or very small)
are not calculated.
[0073] Further savings in the storage required are possible. After
each block has been transformed, only the largest elements are
kept. No storage needs to be used for the elements that are
approximately zero. Many types of block structures, including
irregular blocks and multilevel structures, can also be improved by
the use of this method for storing a block sparse matrix. This will
usually result in a less regular block structure. As an
alternative, it is also possible to store a portion of the
interaction data using composite sources and testers and to store
one or more other portions of the data using another method.
[0074] The non-zero elements of the interaction matrix typically
occur in patterns. After either the step 205 or the step 216, the
process advances to a step 206 where the interaction matrix is
reordered to form regular patterns. For a more uniform case, where
all of the regions have the same number of sources, the resulting
transformed matrix T is shown in FIG. 5. FIG. 5 shows non-zero
elements as shaded and zero elements as unshaded. If only a
compressed storage scheme is desired, the process can stop here.
However, if it is desired to calculate the inverse of this matrix,
or something like its LU (lower-upper triangular) factorization,
then a reordering can be useful.
[0075] The rows and columns of the interaction matrix can be
reordered, to produce a matrix T in the form shown in FIG. 6. This
permutation moves the composite sources that broadcast strongly to
the bottom of the matrix, and it moves the composite testers which
receive strongly to the right side of the matrix. The interaction
between composite sources and composite testers is such that the
sizes of the matrix elements can be estimated au priori. A matrix
element that corresponds to an interaction between a composite
source that radiates strongly and a composite tester that receives
strongly will be relatively large. A matrix element that
corresponds to an interaction between a composite source that
radiates strongly and a composite tester that receives weakly will
be relatively small. Similarly, a matrix element that corresponds
to an interaction between a composite source that radiates weakly
and a composite tester that receives strongly will be relatively
small. A matrix element that corresponds to an interaction between
a composite source that radiates weakly and a composite tester that
receives weakly will be very small.
[0076] The permuted matrix T often will tend to be of a banded
form. That is, the non-zero elements down most of the matrix will
tend to be in a band near the diagonal. For a matrix of this form,
there are many existing sparse-matrix LU factorers and other matrix
solvers, that work well. The order shown in FIG. 6 is one example.
The LU decomposition of the matrix T can be computed very rapidly
by standard sparse matrix solvers. The matrices L and U of the LU
decomposition will themselves be sparse. For problems involving
certain types of excitations, only a part of the matrices L and U
will be needed, and this can result in further savings in the
storage required.
[0077] After reordering, the process 200 advances to a step 207
where the linear matrix problem is solved. The matrix problem to be
solved is written as: T G=H
[0078] where the vector H represents the excitation and the vector
G is the desired solution for composite sources. The excitation is
the physical cause of the sound, temperature, electromagnetic
waves, or whatever phenomenon is being computed. If the excitation
is very distant (for example, as for a plane wave source), H will
have a special form. If the vector H is placed vertically (as a
column vector) alongside the matrix of FIG. 6, the bottom few
elements of H alongside block 602, will be relatively large, and
the remaining elements of H will be approximately equal to zero.
The remaining elements of H are approximately zero because the
composite testers separate the degrees of freedom according to how
strongly they interact with a distant source.
[0079] When T is factored by LU decomposition, then: T =LU;
LUG=H;
[0080] and this is solved by the following two-step process;
[0081] Step I: Find X in LX=H
[0082] Step II: Find G in UG=X
[0083] The matrix L is a lower triangular matrix (meaning elements
above its diagonal are zero). It follows immediately from this that
if only the bottom few elements of H are non-zero, then only the
bottom elements of X are non-zero. As a consequence, only the
bottom right portion of L is needed to compute G. The remaining
parts of L were used in computing this bottom right portion, but
need not be kept throughout the entire process of computing the LU
decomposition. This not only results in reduced storage, but also
results in a faster computation for Step I above.
[0084] If only the far-field scattered by an object needs to be
found, then further efficiencies are possible. In that case, it is
only necessary to find the bottom elements of G, corresponding to
the bottom non-zero elements of H. This can be done using only the
bottom right portion of the upper triangular matrix U. This results
in efficiencies similar to those obtained for L.
