theoretical and computational aspects of the svm, ebcm, and pmm methods in light scatteringby small...
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Theoretical and computational aspectsof the SVM, EBCM, and PMM methods
in light scattering by small particles
V. B. Il’in1,2,3 V. G. Farafonov2 A. A. Vinokurov2,3
1Saint-Petersburg State University, Russia2Saint-Petersburg State University of Aerospace Instrumentation, Russia
3Pulkovo Observatory, Saint-Petersburg, Russia
12th Electromagnetic & Light Scattering Conference
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 1 / 46
Wave Equations and Functions
Maxwell equations
Helmholts equations for E(r), H(r)
∆E(r) + k2(r)E(r) = 0,
where k is the wavenumber
Vector wave functions Fν(r)
For time-harmonic fieldsE(r, t) = E(r) exp(−iωt)
Solutions
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 2 / 46
Wave Equations and Functions
Additional condition divE(r) = 0 leads to:
Fν(r) = Maν(r) = rot(a ψν(r)),
Fν(r) = Naν(r) = rot rot(a ψν(r))/k ,
where a is a vector, ψν(r) are solutions to
∆ψν + k2ψν = 0.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 3 / 46
Field/Potential Expansions
It looks natural to search for unknown fields as
E(r) =∑ν
aνFν(r),
or equivalentlyU,V (r) =
∑ν
aνψν(r),
where U,V are scalar potentials, e.g.
E = rot(bU) + rot rot(cV ).
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 4 / 46
Field/Potential Expansions
In all the methods vector/scalar wave functions are represented as:in spherical coordinates (r , θ, ϕ):
ψν(r) = zn(r)Pmn (θ) exp(imϕ),
in spheroidal coordinates (ξ, η, ϕ):
ψν(r) = Rnm(c , ξ)Snm(c , η) exp(imϕ),
where c is a parameter.
So, separation of variables is actually used in all 3 methods.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 5 / 46
Field/Potential Expansions
In all the methods vector/scalar wave functions are represented as:in spherical coordinates (r , θ, ϕ):
ψν(r) = zn(r)Pmn (θ) exp(imϕ),
in spheroidal coordinates (ξ, η, ϕ):
ψν(r) = Rnm(c , ξ)Snm(c , η) exp(imϕ),
where c is a parameter.
So, separation of variables is actually used in all 3 methods.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 5 / 46
Separation of Variables Method (SVM)
Field expansions are substituted in the boudary conditions
(Einc + Esca)× n = Eint × n, r ∈ ∂Γ,
where n is the outer normal to the particle surface ∂Γ.The conditions are mutiplied by the angular parts of ψν with differentindices and then are integrated over ∂Γ. This yelds the followingsystem: (
A BC D
)(xsca
xint
)=
(EF
)xinc,
where xinc, xsca, xint are vectors of expansion coefficients, A, . . .F —matrices of surface integrals.Generalised SVM1
1see (Kahnert, 2003)Il’in, Farafonov, Vinokurov (Russia) ELS-XII 6 / 46
Extended Boundary Condition Method (EBCM)
Field expansions are substituted in the extended boundary condition:
rot∫∂Γ
n(r)× Eint(r)G(r′, r)ds − . . . =
{−Einc(r′), r′ ∈ Γ−,
Esca(r′), r′ ∈ Γ+.
Due to linear independence of wave functions we get(0 QsI Qr
)(xsca
xint
)=
(I0
)xinc,
where Qs , Qr are matrices, whose elements are surface integrals.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 7 / 46
Generalized Point Matching Method (gPMM)
Residual of the standard boundary conditions
δ =M∑
s=1
(∣∣(Einc + Esca − Eint)× n∣∣2 + . . .
), r = rs ∈ ∂Γ.
