Optimized Schwarz Methods andBest Approximation Problem.
Laurence Halpern
LAGA-Institut Galilée -Université Paris 13
JJJ III × DD16, january 2005 – p. 1/26
Table of contents
Presentation of the problem;
Optimized Schwarz methods for theconvection-diffusion equation;A non-standard problem of bestapproximation.Relevance of the optimization
JJJ III × DD16, january 2005 – p. 2/26
Table of contents
Presentation of the problem;Optimized Schwarz methods for theconvection-diffusion equation;
A non-standard problem of bestapproximation.Relevance of the optimization
JJJ III × DD16, january 2005 – p. 2/26
Table of contents
Presentation of the problem;Optimized Schwarz methods for theconvection-diffusion equation;A non-standard problem of bestapproximation.
Relevance of the optimization
JJJ III × DD16, january 2005 – p. 2/26
The toy problem
Domain of interest
ΙΩ
Ω2Ω1
original problem domain decomposition
JJJ III × DD16, january 2005 – p. 3/26
The Schwarz algorithm
Ω1 = (−∞, L)× Rn , Ω2 = (0,∞)× Rn.
L(uk+11 ) = f in Ω1 × (0, T )
uk+11 (·, 0) = u0 in Ω1
B1uk+11 (L, ·) = B1u
k2(L, ·) in (0, T )
L(uk+12 ) = f in Ω2 × (0, T )
uk+12 (·, 0) = u0 in Ω2
B2uk+12 (0, ·) = B2u
k1(0, ·) in (0, T )
JJJ III × DD16, january 2005 – p. 4/26
The convection-diffusion equation
L(u) := ut − ν∆u+ a∂1u+ b · ∇u+ cu = f
u(·, 0) = u0
ν > 0, a > 0, b ∈ Rn, c > 0.
Fourier transform in time (t↔ ω)and in the tangential variable (y ↔ k)
JJJ III × DD16, january 2005 – p. 5/26
The convection-diffusion equation
L(u) := ut − ν∆u+ a∂1u+ b · ∇u+ cu = f
u(·, 0) = u0
ν > 0, a > 0, b ∈ Rn, c > 0.Fourier transform in time (t↔ ω)
and in the tangential variable (y ↔ k)
JJJ III × DD16, january 2005 – p. 5/26
The optimal Schwarz method
THEOREM The Schwarz method converges in twoiterations with or without overlap when theoperators Bi are given by :
B1 = ∂1 − Λ−, B2 = ∂1 − Λ+
Λ± Pseudo Differential Operator of order 1 in∂t, ∂y.
λ± =a∓ (a2 + 4ν(i(ω + b · k) + ν|k|2 + c))1/2
2ν
JJJ III × DD16, january 2005 – p. 6/26
Approximate transmission conditions and convergence rate
Ba1 = ∂1 − Λ−a , Ba2 = ∂1 − Λ+a
λ− + λ+ =a
ν→ λ−a + λ+
a =a
νCONVERGENCE RATE
ρ(ω, k, λ−a , L) =
(λ− − λ−aλ+ − λ−a
)2
e(λ−−λ+)L
ek+2j (ω, 0, k) = ρ(ω, k, λ−a , L)ekj (ω, 0, k)
JJJ III × DD16, january 2005 – p. 7/26
Variations of the convergence rate
δ(ω, k) = a2 + 4ν((i(ω + b · k) + ν|k|2 + c)
λ− =a− δ1/2
2ν;λ−a =
a− P2ν
;
ρ(ω, k, λ−a , L) =
(P − δ1/2
P + δ1/2
)2
e−2δ1/2L
REMARK :lim
(ω,k)→+∞
∣∣∣∣P − δ1/2
P + δ1/2
∣∣∣∣ = 1
JJJ III × DD16, january 2005 – p. 8/26
Approximate transmission conditions
P = p ∈ P0 →B1 ≡ ∂1 −
a− p2ν
,
P = p+ qx ∈ P1 →
B1 ≡ ∂1 −a− p
2ν+ q(∂t + b · ∇− ν∆S + cI)
.
THEOREM(V. Martin) For p > 0, (resp. p, q > 0, p > a2
4νq ), the
algorithm converges with and without overlap.
JJJ III × DD16, january 2005 – p. 9/26
Approximate transmission conditions
P = p ∈ P0 →B1 ≡ ∂1 −
a− p2ν
,
P = p+ qx ∈ P1 →
B1 ≡ ∂1 −a− p
2ν+ q(∂t + b · ∇− ν∆S + cI)
.