[0085] For other types of excitations, similar savings are also
possible. For example, for many types of antennas, whether acoustic
or electromagnetic, the excitation is localized within one active
region, and the rest of the antenna acts as a passive scatterer. In
that case, the active region can be arranged to be represented in
the matrix of FIG. 6 as far down and as far to the right as
possible. This provides efficiencies similar to those for the
distant excitation.
[0086] A permutation of rows and a permutation of columns of the
matrix T of FIG. 5 brings it to the matrix T of FIG. 6. These
permutations are equivalent to an additional reordering of the
unknowns. Thus, a solution or LU decomposition of the matrix T of
FIG. 6 does not immediately provide a solution to the problem for
the original data. Several additional steps are used. These steps
involve: including the effects of two permutations of the unknowns;
and also including the effect of the transformation from the
original sources and testers to using the composite sources and
composite testers.
[0087] Direct methods (such as LU decomposition) and iterative
methods can both be used to solve the matrix equation herein. An
iterative solution, with the compressed form of the matrix, can
also be used with fewer computer operations than in the prior art.
Many iterative methods require the calculation of the product of a
matrix and a vector for each iteration. Since the compressed matrix
has many zero elements (or elements which may be approximated by
zero), this can be done more quickly using the compressed matrix.
Thus, each iteration can be performed more quickly, and with less
storage, than if the uncompressed matrix were used.
[0088] The compressed format of T has an additional advantage. In
many cases, there is a way to substantially reduce the number of
iterations required, resulting in further increases in speed. For
example, in the method of conjugate gradients, the number of
iterations required to achieve a given accuracy depends on the
condition number of the matrix. (The condition number of a matrix
is defined as its largest singular value divided by its smallest.)
Physical problems have a length scale, and one interpretation of
these composite sources and composite testers involves length
scales. These composite sources and composite testers can be
described in terms of a length scale based on a Fourier transform.
This physical fact can be used to improve the condition number of
the matrix and therefore also improve the speed of convergence of
the iterative method.
[0089] A composite source is a function of spatial position, and
its Fourier transform is a function of "spatial frequency."
Composite sources that broadcast weakly tend to have a Fourier
transform that is large when the absolute value of this spatial
frequency is large. There is a correlation between how large this
spatial frequency is and the smallness of the small singular values
of the matrix. This correlation is used in the present invention to
provide a method to achieve convergence in fewer iterations.
[0090] Two matrices, P.sup.R and P.sup.L are defined as right and
left preconditioning matrices to the compressed matrix. Each column
of the compressed matrix is associated with a composite source.
This composite source can be found using a matrix algebra method,
such as a rank-revealing factorization (e.g., singular value
decomposition and the like). The rank-revealing factorization
method provides some indication of the strength of the interaction
between that composite source and other disturbances. For example,
using a singular value decomposition, the associated singular value
is proportional to this strength. The diagonal matrix P.sup.R is
constructed by choosing each diagonal element according to the
interaction strength for the corresponding composite source. The
diagonal element can be chosen to be the inverse of the square root
of that strength. Similarly, the diagonal matrix P.sup.L can be
constructed by choosing each diagonal element according to the
interaction strength for its associated composite tester. For
example, the diagonal element can be chosen to be the inverse of
the square root of that strength.
[0091] If the compressed matrix is called T, then the
preconditioned matrix is P=P.sup.LTP.sup.R
[0092] The matrix P will often have a better (i.e., smaller)
condition number than the matrix T. There are many iterative
methods that will converge more rapidly when applied to the
preconditioned matrix P rather than to T.
[0093] One embodiment of the composite source compression technique
is used in connection with the computer program NEC2. This program
was written at Lawrence Livermore National Laboratory during the
1970s and early 1980s. The NEC2 computer program itself and manuals
describing its theory and use are freely available over the
Internet. The following development assumes NEC2 is being used to
calculate the electromagnetic fields on a body constructed as a
wire grid.
[0094] NEC2 uses electric currents flowing on a grid of wires to
model electromagnetic scattering and antenna problems. In its
standard use, NEC2 generates an interaction matrix, herein called
the Z matrix. The actual sources used are somewhat complicated.