Minimizing residual in the least squares sense gives(A BC D
)(xsca
xint
)=
(EF
)xinc,
Sum in δ can be replaced with surface integral2
2see (Farafonov & Il’in, 2006)Il’in, Farafonov, Vinokurov (Russia) ELS-XII 8 / 46
Comparison of gPMM and Integral gPMM
1 — PMM, M = N,2 — gPMM, M = 2N,3 — gPMM, M = 4N,5 — iPMM, M = N,6 — iPMM, M = 1.5N.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 9 / 46
Key questions
1 EBCM, SVM, and PMM use the same field expansions. Does it resultin their similar behavior? How close are they?[Yes and Generally close, but in important detail not.]
2 It is well known from numerical experiments that EBCM with aspherical basis [being a widely used approach] gives high accuracyresults for some shapes, while for some others it cannot provide anyreliable results. Why? What can be said about theoretical applicabilityof EBCM?[We have some answers, but not anything is clear.]
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 10 / 46
Key questions
1 EBCM, SVM, and PMM use the same field expansions. Does it resultin their similar behavior? How close are they?[Yes and Generally close, but in important detail not.]
2 It is well known from numerical experiments that EBCM with aspherical basis [being a widely used approach] gives high accuracyresults for some shapes, while for some others it cannot provide anyreliable results. Why? What can be said about theoretical applicabilityof EBCM?[We have some answers, but not anything is clear.]
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 10 / 46
Key questions
1 EBCM, SVM, and PMM use the same field expansions. Does it resultin their similar behavior? How close are they?[Yes and Generally close, but in important detail not.]
2 It is well known from numerical experiments that EBCM with aspherical basis [being a widely used approach] gives high accuracyresults for some shapes, while for some others it cannot provide anyreliable results. Why? What can be said about theoretical applicabilityof EBCM?[We have some answers, but not anything is clear.]
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 10 / 46
Key questions
1 EBCM, SVM, and PMM use the same field expansions. Does it resultin their similar behavior? How close are they?[Yes and Generally close, but in important detail not.]
2 It is well known from numerical experiments that EBCM with aspherical basis [being a widely used approach] gives high accuracyresults for some shapes, while for some others it cannot provide anyreliable results. Why? What can be said about theoretical applicabilityof EBCM?[We have some answers, but not anything is clear.]
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 10 / 46
Singularities of analytic continuations of the fields
Obviously, a plane wave has no such singularities.But a plane wave incident at a scatterer is known to produce scatteredfield outside it and incident field inside it.Generally, analytic continuations of both the scattered field (inside thescatterer) and of the internal field (outside it) may have singularitiesdepending on the scatterer shape.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 11 / 46
Singularities of analytic continuations of the fields
Obviously, a plane wave has no such singularities.But a plane wave incident at a scatterer is known to produce scatteredfield outside it and incident field inside it.Generally, analytic continuations of both the scattered field (inside thescatterer) and of the internal field (outside it) may have singularitiesdepending on the scatterer shape.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 11 / 46
Singularities of analytic continuations of the fields
Obviously, a plane wave has no such singularities.But a plane wave incident at a scatterer is known to produce scatteredfield outside it and incident field inside it.Generally, analytic continuations of both the scattered field (inside thescatterer) and of the internal field (outside it) may have singularitiesdepending on the scatterer shape.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 11 / 46
Singularities for a spheroid and Chebyshev particle
Spheroid
d sca =√
a2 − b2,
d int =∞.
Chebyshev particler(θ, ϕ) = r0(1 + ε cos nθ)
d sca = f (r0, n, ε),
d int = g(r0, n, ε).
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 12 / 46
Near Field
1 In spherical coordinates
ψν(r , θ, ϕ) = zn(r)Pmn (θ) exp(imϕ).
2 For r → 0 or ∞: zn(r) behaves like rk .3 Series U,V (r) =
∑ν aνψν(r) can be transformed into power series.
4 Radius of convergence is determined by the nearest singularity.
For scattered field: r ∈ (max d sca,∞) .
For internal field: r ∈[0,min d int) .
5 In spheroidal coordinates, it is not that simple!
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 13 / 46
Near Field
1 In spherical coordinates
ψν(r , θ, ϕ) = zn(r)Pmn (θ) exp(imϕ).