THEOREM(V. Martin) For p > 0, (resp. p, q > 0, p > a2
4νq ), the
algorithm converges with and without overlap.
JJJ III × DD16, january 2005 – p. 9/26
Approximate transmission conditions
P = p ∈ P0 →B1 ≡ ∂1 −
a− p2ν
,
P = p+ qx ∈ P1 →
B1 ≡ ∂1 −a− p
2ν+ q(∂t + b · ∇− ν∆S + cI)
.
THEOREM(V. Martin) For p > 0, (resp. p, q > 0, p > a2
4νq ), the
algorithm converges with and without overlap.
JJJ III × DD16, january 2005 – p. 9/26
Choice of the coefficients
δ(ω, k) = a2 + 4ν((i(ω + b · k) + ν|k|2 + c)
ρ(ω, k, λ−a , L) =
(P − δ1/2
P + δ1/2
)2
e−2δ1/2L
JJJ III × DD16, january 2005 – p. 10/26
Choice of the coefficients
δ(ω, k) = a2 + 4ν((i(ω + b · k) + ν|k|2 + c)
ρ(ω, k, λ−a , L) =
(P − δ1/2
P + δ1/2
)2
e−2δ1/2L
* Taylor approximation,δ1/2 ≈ x0 =
√a2 + 4νc,
δ1/2 ≈ x0 + 2ν((i(ω + b · k) + ν|k|2 + c)/x0.
JJJ III × DD16, january 2005 – p. 10/26
Choice of the coefficients
δ(ω, k) = a2 + 4ν((i(ω + b · k) + ν|k|2 + c)
ρ(ω, k, λ−a , L) =
(P − δ1/2
P + δ1/2
)2
e−2δ1/2L
* Discrete frequencies : IT = ( πT, π
∆t), Ij = ( π
Xj, π
∆xj).
For a given n, find P in Pn minimizing
supω∈IT ,kj∈Ij
|ρ(ω, k, λ−a , L)|
JJJ III × DD16, january 2005 – p. 10/26
A new best approximation problem
JJJ III × DD16, january 2005 – p. 11/26
The problemK is a compact set in C. f is continuous on K, such that
f(K) ⊂ z ∈ C : Re z > 0.
δn(l) = infp∈Pn
supz∈K
∣∣∣∣p(z)− f(z)
p(z) + f(z)e−lf(z)
∣∣∣∣ ,
Find p∗n such that supz∈K
∣∣∣∣p∗n(z)− f(z)
p∗n(z) + f(z)e−lf(z)
∣∣∣∣ = δn(l)
The classical problem (De la Vallée-Poussin) ♣
infp∈Pn‖p− f‖Lq([a,b])
.JJJ III × DD16, january 2005 – p. 12/26
Existence result
A KEY-GEOMETRICAL LEMMA
D(z0, δ) = z ∈ C,∣∣∣∣z − z0
z + z0
∣∣∣∣ < δ
δ < 1
PSfrag replacementsz0
δ > 1
Z0
JJJ III × DD16, january 2005 – p. 13/26
Existence result
THEOREM For any n ≥ 0, there exists a bestapproximation polynomial p∗n.
Proof :δn = infp∈Pn
supz∈K
∣∣∣∣p(z)− f(z)
p(z) + f(z)
∣∣∣∣ ,
1. δn < 1,2. a minimizing sequence is such that
pk
f (z) ∈ D(1, δn + ε), thus is bounded in Pnby the lemma.
JJJ III × DD16, january 2005 – p. 14/26
Equioscillation result
THEOREM For any n ≥ 0, there exist at least n+ 2points z1, · · · , zn+2 in K such that
∣∣∣∣p∗n(zi)− f(zi)
p∗n(zi) + f(zi)
∣∣∣∣ =
∥∥∥∥p∗n − fp∗n + f
∥∥∥∥∞.
Proof : by contradiction, using that the derivativeof
Aw : Pn → Cm, p 7→(p(wi)− f(wi)
p(wi) + f(wi)
)
1≤i≤m
is continuous and onto.
JJJ III × DD16, january 2005 – p. 15/26
Uniqueness
THEOREM For any n ≥ 0, the best approximationpolynomial p∗n is unique.Proof :
1. The set of best approximation polynomials isconvex,
2. As in the classical case, if P 1 et P 2 are twobest approximation polynomials, so isP = (P 1 + P 2)/2, and equioscillates in atleast n+ 2 points zj in K.