There is at least one source associated with each wire segment.
However, there is overlap so that one source represents current
flowing on more than one wire segment. NEC2 uses an array CURX to
store values of the excitation of each source.
[0095] FIG. 10 is a flowchart 1000 showing the process of using
NEC2 with composite sources and composite testers. The flowchart
1000 begins at a step 1001 where the NEC2 user begins, as usual, by
setting up information on the grid of wires and wire segments. The
process then advances to a step 1002 to obtain from NEC2 the number
of wire segments, their locations (x,y,z coordinates), and a unit
vector {circumflex over (t)} for each segment. The vector
{circumflex over (t)} is tangent along the wire segment, in the
direction of the electric current flow on the wire segment.
[0096] Next, in a step 1003, the wire grid is partitioned into
numbered regions. A number of reference points are chosen. The
reference points are roughly equally spaced over the volume
occupied by the wire grid. Each wire segment is closest to one of
these reference points, and the segment is considered to be in the
region defined by the closest reference point. In one embodiment,
the number of such points (and associated regions) is chosen as the
integer closest to the square root of N (where N is the total
number of segments). This is often an effective choice, although
the optimum number of points (and associated regions) depends on
many factors, and thus other values can also be used. For a set of
N segments, each wire segment has an index, running from 1 to
N.
[0097] After the step 1003, the process advances to a step 1004
where the wires are sorted by region number. After sorting, the
numbering of the wires is different from the numbering used by
NEC2. Mapping between the two numbering systems is stored in a
permutation table that translates between these different indexes
for the wire segments. Using this new numbering of segments, an
array "a" is created, such that a(p) is the index of the last wire
segment of the p.sup.th region (define a(0)=0 for convenience).
[0098] After renumbering, the process advances to a step 1005 where
composite sources and composite testers for all regions are
calculated. Source region p corresponds to unknowns a(p-1)+1
through a(p) in the ordering. Define M as M=a(p)-a(p-1). Choose M
directions substantially equally spaced throughout
three-dimensional space. In other words, place M roughly equally
spaced points on a sphere, and then consider the M directions from
the center of the sphere to each point. The order of the directions
is unimportant. One convenient method for choosing these points is
similar to choosing points on the earth. For example, choose the
North and South poles as points. A number of latitudes are used for
the rest of the points. For each chosen latitude, choose points
equally spaced at a number of longitudes. This is done so that the
distance along the earth between points along a latitude is
approximately the same as the distance between the latitude lines
holding the points. It is desirable that the points are equally
spaced. However, even fairly large deviations from equal spacing
are tolerable.
[0099] Now generate a matrix A of complex numbers with 2M rows and
M columns. For m=1 to M and for n=1 to M, compute elements of this
matrix two at a time: the element at row m and column n and also
the element at row m+M and column n. To compute these two elements,
first fill the NEC2 array CURX with zero in every position. Then,
set position a(p-1)+n of CURX to unity. A value of unity indicates
that only source number a(p-1)+n is excited. This source is
associated with the wire segment of that number, even though it
extends onto neighboring segments. The matrix Z is defined in terms
of these same sources. Then, call the NEC2 subroutine CABC (CURX).
The subroutine CABC generates a different representation of the
source, but the same representation that the NEC2 subroutine FFLD
uses. This representation is automatically stored within NEC2. The
m.sup.th direction previously chosen can be described in spherical
coordinates by the pair of numbers (Theta, Phi). After calculating
Theta and Phi, the NEC2 subroutine FFLD (Theta, Phi, ETH, EPH) is
called. Theta and Phi are inputs, as are the results from CABC. The
outputs from FFLD are the complex numbers ETH and EPH. ETH and EPH
are proportional to the strengths of the electric field in the
far-field (far away from the source) in the theta and phi
directions respectively. ETH is placed in row m and column n,
(m,n), of A. EPH is placed at row m+M and column n of A.
Alternatively, there are other ways to compute the numbers ETH and
EPH produced by FFLD. For example, it will apparent to one of
ordinary skill in the art that the subroutine FFLD can be modified
to produce an answer more quickly by making use of the special form
of the current, since most of the entries in the current are
zero.