2 For r → 0 or ∞: zn(r) behaves like rk .3 Series U,V (r) =
∑ν aνψν(r) can be transformed into power series.
4 Radius of convergence is determined by the nearest singularity.
For scattered field: r ∈ (max d sca,∞) .
For internal field: r ∈[0,min d int) .
5 In spheroidal coordinates, it is not that simple!
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 13 / 46
Near Field
1 In spherical coordinates
ψν(r , θ, ϕ) = zn(r)Pmn (θ) exp(imϕ).
2 For r → 0 or ∞: zn(r) behaves like rk .3 Series U,V (r) =
∑ν aνψν(r) can be transformed into power series.
4 Radius of convergence is determined by the nearest singularity.
For scattered field: r ∈ (max d sca,∞) .
For internal field: r ∈[0,min d int) .
5 In spheroidal coordinates, it is not that simple!
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 13 / 46
Near Field
1 In spherical coordinates
ψν(r , θ, ϕ) = zn(r)Pmn (θ) exp(imϕ).
2 For r → 0 or ∞: zn(r) behaves like rk .3 Series U,V (r) =
∑ν aνψν(r) can be transformed into power series.
4 Radius of convergence is determined by the nearest singularity.
For scattered field: r ∈ (max d sca,∞) .
For internal field: r ∈[0,min d int) .
5 In spheroidal coordinates, it is not that simple!
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 13 / 46
Near Field
1 In spherical coordinates
ψν(r , θ, ϕ) = zn(r)Pmn (θ) exp(imϕ).
2 For r → 0 or ∞: zn(r) behaves like rk .3 Series U,V (r) =
∑ν aνψν(r) can be transformed into power series.
4 Radius of convergence is determined by the nearest singularity.
For scattered field: r ∈ (max d sca,∞) .
For internal field: r ∈[0,min d int) .
5 In spheroidal coordinates, it is not that simple!
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 13 / 46
Spheroid singularities, a/b = 1.4
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 14 / 46
Spheroid singularities, a/b = 2.0
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 15 / 46
Spheroid singularities, a/b = 2.5
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 16 / 46
Spheroid convergence in the near field, a/b = 1.4
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 17 / 46
Spheroid convergence in the near field, a/b = 1.8
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 18 / 46
Spheroid convergence in the near field, a/b = 2.5
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 19 / 46
Spheroid convergence in the near field, a/b = 3.5
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 20 / 46
Rayleigh hypothesis
We use a generalized and simplified definition of the term as anassumption that field expansions in terms of wave functionsconverge everywhere up to the scatterer surface.In spherical coordinates and for spherical basis:
max d sca < min r(θ, ϕ) and max r(θ, ϕ) < min d int.
As field expansions are substituted in the boundary conditions,Rayleigh hypothesis seems to be required to be valid.However, we know that the methods provide accurate results whenRayleigh hypothesis is not valid.Do we realy need Rayleigh hypothesis to be valid?
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 21 / 46
Rayleigh hypothesis
We use a generalized and simplified definition of the term as anassumption that field expansions in terms of wave functionsconverge everywhere up to the scatterer surface.In spherical coordinates and for spherical basis:
max d sca < min r(θ, ϕ) and max r(θ, ϕ) < min d int.
As field expansions are substituted in the boundary conditions,Rayleigh hypothesis seems to be required to be valid.However, we know that the methods provide accurate results whenRayleigh hypothesis is not valid.Do we realy need Rayleigh hypothesis to be valid?
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 21 / 46
Rayleigh hypothesis
We use a generalized and simplified definition of the term as anassumption that field expansions in terms of wave functionsconverge everywhere up to the scatterer surface.In spherical coordinates and for spherical basis:
max d sca < min r(θ, ϕ) and max r(θ, ϕ) < min d int.
As field expansions are substituted in the boundary conditions,Rayleigh hypothesis seems to be required to be valid.However, we know that the methods provide accurate results whenRayleigh hypothesis is not valid.Do we realy need Rayleigh hypothesis to be valid?
