superconvergence of projection methods for weakly singular
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General factsProjection approximations
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Superconvergence of projection methods forweakly singular integral operators
Mario Ahues, Alain Largillier,Laboratoire de Mathematiques
Universite de Saint-Etienne, France&
Andrey AmosovInstitute of Energy at Moscow, Russia
February 10, 2007
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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General factsProjection approximations
ComputationsForthcoming research
References
General facts
Projection approximations
Computations
Forthcoming research
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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I.W. Busbridge, The Mathematics of radiative transfer, CambridgeUniversity Press, 1960.
I.H. Sloan, “Superconvergence and the Galerkin method forintegral equations of the second kind”, In: Treatment of IntegralEquations by Numerical Methods, Academic Press, Inc. 1982, pp.197-206.
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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A Weakly Singular Integral Equation
Given τ∗ > 0, ω0 ∈ [0, 1[, g such that
• g(0+) = +∞,
• g is continuous, positive and decreasing on ]0,∞[,
• g ∈ L1(R) ∩W 1,1(δ,+∞) for all δ > 0,
• ‖g‖L1(R+) ≤ 12,
and a function f , find a function ϕ such that
ϕ(τ) = ω0
∫ τ∗
0g(|τ − τ ′|)ϕ(τ ′) dτ ′ + f (τ), τ ∈ [0, τ∗].
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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An example: The Transfer Equation
ϕ(τ) =ω0
2
∫ τ∗
0E1(|τ − τ ′|)ϕ(τ ′) dτ ′ + f (τ), τ ∈ [0, τ∗],
E1(τ) :=
∫ 1
0µ−1e−τ/µ dµ, τ > 0,
f , the source term, belongs to L1(0, τ∗),τ∗, the optical depth, is a very large number,ω0 , the albedo, may be very close to 1.
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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Goal
For a class of four numerical solutions based on projections,find acurate error estimates which
• are independent of the grid regularity,
• are independent of τ∗,
• depend on ω0 in an explicit way, and
• suggest global superconvergence phenomena.
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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Abstract framework
Set
(Λϕ)(τ) :=
∫ τ∗
0g(|τ − τ ′|)ϕ(τ ′) dτ ′, τ ∈ [0, τ∗].
For X and Y , suitable Banach spaces, the problem reads:
Given f ∈ Y , find ϕ ∈ X such that ϕ = ω0Λϕ+ f .
Remark:
‖(I − ω0Λ)−1‖ ≤ γ0 :=1
1− ω0
.
The equality holds for X = Y = L2(0, τ∗).
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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Projecting onto piecewise constant functions
Let be a grid of n + 1 points in [0, τ∗] :
0 =: τ0 < τ1 < . . . < τn−1 < τn := τ∗,
hi := τi − τi−1, i ∈ [[1, n ]],
h := (h1, h2, . . . , hn),
Gh := (τ0, τ1, . . . , τn),
h := maxi∈[[1,n ]]
hi ,
Ii− 12
:= ]τi−1, τi [, i ∈ [[1, n ]],
τi− 12
:= (τi−1 + τi )/2, i ∈ [[1, n ]].
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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The approximating space Ph0(0, τ∗)
f ∈ Ph0(0, τ∗) ⇐⇒ ∀i ∈ [[1, n ]], ∀τ ∈ Ii− 1
2, f (τ) = f (τi− 1
2).
The family of projections πh:
πh : Lp(0, τ∗) → Lp(0, τ∗)
∀i ∈ [[1, n ]],∀τ ∈ Ii− 12,
(πhϕ)(τ) :=1
hi
∫ τi
τi−1
ϕ(τ ′) dτ ′.
Henceπh(L
p(0, τ∗)) = Ph0(0, τ∗).
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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1
0 τi−1 τi− 12
τi τ∗
A basis function in the range of πh
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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Four approximations based on πh
The classical Galerkin approximation ϕGh solves
ϕGh = ω0πhΛϕ
Gh + πhf .
The Sloan approximation ϕSh (iterated Galerkin) solves
ϕSh = ω0Λπhϕ
Sh + f .
The Kantorovich approximation ϕKh solves
ϕKh = ω0πhΛϕ
Kh + f .
The Authors’ approximation ϕAh (iterated Kantorovich) solves
ϕAh = ω0Λπhϕ
Ah + f + ω0Λ(I − πh)f .
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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Useful remarks and relationships
• ϕGh ∈ Ph
0(0, τ∗),
• ϕGh is computed through an algebraic linear system,
• ϕSh = ω0Λϕ
Gh + f ,
• ψh = ω0πhΛψh + πhΛf =⇒ ϕKh = ω0ψh + f ,
• φh = ω0Λπhφh + Λf =⇒ ϕAh = ω0φh + f .
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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The absolute errors and superconvergence
εNh := ϕNh − ϕ for N ∈ {G,K,S,A}.
Superconvergence is understood with respect to dist(ϕ,Ph0(0, τ∗)).
TheoremIn any Hilbert space setting, ∃ α such that
‖ϕ− πhϕ‖ ≤ ‖εGh ‖ ≤ α‖ϕ− πhϕ‖.