[0100] Next, a singular value decomposition of A is performed, such
that: A=UDV.sup.h
[0101] where U and V are unitary matrices, and D is a diagonal
matrix. The matrix U will not be used, so one can save on computer
operations by not actually calculating U. The matrix V has M rows
and M columns. Since these calculations were performed for the
p.sup.th region, the square matrix d.sub.p.sup.R is defined by
d.sub.p.sup.R=V
[0102] The reason for this choice comes from the fact that
AV=UD
[0103] and that each successive columns of the product UD tends to
become smaller in magnitude. They become smaller because U is
unitary and the singular values on the diagonal of D decrease going
down the diagonal.
[0104] Next, assemble an N by N block diagonal matrix D.sup.R. That
is, along the diagonal the first block corresponds to p=1. Starting
at the bottom right corner of that block, attach the block for p=2,
etc., as shown in FIG. 7.
[0105] Next a similar procedure is followed to find the block
diagonal matrix D.sup.L. For each region p, a matrix A is filled as
before. However, this time this region is considered as receiving
rather than as transmitting. Again A will have 2M rows and M
columns, where M=a(p)-a(p-1). Again there are M directions, but now
those are considered to be the receiving directions.
[0106] To understand what is to be put into A, it is instructive to
note how the NEC2 computer program defines the interaction matrix
Z. When used with wire grid models, the sources radiate electric
and magnetic fields. However, it is the electric field reaching
another segment that is used in NEC2. Each matrix element of Z is
computed by computing the component of that electric field which is
in the direction of the tangent to the wire segment.
[0107] For the pair of numbers (m,n), where m=1, . . . , M and n=1,
. . . , M, the matrix entries for A at (m,n) and (m+M,n) are
calculated as follows. Compute a unit vector {circumflex over (k)}
in the m.sup.th direction. Find the unit vector tangent to segment
number n, and call it {circumflex over (t)}. The position of the
center of wire segment number n is found and is designated as the
vector X. Then compute the vector {overscore (f)}=({circumflex over
(t)}-({circumflex over (k)}{circumflex over (t)}){circumflex over
(k)})e.sup.j2.pi.{circumflex over (k)}{overscore (X)}/.lamda.
[0108] where the wavelength is given by .lamda. (NEC2 uses units
where .lamda.=1).
[0109] Note that the Green's function for this problem has a minus
sign in the exponential, and the foregoing expression does not.
This is because the direction of {circumflex over (k)} is outward,
which is opposite to the direction of propagation of the
radiation.
[0110] For problems in electromagnetics, the physical wavelength
.lamda. is greater than zero. If a problem in electrostatics were
being solved instead, electrostatics can be considered as the limit
when the wavelength becomes arbitrarily large. The complex
exponential above can then be replaced by unity. Also, for
electrostatics, the relevant field quantity can be longitudinal
(meaning f would be parallel to {circumflex over (k)}).
[0111] For this value of m (and associated direction {circumflex
over (k)}), spherical coordinates define two directions called the
theta and the phi directions. These directions are both
perpendicular to the direction of {circumflex over (k)}. Compute
the components of f in each of these directions, and designate them
as fTheta and fPhi. These are complex numbers. Then place fTheta in
row m and column n of A and place fPhi in row m+M and column n of
A.
[0112] The matrix A is a matrix of complex numbers. Take the
complex conjugate of A, (A*), and perform a singular value
decomposition on it, such that: A*=UDV.sup.h
[0113] Now define the left diagonal block for region p,
d.sub.p.sup.L, as d.sub.p.sup.L=V.sup.h
[0114] The superscript h on V, indicates Hermitian conjugate. The
definition of the blocks for the right side did not have this
Hermitian conjugate. From these diagonal blocks, assemble an N by N
matrix D.sup.L in the same way as D.sup.R was assembled earlier.
The motivation for these choices is partly that the matrix DU.sup.h
has rows that tend to become smaller. Further, it is expected that
the Green's function that was used in creating Z has properties
similar to the far-field form used in creating A.sup.t. The formula
V.sup.hA.sup.t=DU.sup.h
[0115] shows that V.sup.h A.sup.t will also have successive rows
that tend to become smaller. The choices described above suggest
that successive rows of each block of the compressed matrix will
also have that property.