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 21 / 46
Rayleigh hypothesis
We use a generalized and simplified definition of the term as anassumption that field expansions in terms of wave functionsconverge everywhere up to the scatterer surface.In spherical coordinates and for spherical basis:
max d sca < min r(θ, ϕ) and max r(θ, ϕ) < min d int.
As field expansions are substituted in the boundary conditions,Rayleigh hypothesis seems to be required to be valid.However, we know that the methods provide accurate results whenRayleigh hypothesis is not valid.Do we realy need Rayleigh hypothesis to be valid?
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 21 / 46
Rayleigh hypothesis
We use a generalized and simplified definition of the term as anassumption that field expansions in terms of wave functionsconverge everywhere up to the scatterer surface.In spherical coordinates and for spherical basis:
max d sca < min r(θ, ϕ) and max r(θ, ϕ) < min d int.
As field expansions are substituted in the boundary conditions,Rayleigh hypothesis seems to be required to be valid.However, we know that the methods provide accurate results whenRayleigh hypothesis is not valid.Do we realy need Rayleigh hypothesis to be valid?
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 21 / 46
Infinite linear systems analysis
a11 a12 · · ·a21 a22 · · ·...
.... . .
x1
x2...
=
b1b2...
Kantorovich & Krylov (1958)Regular systems: ρi = 1−
∑∞k=1 |aik | > 0, (i = 1, 2, . . .).
Theorem. If ∃K > 0 : |bi | < Kρi , (i = 1, 2, . . .), thenI regular system is solvable,I it has the only solution,I solutions of truncated systems converge to it.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 22 / 46
Infinite linear systems analysis
a11 a12 · · ·a21 a22 · · ·...
.... . .
x1
x2...
=
b1b2...
Kantorovich & Krylov (1958)Regular systems: ρi = 1−
∑∞k=1 |aik | > 0, (i = 1, 2, . . .).
Theorem. If ∃K > 0 : |bi | < Kρi , (i = 1, 2, . . .), thenI regular system is solvable,I it has the only solution,I solutions of truncated systems converge to it.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 22 / 46
Infinite linear systems analysis
a11 a12 · · ·a21 a22 · · ·...
.... . .
x1
x2...
=
b1b2...
Kantorovich & Krylov (1958)Regular systems: ρi = 1−
∑∞k=1 |aik | > 0, (i = 1, 2, . . .).
Theorem. If ∃K > 0 : |bi | < Kρi , (i = 1, 2, . . .), thenI regular system is solvable,I it has the only solution,I solutions of truncated systems converge to it.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 22 / 46
Analysis of EBCM, gSVM and gPMM systems
gPMMSystem has positively determined matrix and hence has always the onlysolution.
EBCMSystem is regular and satisfies solvability condition if
max d sca < min d int.
gSVMThere is no such condition for SVM.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 23 / 46
Analysis of EBCM, gSVM and gPMM systems
gPMMSystem has positively determined matrix and hence has always the onlysolution.
EBCMSystem is regular and satisfies solvability condition if
max d sca < min d int.
gSVMThere is no such condition for SVM.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 23 / 46
Analysis of EBCM, gSVM and gPMM systems
gPMMSystem has positively determined matrix and hence has always the onlysolution.
EBCMSystem is regular and satisfies solvability condition if
max d sca < min d int.
gSVMThere is no such condition for SVM.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 23 / 46
Chebyshev particle singularities, n = 5, ε = 0.07
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 24 / 46
Chebyshev particle singularities, n = 5, ε = 0.14
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 25 / 46
Chebyshev particle singularities, n = 5, ε = 0.21
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 26 / 46
Solvability condition, EBCM
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 27 / 46
Solvability condition, SVM
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 28 / 46
Convergence of results, Chebyshev particle, n = 5, ε = 0.07
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 29 / 46
Convergence of results, Chebyshev particle, n = 5, ε = 0.14
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 30 / 46
Convergence of results, Chebyshev particle, n = 5, ε = 0.21
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 31 / 46
Paradox of the EBCM
EBCM solutions converge even if field expansions used in theboundary conditions diverge.How is this possible?Let’s consider Pattern Equation Method3.Search for far field pattern in terms of angular parts of ψν (as r →∞)As the patterns are defined only in the far field zone, one does notneed convergence of any expansions at scatterer boundary.We found that the infinite systems arisen in EBCM coincide withthose arisen in the PEM.When Rayleigh hypothesis is not valid, EBCM is notmathematically correct, but its applicability is extended in thefar field due to lucky coincindence with PEM.