JJJ III × DD16, january 2005 – p. 16/26
Uniqueness
PSfrag replacements
P 1(zk)
f(zk)
P 2(zk)
f(zk)
P (zk)
f(zk)
1
2
P 1(zk) + P 2(zk)
f(zk)
D(1, δn)
P = (P 1 + P 2)/2
JJJ III × DD16, january 2005 – p. 17/26
A local result without overlap
THEOREM
For any n ≥ 0, any strict local minimum forδn =
∣∣∣∣∣∣∣∣p− fp+ f
∣∣∣∣∣∣∣∣ is a global minimum.
JJJ III × DD16, january 2005 – p. 18/26
The 1d case
n = 0 : calculations by hand (with maple!!).n = 1 :1. we calculate by hand a solution
equioscillating in 3 real points,2. asymptotic expansions in 1/∆t prove it to be
a strict local minimum,3. conclusion by previous theorem.And the constrains on the coefficients forwell-posedness are satisfied.
JJJ III × DD16, january 2005 – p. 19/26
Asymptotic convergence ratesL ≈ C1∆x and ∆t ≈ C2∆xβ , β ≥ 1.
method β = 1 β = 2
Taylor order 0 1−O(∆x1/2) 1−O(∆x1/2)
P0 approximation 1−O(∆x1/4) 1−O(∆x1/3)
Taylor order 1 1−O(∆x1/2) 1−O(∆x1/2)
P1 approximation 1−O(∆x1/8) 1−O(∆x1/5)
WITH OVERLAPmethod asymptotic convergence rate
Taylor order 0 1−O(∆t1/2)
P0 approximation 1−O(∆t1/4)
Taylor order 1 1−O(∆t1/2)
P1 approximation 1−O(∆t1/8)
WITHOUT OVERLAP
JJJ III × DD16, january 2005 – p. 20/26
Comparison
The best method: P1 approximation,
∆t ≈ ∆x : performances with or without overlapare comparable,
∆t ≈ ∆x2 : performances with overlap are higher.
JJJ III × DD16, january 2005 – p. 21/26
Numerical validation in 1-D
Dataν = 0.2, a = 1, c = 0.
u(0, t) = 0 and u(6, t) = 0.Ω = (0, 6), T = 2.5. u(x, 0) = e−3(1.2−x)2.Ω1 = (0, 3.04), Ω2 = (2.96, 6), L = 0.08.
Numerical scheme: upwind in space +backwardEuler in time.
∆x = 0.02,∆t = 0.005.Initial guess : random.
JJJ III × DD16, january 2005 – p. 22/26
Comparison
0 1 2 3 4 5 6 7 8 9 1010−12
10−10
10−8
10−6
10−4
10−2
100
102
iteration
erro
r
ClassicalTaylor 0Optimized 0Taylor 1Optimized 1
JJJ III × DD16, january 2005 – p. 23/26
Relevance of the optimization
0 0.5 1 1.50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
q
p
−9
−8 −8
−7
−7
−7
−6
−6
−6
−6
−5
−5
−5
−5
−5
−4
−4
−4
−4
−4
−3
−3
−3
−3
−3
−2
−2
−10
Error after 5 iterations as a function of p and q.
JJJ III × DD16, january 2005 – p. 24/26
Conclusionperspectives
JJJ III × DD16, january 2005 – p. 25/26
Extensions
The 2D case for P0 and P1 optimization,
Approximation by rational fractions as forabsorbing boundary conditions, ♣The wave equation,The case of systems of equations,etc..
JJJ III × DD16, january 2005 – p. 26/26
Extensions
The 2D case for P0 and P1 optimization,Approximation by rational fractions as forabsorbing boundary conditions, ♣
The wave equation,The case of systems of equations,etc..
JJJ III × DD16, january 2005 – p. 26/26
Extensions
The 2D case for P0 and P1 optimization,Approximation by rational fractions as forabsorbing boundary conditions, ♣The wave equation,
The case of systems of equations,etc..
JJJ III × DD16, january 2005 – p. 26/26
Extensions
The 2D case for P0 and P1 optimization,Approximation by rational fractions as forabsorbing boundary conditions, ♣The wave equation,The case of systems of equations,
etc..
JJJ III × DD16, january 2005 – p. 26/26
Extensions
The 2D case for P0 and P1 optimization,Approximation by rational fractions as forabsorbing boundary conditions, ♣The wave equation,The case of systems of equations,etc..
JJJ III × DD16, january 2005 – p. 26/26