[0116] It should be noted that the matrix A, whether used for the
right side or for the left side, can be filled in other ways as
well. For example, with an appropriate (consecutive in space)
ordering of the angles, A can be made as an M by M matrix by using
theta polarization (fTheta) values for one angle and phi
polarization values (fPhi) for the next. Usually, it is desirable
that A does not leave large gaps in angle for any component of the
electric field, which is important far from the source or
receiver.
[0117] In performing the singular value decompositions for the
right and left sides, singular values are found each time. FIGS. 8
and 9 show the singular values found for blocks of size 67 by 93
and 483 by 487, respectively. These calculations were done for a
wire grid model with NEC2. The singular values are plotted in terms
of how many orders of magnitude they are smaller than the largest
singular value, and this is called "Digits of Accuracy" on the
plots. FIGS. 8 and 9 show the accuracy that is achieved when
truncating to a smaller number of composite sources or composite
testers for regions that are relatively far apart. For regions that
are closer together, the desired accuracy often requires the
information from more composite sources and composite testers to be
kept.
[0118] After computing composite sources and composite testers, the
process advances to a step 1006 where a new matrix T, which uses
the composite sources and testers associated with D.sup.L and
D.sup.R, is computed. The matrix is T given by the equation
T=D.sup.LZD.sup.R
[0119] T can be efficiently generated by using the numbering of the
wire segments developed herein (rather than the numbering used in
NEC2). The matrix Z is computed by NEC2 and renumbered to use the
numbering described herein. Note that a block structure has been
overlaid on Z and T. This block structure follows from the choice
of regions. FIG. 4 shows one example of a block structure. Block
{p, q} of the matrix T, to be called T{p,q}, is the part of T for
the rows in region number p and the columns in region number q. The
formula for T given above is such that T{p,q} only depends on
Z{p,q}. Thus, only one block of Z at a time needs to be stored.
[0120] In the step 1006, T is assembled one block at a time. For
each block of T, first obtain from NEC2 the corresponding block of
Z. The wire segments within a block are numbered consecutively
herein (NEC2 numbers them differently). Thus, first renumber Z
using the renumber mapping from step 1004, and then perform the
calculation: T{p,q}=d.sub.p.sup.LZ{p,q}d.sub.q.sup.R
[0121] Many of the numbers in T{p,q} will be relatively small. An
appropriate rule based on a desired accuracy is used to choose
which ones may be approximated by zero. The remaining non-zero
numbers are stored. Storage associated with the zero-valued
elements of T{p,q} and of Z{p,q} can be released before the next
block is calculated. The top left portion of T{p,q} has matrix
elements which will be kept. Anticipating this, the calculation
speed can be increased by not calculating either the right portion
or the bottom portion of T{p,q}.
[0122] The matrix T is a sparse matrix, and it can be stored using
an appropriate data structure for a sparse matrix. For a matrix
with N.sub.z non-zero elements, an array Z.sub.z(i) for i=1, . . .
, N.sub.z, can be used, where Z.sub.z(i) is the complex value of
the i.sup.th matrix element. There are two integer valued arrays,
I.sub.z(i) and J.sub.z(i) for i=1, . . . , N.sub.z, I.sub.z(i)
gives the row number where the i.sup.th matrix element occurs in T
and J.sub.z(i) its column number.
[0123] After calculation of T, the process proceeds to a process
block 1007 where the rows and columns of the matrix T are reordered
to produce a matrix T . The matrix T is reordered into a matrix T
so that the top left corner of every block of T ends up in the
bottom right corner of the whole matrix. The T form is more
amenable to LU factorization. FIG. 5 shows an example of a matrix
T, and FIG. 6 shows an example of a matrix T after reordering. One
embodiment uses a solver that has its own reordering algorithms
thus negating the need for an explicit reordering from T to T .
[0124] After reordering, the process advances to a step 1008 where
the matrix T is passed to a sparse matrix solver, such as, for
example, the computer program "Sparse," from the Electrical
Engineering Department of University of California at Berkeley. The
program Sparse can be used to factor the matrix T into a sparse LU
decomposition.