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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Main technical notions and resultsDefinitions
∆εg(τ) := g(τ + ε)− g(τ),
ω1(g , δ) := sup0<ε<δ
‖∆εg(| · |)‖L1(R)
= sup0<ε<δ
∫R
|g(|τ + ε|)− g(|τ |)| dτ.
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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Main technical notions and resultsThe key for estimating errors
For α ∈ [0, 1] and i ∈ [[1, n ]]:
∆hαf (τ) :=
{∆αhi
f (τ) for τ ∈ [τi−1, τi − αhi ],0 for τ ∈ ]τi − αhi , τi ],
ωp(f ,Gh) := 21/p[ ∫ 1
0‖∆h
αf ‖ppdα
]1/pfor p < +∞,
ω∞(f ,Gh) := maxi∈[[1,n ]]
essup(τ,τ ′)∈[τi−1,τi ]2
|f (τ)− f (τ ′)|,
ω1(g ,Gh) := sup0<τ ′<τ∗
ω1(g(| · −τ ′|,Gh).
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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Global superconvergenceThe main result
TheoremIf 0 6= f ∈ Lp(0, τ∗) for some p ∈ [1,+∞], then 0 6= ϕ ∈ Lp(0, τ∗)and
‖εKh ‖p
‖ϕ‖p
≤ 4 · 31p · ω0 · γ0 ·
∫ h
0g(τ) dτ,
‖εSh‖p
‖ϕ‖p
≤ 12 · 3−1p · ω0 · γ0 ·
∫ h
0g(τ) dτ,
‖εAh ‖p
‖ϕ‖p
≤ 48 · ω20· γ0 ·
[ ∫ h
0g(τ) dτ
]2.
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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The proofFirst step
For all f ∈ Lp(0, τ∗),
‖(I − πh)f ‖p ≤ ωp(f ,Gh).
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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The proofSecond step
‖εKh ‖p
‖ϕ‖p
≤ ω0 · γ0 · ‖(I − πh)Λ‖p ,
‖εSh‖p
‖ϕ‖p
≤ ω0 · γ0 · ‖Λ(I − πh)‖p ,
‖εAh ‖p
‖ϕ‖p
≤ ω20· γ0 · ‖Λ(I − πh)Λ‖p .
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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The proofThird step
‖(I − πh)Λ‖p ≤ ω1(g , h)1−1p ω1(g ,Gh)
1p ,
‖Λ(I − πh)‖p ≤ ω1(g , h)1p ω1(g ,Gh)1−
1p ,
‖Λ(I − πh)Λ‖p ≤ ω1(g , h) ω1(g ,Gh).
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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The proofFourth step
ω1(g , h) ≤ 4
∫ h
0g(τ) dτ,
ω1(g ,Gh) ≤ 12
∫ h
0g(τ) dτ.
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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The proofDesired conclusion
‖εKh ‖p
‖ϕ‖p
≤ 4 · 31p · ω0 · γ0 ·
∫ h
0g(τ) dτ,
‖εSh‖p
‖ϕ‖p
≤ 12 · 3−1p · ω0 · γ0 ·
∫ h
0g(τ) dτ,
‖εAh ‖p
‖ϕ‖p
≤ 48 · ω20· γ0 ·
[ ∫ h
0g(τ) dτ
]2.
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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Numerical evidence
• g = 12E1 =⇒
∫ h
0g(τ) dτ = O(h ln h)
• Data: ω0 = 0.5, τ∗ = 500, f (τ) = 0.5
• Number of grid points: n = 1000
• First 5 points equally spaced by 0.1
• Last 5 points equally spaced by 0.1
• Between 0.5 and 499.5: a uniform grid with 990 points
• h = 0.504
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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Plotting residuals on [0, 5]
0
0.0005
0.0010
0.0015
0 1 2 3 4 5
•
••••
•
•
••
• • • • •
Kantorovich Residual •Authors Residual
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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Looking for generalizationsA joint work with F. d’Almeida (Porto) and R. Fernandes (Braga)
X , a complex Banach space,
T , a bounded linear operator from X into itself,
0 6= ζ ∈ re(T ), the resolvent set of T , and f ∈ X .
Problem:
Find ϕ ∈ X such that (T − ζI )ϕ = f .
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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Looking for generalizationsA family of numerical schemes
Use an approximate operator Tn such that re(T ) ⊆ re(Tn) andan approximate second member fn to solve exactly
(Tn − ζI )ϕn = fn.
Remark that the resolvents
R(ζ) := (T − ζI )−1, Rn(ζ) := (Tn − ζI )−1
satisfy
Rn(ζ)− R(ζ) = R(ζ)(T − Tn)Rn(ζ),
hence the error,
εn := ϕn − ϕ = R(ζ)[fn − f + (T − Tn)ϕn].
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL
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Looking for generalizationsIterating
A fixed point approach leads to build the so-called
Iterated approximate solution ϕn :=1
ζ(Tϕn − f ).
The error of ϕn is related with the error ϕn by
εn =1
ζTεn =
1
ζR(ζ)T [fn − f + (T − Tn)ϕn].
Under some conditions on T and Tn we may expect that ‖ϕn‖ isuniformly bounded in n, and that T (T − Tn), T (fn − f ) tend to 0faster than (T − Tn)ϕn and fn − f , respectively.
M. Ahues, A. Amosov & A. Largillier CIM at Coimbra, PORTUGAL