3see works by Kyurkchan and SmirnovaIl’in, Farafonov, Vinokurov (Russia) ELS-XII 32 / 46
Paradox of the EBCM
EBCM solutions converge even if field expansions used in theboundary conditions diverge.How is this possible?Let’s consider Pattern Equation Method3.Search for far field pattern in terms of angular parts of ψν (as r →∞)As the patterns are defined only in the far field zone, one does notneed convergence of any expansions at scatterer boundary.We found that the infinite systems arisen in EBCM coincide withthose arisen in the PEM.When Rayleigh hypothesis is not valid, EBCM is notmathematically correct, but its applicability is extended in thefar field due to lucky coincindence with PEM.
3see works by Kyurkchan and SmirnovaIl’in, Farafonov, Vinokurov (Russia) ELS-XII 32 / 46
Paradox of the EBCM
EBCM solutions converge even if field expansions used in theboundary conditions diverge.How is this possible?Let’s consider Pattern Equation Method3.Search for far field pattern in terms of angular parts of ψν (as r →∞)As the patterns are defined only in the far field zone, one does notneed convergence of any expansions at scatterer boundary.We found that the infinite systems arisen in EBCM coincide withthose arisen in the PEM.When Rayleigh hypothesis is not valid, EBCM is notmathematically correct, but its applicability is extended in thefar field due to lucky coincindence with PEM.
3see works by Kyurkchan and SmirnovaIl’in, Farafonov, Vinokurov (Russia) ELS-XII 32 / 46
Paradox of the EBCM
EBCM solutions converge even if field expansions used in theboundary conditions diverge.How is this possible?Let’s consider Pattern Equation Method3.Search for far field pattern in terms of angular parts of ψν (as r →∞)As the patterns are defined only in the far field zone, one does notneed convergence of any expansions at scatterer boundary.We found that the infinite systems arisen in EBCM coincide withthose arisen in the PEM.When Rayleigh hypothesis is not valid, EBCM is notmathematically correct, but its applicability is extended in thefar field due to lucky coincindence with PEM.
3see works by Kyurkchan and SmirnovaIl’in, Farafonov, Vinokurov (Russia) ELS-XII 32 / 46
Paradox of the EBCM
EBCM solutions converge even if field expansions used in theboundary conditions diverge.How is this possible?Let’s consider Pattern Equation Method3.Search for far field pattern in terms of angular parts of ψν (as r →∞)As the patterns are defined only in the far field zone, one does notneed convergence of any expansions at scatterer boundary.We found that the infinite systems arisen in EBCM coincide withthose arisen in the PEM.When Rayleigh hypothesis is not valid, EBCM is notmathematically correct, but its applicability is extended in thefar field due to lucky coincindence with PEM.
3see works by Kyurkchan and SmirnovaIl’in, Farafonov, Vinokurov (Russia) ELS-XII 32 / 46
Equivalence of the EBCM and gSVM systems
It was generally shown earlier (e.g., Schmidt et al., 1998).We have strictly demonstrated that the matrix of EBCM infinitesystem can be transformed into the matrix of gSVM system and viceversa.
Qs = i[CTB− ATD
], Qr = i
[FTB− ETD
],
where A = ASVM , B = BSVM , . . .
If in iPMM residual ∆ = 0, then
APMM = AT∗A + CT∗A, . . .