[0125] NEC2 solves the equation J=Z.sup.-1E
[0126] for various vectors E. In FIG. 10, the solution of the above
matrix equation is done in steps 1009-1016 or, alternatively, in
steps 1017-1023. The sequence of steps 1009-1016 is used with a
matrix equation solver that does not provide reordering. The
sequence of steps 1017-1023 is used with a matrix equation solver
that does provide reordering.
[0127] In the step 1009, the vector E is computed by NEC2. Then, in
the step 1010, the elements of E are permutated (using the same
permutation as that used in the step 1004) to produce a vector E'.
This permutation is called the region permutation. Next, in the
step 1011, E' is expressed in terms of composite testers by
multiplying E' by D.sup.L, giving D.sup.LE'. Then, in the step
1012, the same permutation used in the step 1007 is applied to
D.sup.LE' to yield (D.sup.LE') . (This permutation is called the
solver permutation.) Then, in the step 1013, a matrix equation
solver (such as, for example, the solver known as "SPARSE") is used
to solve for the vector Y from the equation T (Y )=(D.sup.LE')
[0128] Then, in the step 1014, the inverse of the solver
permutation is applied to Y to yield Y. In the subsequent step
1015, J' is computed from the equation J'=D.sup.RY
[0129] In the subsequent, and final, step 1016, the inverse of the
region permutation is applied to J' to yield the desired answer
J.
[0130] Alternatively, the embodiment shown in steps 1017-1023 is
conveniently used when the matrix equation solver provides its own
reordering algorithms, thus eliminating the need to reorder from T
to T (as is done in the step 1007 above). In the step 1017, a
reordering matrix solver is used to solve the matrix T. In the
subsequent step 1018, the vector E is computed by NEC2. Then, in
the step 1019, the elements of E are permutated using the region
permutation to produce a vector E'. Then, in the step 1020,
D.sup.LE' is computed. The process then proceeds to the step 1021
where the equation TY=D.sup.LE'
[0131] is solved for Y. After Y is computed, the process advances
to the step 1022 where J' is calculated from the equation
J'=D.sup.RY
[0132] Finally, in the step 1023, the inverse of the region
permutation is applied to J' to yield the desired answer J.
[0133] Many matrix elements are made small by this method. FIGS. 11
and 12 show before and after results for a problem using a wire
grid model in NEC2, with a matrix Z of size 2022 by 2022 and a
block of size 67 by 93. FIG. 11 shows the magnitudes of the matrix
elements before changing the sources and testers, meaning it shows
a 67 by 93 block of the renumbered Z. FIG. 12 shows this same block
of T. The matrix T has a regular structure wherein the large
elements are in the top left corner. This is a general property of
the transformed matrix. For larger blocks, the relative number of
small matrix elements is even better.
[0134] The algorithms expressed by the flowchart shown in FIG. 2
can be implemented in software and loaded into a computer memory
attached to a computer processor to calculate, for example,
propagation of energy, pressure, vibration, electric fields,
magnetic fields, strong nuclear forces, weak nuclear forces, etc.
Similarly, the algorithms expressed by the flowchart shown in FIG.
10 can be implemented in software and loaded into a computer memory
attached to a computer processor to calculate, for example,
electromagnetic radiation by an antenna, electromagnetic
scattering, antenna properties, etc.
[0135] Although the foregoing has been a description and
illustration of specific embodiments of the invention, various
modifications and changes can be made thereto by persons skilled in
the art without departing from the scope and spirit of the
invention. For example, in addition to electromagnetic fields, the
techniques described above can also be used to compress interaction
data for physical disturbances involving a heat flux, an electric
field, a magnetic field, a vector potential, a pressure field, a
sound wave, a particle flux, a weak nuclear force, a strong nuclear
force, a gravity force, etc. The techniques described above can
also be used for lattice gauge calculations, economic forecasting,
state space reconstruction, and image processing (e.g., image
formation for synthetic aperture radar, medical, or sonar images).
Accordingly, the invention is limited only by the claims that
follow.
* * * * *