Hence, iPMM infinite system is also equivalent to gSVM system.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 33 / 46
Equivalence of the EBCM and gSVM systems
It was generally shown earlier (e.g., Schmidt et al., 1998).We have strictly demonstrated that the matrix of EBCM infinitesystem can be transformed into the matrix of gSVM system and viceversa.
Qs = i[CTB− ATD
], Qr = i
[FTB− ETD
],
where A = ASVM , B = BSVM , . . .
If in iPMM residual ∆ = 0, then
APMM = AT∗A + CT∗A, . . .
Hence, iPMM infinite system is also equivalent to gSVM system.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 33 / 46
Equivalence of the EBCM and gSVM systems
It was generally shown earlier (e.g., Schmidt et al., 1998).We have strictly demonstrated that the matrix of EBCM infinitesystem can be transformed into the matrix of gSVM system and viceversa.
Qs = i[CTB− ATD
], Qr = i
[FTB− ETD
],
where A = ASVM , B = BSVM , . . .
If in iPMM residual ∆ = 0, then
APMM = AT∗A + CT∗A, . . .
Hence, iPMM infinite system is also equivalent to gSVM system.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 33 / 46
Equivalence of the EBCM and gSVM systems
It was generally shown earlier (e.g., Schmidt et al., 1998).We have strictly demonstrated that the matrix of EBCM infinitesystem can be transformed into the matrix of gSVM system and viceversa.
Qs = i[CTB− ATD
], Qr = i
[FTB− ETD
],
where A = ASVM , B = BSVM , . . .
If in iPMM residual ∆ = 0, then
APMM = AT∗A + CT∗A, . . .
Hence, iPMM infinite system is also equivalent to gSVM system.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 33 / 46
Truncation of infinite systems
For truncated systems the proof of equivalence is not correct.For EBCM and iPMM we have regular systems.For gSVM we couldn’t prove that systems are regular.Infinite EBCM and gSVM systems are equivalent, buttruncated are not.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 34 / 46
Truncation of infinite systems
For truncated systems the proof of equivalence is not correct.For EBCM and iPMM we have regular systems.For gSVM we couldn’t prove that systems are regular.Infinite EBCM and gSVM systems are equivalent, buttruncated are not.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 34 / 46
Truncation of infinite systems
For truncated systems the proof of equivalence is not correct.For EBCM and iPMM we have regular systems.For gSVM we couldn’t prove that systems are regular.Infinite EBCM and gSVM systems are equivalent, buttruncated are not.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 34 / 46
Truncation of infinite systems
For truncated systems the proof of equivalence is not correct.For EBCM and iPMM we have regular systems.For gSVM we couldn’t prove that systems are regular.Infinite EBCM and gSVM systems are equivalent, buttruncated are not.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 34 / 46
Numerical comparison, prolate spheroid, a/b = 1.5
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 35 / 46
Numerical comparison, prolate spheroid, a/b = 2.0
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 36 / 46
Numerical comparison, prolate spheroid, a/b = 2.5
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 37 / 46
Numerical comparison, Chebyshev particle, n = 5, ε = 0.07
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 38 / 46
Numerical comparison, Chebyshev particle, n = 5, ε = 0.14
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 39 / 46
Numerical comparison, Chebyshev particle, n = 5, ε = 0.21
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 40 / 46
Condition number for gSVM, EBCM, iPMM systems
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 41 / 46
System matrix elements, SVM
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 42 / 46
System matrix elements, EBCM
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 43 / 46
System matrix elements, PMM
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 44 / 46
Layered scatterers
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 45 / 46
Conclusions
1 Methods are very similar, but have key differencies.2 Methods applicability ranges are affected by singularities.3 Rayleigh hypothesis is required for near field computations.4 EBCM has solvability condition for far field.5 Infinite matrices of the methods’ systems are equivalent.6 Truncated matrices are not.7 Different methods are good for different particles.8 Systems are ill-conditioned from the beginning.9 Because of solvability condition EBCM doesn’t provide accurate
results for layered scatterers.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 46 / 46
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